Chapter 7 Algebra

In algebra, when a number is written directly next to a variable, it indicates multiplication.

The expression 4a means 4 multiplied by the variable a.

Let's look at the options:

(A) 4 + a means 4 plus a.

(B) 4 × a means 4 multiplied by a.

(C) a × a × a × a means a multiplied by itself four times, which is written as $a^4$.

(D) 4 ÷ a means 4 divided by a.

Comparing the given expression 4a with the options, we see that it is equal to $4 \times a$.

The correct option is (B).

We need to translate the given verbal statement into an algebraic expression.

The statement is "8 more than three times the number x".

Let the number be x.

"Three times the number x" means 3 multiplied by x, which is written as 3x.

"8 more than three times the number x" means we add 8 to three times the number x.

Let's look at the options:

(A) 8 + x + 3 simplifies to $11 + x$.

(B) 3x – 8 represents 8 less than three times the number x.

(C) 3x + 8 represents 8 more than three times the number x.

(D) 8x + 3 represents 3 more than eight times the number x.

Comparing our expression $3x + 8$ with the options, we find the correct representation.

The correct option is (C).

An equation is a mathematical statement that two expressions are equal. It always contains an equality sign (=) between the two expressions.

Let's examine each option:

(A) $x + 7$: This is an algebraic expression, as it does not contain an equality sign.

(B) $2y + 3 = 7$: This is a mathematical statement that asserts the expression $2y + 3$ is equal to the expression $7$. It contains an equality sign (=). Therefore, this is an equation.

(C) $2p < 10$: This statement contains an inequality sign (<). This is an inequation, not an equation.

(D) $12x$: This is an algebraic expression, as it does not contain an equality sign.

Based on the definition, the option that represents an equation is (B).

We need to translate the given verbal statement into an algebraic expression.

The statement is "7 times of y subtracted from 50".

"7 times of y" means 7 multiplied by y, which is written as 7y.

"7 times of y subtracted from 50" means we take 50 and subtract 7y from it.

So, 7 times of y subtracted from 50 can be expressed as $50 - 7y$.

The expression is $\boxed{50 - 7y}$.

To check if $x = 5$ is a solution of the equation $3 - x = 8$, we need to substitute $x = 5$ into the equation and see if the left side is equal to the right side.

The given equation is $3 - x = 8$.

Substitute $x = 5$ into the left side (LHS) of the equation:

The right side (RHS) of the equation is 8.

Compare the LHS and RHS:

Since $-2 \neq 8$, the LHS is not equal to the RHS when $x = 5$.

Therefore, $x = 5$ is not a solution of the equation $3 - x = 8$.

The statement "x = 5 is a solution of the equation 3 – x = 8" is false.

The statement is $\boxed{False}$.

Let the number be represented by a variable, say $n$.

"Thrice of a number" means 3 times the number, which is $3 \times n = 3n$.

"13 subtracted from thrice of a number" means we take thrice of the number and subtract 13 from it.

The expression is $\boxed{3n - 13}$.

Let the age of Megha's daughter (in years) be represented by a variable, say $d$.

"5 times her daughter’s age" means 5 multiplied by her daughter's age, which is $5 \times d = 5d$.

"2 more than 5 times her daughter’s age" means we add 2 to 5 times her daughter's age.

Let Megha's age (in years) be $M$. Then the expression for Megha's age based on her daughter's age is $5d + 2$.

So, the relationship can be expressed as an equation: $M = 5d + 2$. However, the question asks for an expression representing Megha's age in terms of her daughter's age.

The expression for Megha's age in terms of her daughter's age is $\boxed{5d + 2}$, where $d$ is the daughter's age in years.

Given:

Anagha is at step $p$.

Sushant is 10 steps ahead of Anagha.

Faizal is 6 steps behind Anagha.

The total number of steps to the hill top is 3 steps less than 8 times what Anagha has reached.

To Find:

1. The position of Sushant in terms of $p$.

2. The position of Faizal in terms of $p$.

3. An expression for the total number of steps using $p$.

Solution:

Anagha is at step $p$.

1. Position of Sushant:

Sushant is 10 steps ahead of Anagha. This means Sushant is at the step number that is 10 more than Anagha's step number.

Position of Sushant = Anagha's step number + 10

Sushant is at step $p + 10$.

2. Position of Faizal:

Faizal is 6 steps behind Anagha. This means Faizal is at the step number that is 6 less than Anagha's step number.

Position of Faizal = Anagha's step number - 6

Faizal is at step $p - 6$.

3. Expression for the total number of steps:

The total number of steps is 3 steps less than 8 times what Anagha has reached.

Anagha has reached step $p$.

"8 times what Anagha has reached" means 8 multiplied by Anagha's step number, which is $8 \times p = 8p$.

"3 steps less than 8 times what Anagha has reached" means we subtract 3 from 8 times what Anagha has reached.

Sushant is at step $p + 10$.

Faizal is at step $p - 6$.

The expression for the total number of steps using $p$ is $\boxed{8p - 3}$.

We are given the cost of a pencil is $\textsf{₹}x$.

The cost of a pen is given as $\textsf{₹}6x$.

The expression $6x$ means 6 multiplied by $x$.

So, the cost of a pen is 6 times the cost of a pencil.

We need to express this relationship in ordinary language.

Statement in ordinary language:

The cost of a pen is 6 times the cost of a pencil.

We are given that Manisha's age is $z$ years.

Her uncle's age is given as $5z$ years. The expression $5z$ means 5 multiplied by $z$. This means the uncle's age is 5 times Manisha's age.

Her aunt's age is given as $(5z - 4)$ years. The expression $5z - 4$ means 4 is subtracted from $5z$. Since $5z$ is the uncle's age, this means the aunt's age is 4 less than the uncle's age.

We need to express these relationships in ordinary language.

Statements in ordinary language:

Manisha's uncle is 5 times as old as Manisha.

Manisha's aunt is 4 years younger than her uncle.

Given:

Number of matchsticks in each box = 50

Number of boxes = n

To Find:

The total number of matchsticks required to fill n such boxes.

Solution:

Let M be the total number of matchsticks required.

The total number of matchsticks is the product of the number of matchsticks per box and the number of boxes.

$M = (\text{Number of matchsticks per box}) \times (\text{Number of boxes})$

$M = 50 \times n$

$M = 50n$

Thus, the number of matchsticks required to fill n such boxes is $50n$.

Comparing this with the given options, the correct option is (B).

The correct answer is (B) 50n.

Given:

Amulya's current age = x years

To Find:

Amulya's age 5 years ago.

Solution:

To find someone's age a certain number of years ago, we subtract that number of years from their current age.

Amulya's age 5 years ago = (Current age) - (Number of years ago)

Amulya's age 5 years ago = x - 5 years

Thus, Amulya's age 5 years ago was $(x - 5)$ years.

Comparing this with the given options, the correct option is (C).

The correct answer is (C) (x – 5) years.

Given:

The expression $6 \times x$.

To Find:

Which of the given options represents the expression $6 \times x$.

Solution:

In algebra, when a number is multiplied by a variable, the multiplication sign ($\times$) is often omitted for brevity.

The convention is to write the number (coefficient) before the variable.

So, the expression $6 \times x$ is written as $6x$.

Comparing this with the given options:

(A) $6x$ represents $6 \times x$

(B) $\frac{x}{6}$ represents $x$ divided by 6 or $x \div 6$

(C) $6 + x$ represents 6 added to $x$

(D) $6 – x$ represents 6 minus $x$

Therefore, the expression $6 \times x$ is represented by $6x$.

The correct option is (A).

The correct answer is (A) 6x.

Given:

The options provided are mathematical statements or expressions.

(A) $x + 1$

(B) $x – 1$

(C) $x – 1 = 0$

(D) $x + 1 > 0$

To Find:

Which of the given options is an equation.

Solution:

An equation is a mathematical statement that shows that two expressions are equal. It always contains an equality sign (=).

Let's examine each option:

(A) $x + 1$: This is an algebraic expression. It does not contain an equality sign. Therefore, it is not an equation.

(B) $x – 1$: This is an algebraic expression. It does not contain an equality sign. Therefore, it is not an equation.

(C) $x – 1 = 0$: This is a mathematical statement where the expression $x - 1$ is stated to be equal to the expression $0$. It contains an equality sign (=). Therefore, this is an equation.

(D) $x + 1 > 0$: This is a mathematical statement that shows that the expression $x + 1$ is greater than the expression $0$. It contains an inequality sign ($>$). Therefore, it is an inequality, not an equation.

Based on the definition of an equation, only option (C) satisfies the condition of containing an equality sign and stating that two expressions are equal.

The correct option is (C).

The correct answer is (C) x – 1 = 0.

Given:

The expression is $x + 10$.

The value of x is given as 2.

To Find:

The value of the expression $x + 10$ when $x=2$.

Solution:

To find the value of the expression $x + 10$ when $x$ takes the value 2, we substitute 2 in place of $x$ in the expression.

Value of $x + 10$ when $x = 2$ is:

$2 + 10$

Adding the numbers:

$2 + 10 = 12$

So, when $x=2$, the value of $x+10$ is 12.

Comparing this result with the given options, we find that option (B) is 12.

The correct answer is (B) 12.

Given:

The figure is a regular hexagon.

The perimeter of the regular hexagon is $x$ metres.

To Find:

The length of each side of the regular hexagon.

Solution:

A regular hexagon is a polygon with 6 sides of equal length and 6 equal angles.

The perimeter of any polygon is the sum of the lengths of its sides.

For a regular polygon, the perimeter is equal to the number of sides multiplied by the length of one side.

Let $s$ be the length of each side of the regular hexagon in metres.

Number of sides of a hexagon = 6.

Perimeter of the regular hexagon = (Number of sides) $\times$ (Length of each side)

We are given that the perimeter is $x$ metres.

So, $x = 6 \times s$

To find the length of each side, $s$, we need to isolate $s$ in the equation.

Divide both sides of the equation by 6:

So, the length of each side is $\frac{x}{6}$ metres.

This can also be written as $x \div 6$ metres.

Comparing this result with the given options:

(A) $(x + 6)$ metres represents the sum of the perimeter and the number of sides.

(B) $(x \div 6)$ metres represents the perimeter divided by the number of sides.

(C) $(x – 6)$ metres represents the difference between the perimeter and the number of sides.

(D) $(6 \div x)$ metres represents the number of sides divided by the perimeter.

The length of each side is given by $(x \div 6)$ metres, which matches option (B).

The correct answer is (B) (x ÷ 6) metres.

Given:

The value $x = 2$.

The given equations are:

(A) $x + 2 = 5$

(B) $x – 2 = 0$

(C) $2x + 1 = 0$

(D) $x + 3 = 6$

To Find:

Which of the given equations has $x=2$ as a solution.

Solution:

A value of a variable is a solution to an equation if, when substituted into the equation, it makes the equation a true statement (i.e., the left side of the equation becomes equal to the right side).

We will substitute $x=2$ into each equation and check if the equality holds true.

Let's test option (A): $x + 2 = 5$

Substitute $x=2$ into the left side (LHS):

LHS = $2 + 2 = 4$

The right side (RHS) is 5.

Since $4 \neq 5$, the equation $x + 2 = 5$ is not true for $x=2$. So, $x=2$ is not a solution for equation (A).

Let's test option (B): $x – 2 = 0$

Substitute $x=2$ into the left side (LHS):

LHS = $2 – 2 = 0$

The right side (RHS) is 0.

Since $0 = 0$, the equation $x – 2 = 0$ is true for $x=2$. So, $x=2$ is a solution for equation (B).

Let's test option (C): $2x + 1 = 0$

Substitute $x=2$ into the left side (LHS):

LHS = $2 \times 2 + 1 = 4 + 1 = 5$

The right side (RHS) is 0.

Since $5 \neq 0$, the equation $2x + 1 = 0$ is not true for $x=2$. So, $x=2$ is not a solution for equation (C).

Let's test option (D): $x + 3 = 6$

Substitute $x=2$ into the left side (LHS):

LHS = $2 + 3 = 5$

The right side (RHS) is 6.

Since $5 \neq 6$, the equation $x + 3 = 6$ is not true for $x=2$. So, $x=2$ is not a solution for equation (D).

Only equation (B) is satisfied when $x=2$.

The correct option is (B).

The correct answer is (B) x – 2 = 0.

Given:

Two integers, x and y.

Options representing mathematical relationships between x and y.

To Find:

The option that suggests the commutative property of addition for integers.

Solution:

The commutative property for an operation states that the order of the operands does not affect the result of the operation.

For addition, the commutative property states that for any two numbers (or variables), the sum remains the same regardless of the order in which they are added.

In mathematical terms, for any two integers x and y, the commutative property of addition is represented as:

Let's examine the given options:

(A) $x + y = y + x$:

This statement shows that the sum of x and y is the same as the sum of y and x. This matches the definition of the commutative property of addition.

(B) $x + y > x$:

This is an inequality which states that the sum of x and y is greater than x. This is not a statement about the order of operation.

(C) $x – y = y – x$:

This statement shows if subtraction is commutative. However, subtraction is generally not commutative. For example, $5 - 3 = 2$ but $3 - 5 = -2$, and $2 \neq -2$. So, this option is incorrect and does not represent the commutative property of addition.

(D) $x \times y = y \times x$:

This statement shows that the product of x and y is the same as the product of y and x. This represents the commutative property of multiplication, not addition.

Based on the analysis, only option (A) represents the commutative property of addition.

The correct option is (A).

The correct answer is (A) x + y = y + x.

Given:

The given equations are:

(A) $x + 1 = 1$

(B) $x – 1 = 3$

(C) $2x + 1 = 6$

(D) $1 – x = 5$

To Find:

The equation which does not have a solution that is an integer.

Solution:

We need to solve each equation for $x$ and check if the value of $x$ is an integer. An integer is a whole number (positive, negative, or zero), e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...

Let's solve each equation:

Option (A): $x + 1 = 1$

Subtract 1 from both sides:

$x + 1 - 1 = 1 - 1$

$x = 0$

Since 0 is an integer, this equation has an integer solution.

Option (B): $x – 1 = 3$

Add 1 to both sides:

$x – 1 + 1 = 3 + 1$

$x = 4$

Since 4 is an integer, this equation has an integer solution.

Option (C): $2x + 1 = 6$

Subtract 1 from both sides:

$2x + 1 - 1 = 6 - 1$

$2x = 5$

Divide both sides by 2:

$x = \frac{5}{2}$

$x = 2.5$

Since 2.5 is not an integer (it is a fraction/decimal), this equation does not have an integer solution.

Option (D): $1 – x = 5$

Subtract 1 from both sides:

$1 – x - 1 = 5 - 1$

$-x = 4$

Multiply both sides by -1:

$x = -4$

Since -4 is an integer, this equation has an integer solution.

From the analysis, only equation (C) has a solution that is not an integer.

The correct option is (C).

The correct answer is (C) 2x + 1 = 6.

Given:

The arithmetic expression $3 \times 5$.

To Find:

The value of the arithmetic expression $3 \times 5$.

Solution:

In arithmetic, the multiplication symbol ($\times$) represents the operation of finding the product of two numbers.

To calculate $3 \times 5$, we multiply 3 by 5.

$3 \times 5 = 15$

The question highlights the difference in notation between algebra and arithmetic. In algebra, the multiplication of variables or a number and a variable is often represented by writing them next to each other (e.g., $a \times b = ab$, $6 \times x = 6x$). However, in arithmetic, when multiplying specific numbers, the multiplication symbol ($\times$) is typically used, and the result is the numerical product.

Comparing the result with the given options:

(A) 35

(B) 53

(C) 15

(D) 8

The calculated product is 15, which matches option (C).

The correct answer is (C) 15.

Given:

The question asks about the role of letters in algebra.

To Find:

What letters represent in algebra from the given options.

Solution:

In algebra, letters are used as symbols to represent quantities that can change or whose value is not yet determined. These letters are called variables.

Variables are frequently used to represent unknown quantities in equations or expressions. For example, in the equation $x + 5 = 12$, the letter $x$ represents an unknown number that makes the equation true.

Letters can also be used to represent quantities that can take on different values, such as in formulas (e.g., the formula for the perimeter of a square is $P = 4s$, where $s$ can be any side length) or to express general relationships.

While letters can sometimes represent constants (fixed numbers) in certain contexts (like $a$ and $b$ in $ax + b = 0$), their primary and most general role in algebra is to stand for quantities whose value is either unknown or variable.

Let's consider the options:

(A) known quantities: While a variable might eventually represent a known quantity after solving an equation, the variable itself initially represents a quantity that is not known.

(B) unknown quantities: This is a fundamental use of variables in algebra, particularly in solving equations.

(C) fixed numbers: Letters can represent fixed numbers (constants), but the term 'variable' emphasizes the potential for the quantity to vary or be unknown.

(D) none of these: This is incorrect as letters do represent quantities in algebra.

Among the given options, "unknown quantities" is the most accurate description of what letters primarily stand for when introduced in basic algebra, especially in the context of solving problems and equations.

The correct answer is (B) unknown quantities.

Given:

The term "Variable".

To Find:

The meaning of the term "Variable" in the context of algebra.

Solution:

In algebra, a variable is a symbol (usually a letter) used to represent a quantity that may change or vary, or whose value is unknown. The defining characteristic of a variable is its ability to assume various values from a given set (like integers, real numbers, etc.).

Let's examine the given options:

(A) can take different values: This accurately describes the nature of a variable. A variable is called "variable" precisely because its value is not fixed and can change.

(B) has a fixed value: A quantity that has a fixed value is called a constant, not a variable.

(C) can take only 2 values: While some variables in specific contexts (like boolean algebra) might be restricted to only two values (e.g., true/false or 0/1), the general definition of a variable in algebra does not limit the number of possible values to just two. It can take infinitely many values depending on the domain.

(D) can take only three values: Similar to option (C), this imposes an arbitrary limit on the number of values a variable can take, which is not part of the general definition of a variable.

Therefore, the fundamental meaning of a "variable" is that it can take different values.

The correct option is (A).

The correct answer is (A) can take different values.

Given:

The algebraic expression $10 – x$.

To Find:

The correct interpretation of the expression $10 – x$ among the given options.

Solution:

In an algebraic expression involving subtraction, such as $a - b$, the term '$- b$' signifies that the quantity represented by $b$ is being subtracted from the quantity represented by $a$.

In the expression $10 – x$, the number $10$ is the minuend (the quantity from which another is subtracted), and $x$ is the subtrahend (the quantity being subtracted).

Therefore, the expression $10 – x$ means that $x$ is subtracted from $10$.

Let's examine the given options:

(A) 10 is subtracted x times: This would typically involve multiplication or repeated subtraction, like $10 \times x$ or $10 - 10 - ... - 10$ (x times), which is not what $10 - x$ represents.

(B) x is subtracted 10 times: This interpretation does not match the structure of $10 - x$. Repeated subtraction of $x$ from an initial value, say $A$, 10 times would be $A - 10x$.

(C) x is subtracted from 10: This accurately describes the operation in $10 - x$. The quantity $x$ is taken away from $10$.

(D) 10 is subtracted from x: This operation would be represented as $x - 10$.

Based on the standard interpretation of subtraction in algebra, $10 - x$ means that $x$ is subtracted from $10$.

The correct option is (C).

The correct answer is (C) x is subtracted from 10.

Given:

Initial amount Savitri has = $\textsf{₹} x$

Amount spent on grocery = $\textsf{₹} 1000$

Amount spent on clothes = $\textsf{₹} 500$

Amount spent on education = $\textsf{₹} 400$

Amount received as a gift = $\textsf{₹} 200$

To Find:

The amount of money left with Savitri (in $\textsf{₹}$).

Solution:

First, calculate the total amount Savitri spent.

Total amount spent = (Amount on grocery) + (Amount on clothes) + (Amount on education)

Total amount spent = $\textsf{₹} 1000 + \textsf{₹} 500 + \textsf{₹} 400$

Total amount spent = $\textsf{₹} 1900$

Next, consider the initial amount she had and the transactions (spending and receiving).

Money left = (Initial amount) - (Total amount spent) + (Amount received)

Money left = $\textsf{₹} x - \textsf{₹} 1900 + \textsf{₹} 200$

Combine the constant terms:

$-1900 + 200 = -1700$

Money left = $\textsf{₹} (x - 1700)$

Thus, the amount of money left with Savitri is $\textsf{₹} (x - 1700)$.

Comparing this expression with the given options:

(A) $x – 1700$

(B) $x – 1900$

(C) $x + 200$

(D) $x – 2100$

The expression we found matches option (A).

The correct answer is (A) x – 1700.

Given:

A triangle with side lengths $x$, $x$, and $y$. The image shows a triangle with two sides marked with length $x$ and one side marked with length $y$.

To Find:

The perimeter of the triangle.

Solution:

The perimeter of a triangle (or any polygon) is the total length of its boundary, which is found by adding the lengths of all its sides.

In the given triangle, the lengths of the three sides are $x$, $x$, and $y$.

Perimeter = Sum of the lengths of the sides

Perimeter = $x + x + y$

Combining the terms involving $x$:

$x + x = 2x$

So, the perimeter of the triangle is $2x + y$.

Comparing this expression with the given options:

(A) $2x + y$

(B) $x + 2y$

(C) $x + y$

(D) $2x – y$

The calculated perimeter matches option (A).

The correct answer is (A) 2x + y.

Given:

A square with each side of length $x$.

To Find:

The area of the square.

Solution:

A square is a quadrilateral with four equal sides and four right angles.

The area of a square is calculated by multiplying the length of one side by itself.

Area of a square = (Side length) $\times$ (Side length)

Given that the length of each side is $x$, the area is:

Area = $x \times x$

In algebra, the product of a variable multiplied by itself is usually written using exponents.

$x \times x = x^2$

However, the options provided do not use exponential notation, so we compare with the given forms.

Let's examine the given options:

(A) $x \times x$: This directly represents the formula for the area of a square with side $x$.

(B) $4x$: This represents the perimeter of a square with side $x$ (sum of the lengths of the four sides: $x + x + x + x = 4x$).

(C) $x + x$: This represents the sum of two sides of the square, or $2x$.

(D) $4 + x$: This represents the sum of 4 and the side length $x$, which does not correspond to a standard geometric property of the square related to its side length.

The formula for the area of a square with side $x$ is $x \times x$.

The correct answer is (A) x × x.

Given:

A description of operations involving a variable x.

To Find:

The algebraic expression that represents the described operations.

Solution:

The description involves two steps:

Step 1: "$x$ is multiplied by 2"

Multiplying $x$ by 2 gives $2 \times x$. In algebraic notation, this is written as $2x$.

Step 2: "then subtracted from 3"

This means the result from Step 1 (which is $2x$) is taken away from 3. In subtraction, "subtracted from" means the second quantity is the subtrahend and the first quantity is the minuend.

So, we subtract $2x$ from 3, which is written as $3 - 2x$.

The expression obtained is $3 - 2x$.

Let's compare this with the given options:

(A) $2x – 3$: This means 3 is subtracted from 2x.

(B) $2x + 3$: This means 3 is added to 2x.

(C) $3 – 2x$: This means 2x is subtracted from 3.

(D) $3x – 2$: This means 2 is subtracted from 3x.

The expression we derived matches option (C).

The correct answer is (C) 3 – 2x.

Given:

The equation $\frac{q}{2} = 3$.

To Find:

The solution to the given equation from the options.

Solution:

We are given the equation:

To solve for $q$, we need to isolate $q$ on one side of the equation. Currently, $q$ is being divided by 2.

To undo the division by 2, we multiply both sides of the equation by 2.

On the left side, the multiplication by 2 cancels out the division by 2:

$q = 3 \times 2$

Perform the multiplication on the right side:

$q = 6$

So, the solution to the equation $\frac{q}{2} = 3$ is $q = 6$.

We can check the solution by substituting $q=6$ back into the original equation:

LHS = $\frac{6}{2} = 3$

RHS = 3

Since LHS = RHS, the solution $q=6$ is correct.

Comparing the solution with the given options:

(A) 6

(B) 8

(C) 3

(D) 2

Our solution matches option (A).

The correct answer is (A) 6.

Given:

The equation $x – 4 = – 2$.

To Find:

The solution to the given equation from the options.

Solution:

We are given the equation:

To solve for $x$, we need to isolate $x$ on one side of the equation. Currently, 4 is being subtracted from $x$.

To undo the subtraction of 4, we add 4 to both sides of the equation.

On the left side, $-4 + 4$ equals 0, leaving just $x$. On the right side, $-2 + 4$ equals 2.

$x = 2$

So, the solution to the equation $x – 4 = – 2$ is $x = 2$.

We can check the solution by substituting $x=2$ back into the original equation:

LHS = $2 – 4 = -2$

RHS = $-2$

Since LHS = RHS, the solution $x=2$ is correct.

Comparing the solution with the given options:

(A) 6

(B) 2

(C) – 6

(D) – 2

Our solution matches option (B).

The correct answer is (B) 2.

Given:

The mathematical statement $\frac{4}{2} = 2$.

To Find:

What the statement $\frac{4}{2} = 2$ represents from the given options.

Solution:

Let's analyze the statement $\frac{4}{2} = 2$ based on the options provided:

(A) numerical equation:

An equation is a statement that two mathematical expressions are equal, indicated by the equality sign (=). In the given statement, we have $\frac{4}{2}$ on the left side and $2$ on the right side, connected by an equality sign. Both $\frac{4}{2}$ and $2$ are numerical expressions (they consist only of numbers and arithmetic operations). Since the equality sign is present and it involves only numbers, this is a numerical equation.

Let's verify if the equation is true: $\frac{4}{2} = 2$, and $2 = 2$. The statement is true.

(B) algebraic expression:

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation signs. The given statement contains an equality sign and does not contain any variables. It is a statement of equality between two numerical expressions, not just an expression itself.

(C) equation with a variable:

An equation with a variable contains one or more variables. The given statement $\frac{4}{2} = 2$ contains only numbers and the equality sign; there are no variables present.

(D) false statement:

A false statement is a mathematical statement that is not true. We verified that $\frac{4}{2} = 2$ is a true statement, as $4 \div 2 = 2$.

Based on the analysis, the statement $\frac{4}{2} = 2$ is a numerical equation because it is a statement of equality between two numerical expressions.

The correct option is (A).

The correct answer is (A) numerical equation.

Given:

Initial number of pencils Kanta has = p

Number of pencils she adds = q

To Find:

The total number of pencils with Kanta after adding the new pencils.

Solution:

To find the total number of pencils, we need to add the initial number of pencils she had to the number of pencils she added.

Total number of pencils = (Initial number of pencils) + (Number of pencils added)

Total number of pencils = p + q

Let's examine the given options:

(A) p + q: This represents the sum of the initial pencils and the added pencils.

(B) pq: This represents the product of the initial pencils and the added pencils (p $\times$ q).

(C) p – q: This represents the difference between the initial pencils and the added pencils (subtracting q from p).

(D) $\frac{p}{q}$: This represents the initial pencils divided by the added pencils (p $\div$ q).

The total number of pencils is obtained by adding the initial number and the added number, which is $p + q$.

The correct option is (A).

The correct answer is (A) p + q.

Given:

The equation $4x = 16$.

To Find:

The value of x that satisfies the given equation.

Solution:

We are given the equation:

The term $4x$ means 4 multiplied by $x$. To find the value of $x$, we need to isolate $x$ on one side of the equation.

To undo the multiplication by 4, we divide both sides of the equation by 4.

On the left side, the division by 4 cancels out the multiplication by 4, leaving just $x$. On the right side, we perform the division $16 \div 4$.

$x = 4$

So, the value of x that satisfies the equation $4x = 16$ is $x = 4$.

We can verify this solution by substituting $x=4$ back into the original equation:

LHS = $4 \times x = 4 \times 4 = 16$

RHS = 16

Since LHS = RHS, the equation is satisfied by $x=4$.

Comparing the solution with the given options:

(A) 4

(B) 2

(C) 12

(D) –12

Our solution $x=4$ matches option (A).

The correct answer is (A) 4.

Given:

A statement describing a relationship between an unknown number and the result of an operation.

The operations are: thinking of a number, adding 13 to it, and getting 27 as the result.

To Find:

The equation that represents the given statement.

Solution:

Let the unknown number be represented by the variable $x$.

The statement says "on adding 13 to it". This translates to adding 13 to the number $x$, which is written as $x + 13$.

The statement then says "I get 27". This means the result of the operation $x + 13$ is equal to 27.

So, the equation that represents the statement is:

Now, let's compare this equation with the given options:

(A) $x – 27 = 13$: This represents "subtracting 27 from x gives 13".

(B) $x – 13 = 27$: This represents "subtracting 13 from x gives 27".

(C) $x + 27 = 13$: This represents "adding 27 to x gives 13".

(D) $x + 13 = 27$: This represents "adding 13 to x gives 27".

The equation we derived, $x + 13 = 27$, matches option (D).

The correct answer is (D) x + 13 = 27.

Given:

Constant speed = 40 km/hour

Time taken = h hours

To Find:

The distance travelled in km.

Solution:

The formula for distance when speed and time are known is:

Distance = Speed $\times$ Time

Substitute the given values into the formula:

Distance = 40 km/hour $\times$ h hours

Distance = $(40 \times h)$ km

In algebraic form, $40 \times h$ is written as $40h$.

Distance = $40h$ km

The distance travelled is $40h$ km.

So, the blank should be filled with the expression $40h$.

The distance (in km) travelled in h hours at a constant speed of 40km per hour is $40h$.

Given:

Quantity of potatoes bought = p kg

Total cost of p kg potatoes = $\textsf{₹} 70$

To Find:

The cost of 1 kg of potatoes (in $\textsf{₹}$).

Solution:

To find the cost of 1 kg of potatoes, we need to divide the total cost by the total quantity of potatoes bought.

Cost of 1 kg potatoes = $\frac{\text{Total cost}}{\text{Quantity of potatoes}}$

Cost of 1 kg potatoes = $\frac{\textsf{₹} 70}{\text{p kg}}$

Cost of 1 kg potatoes = $\frac{70}{p}$ $\textsf{₹}$/kg

The cost of 1 kg of potatoes in $\textsf{₹}$ is $\frac{70}{p}$.

The blank should be filled with the expression $\frac{70}{p}$.

p kg of potatoes are bought for Rs 70. Cost of 1kg of potatoes (in Rs) is $\frac{70}{p}$.

Given:

Charge for the first kilometre = $\textsf{₹} 10$

Charge for each subsequent kilometre = $\textsf{₹} 8$

Total distance travelled = d kilometres (where $d \ge 1$)

To Find:

The total charge in Rupees for travelling d kilometres.

Solution:

The total distance travelled is d kilometres. We need to consider the charge for the first kilometre separately from the charge for the remaining kilometres.

The charge for the first 1 kilometre is fixed at $\textsf{₹} 10$.

The remaining distance is the total distance minus the first kilometre. This remaining distance is $(d - 1)$ kilometres.

The charge for each of these subsequent kilometres is $\textsf{₹} 8$.

So, the charge for the $(d - 1)$ subsequent kilometres is $8 \times (d - 1)$.

Charge for subsequent km = $\textsf{₹} 8(d - 1)$

The total charge is the sum of the charge for the first kilometre and the charge for the subsequent kilometres.

Total charge = (Charge for first km) + (Charge for subsequent km)

Total charge = $\textsf{₹} 10 + \textsf{₹} 8(d - 1)$

Now, we can simplify the expression:

Total charge = $10 + 8d - 8$

Total charge = $8d + 10 - 8$

Total charge = $8d + 2$

This expression is valid for $d \ge 1$. If $d=1$, the charge is $8(1) + 2 = 10$, which is correct.

The total charge for d kilometres is $\textsf{₹} (8d + 2)$.

The blank should be filled with the expression $8d + 2$.

The distance (in km) travelled in h hours at a constant speed of 40km per hour is $8d + 2$.

Given:

The equation $7x + 4 = 25$.

To Find:

The value of x that satisfies the given equation.

Solution:

We are given the equation:

To solve for $x$, we first isolate the term containing $x$ by subtracting 4 from both sides of the equation.

Now, the equation is $7x = 21$, which means 7 multiplied by $x$ is 21. To find $x$, we divide both sides of the equation by 7.

We can verify the solution by substituting $x=3$ back into the original equation:

LHS = $7(3) + 4 = 21 + 4 = 25$.

RHS = 25.

Since LHS = RHS, the value $x=3$ is correct.

The value of x is 3.

The blank should be filled with the value 3.

If 7x + 4 = 25, then the value of x is 3.

Given:

The equation $3x + 7 = –20$.

To Find:

The solution of the given equation.

Solution:

We are given the equation:

To solve for $x$, we first isolate the term containing $x$ by subtracting 7 from both sides of the equation.

On the left side, $7 - 7$ equals 0, leaving just $3x$. On the right side, $-20 - 7$ equals $-27$.

Now, the equation is $3x = –27$, which means 3 multiplied by $x$ is –27. To find $x$, we divide both sides of the equation by 3.

We can verify the solution by substituting $x=-9$ back into the original equation:

LHS = $3(-9) + 7 = -27 + 7 = -20$.

RHS = $-20$.

Since LHS = RHS, the solution $x=-9$ is correct.

The solution of the equation is $x = -9$.

The blank should be filled with the value $-9$.

The solution of the equation 3x + 7 = –20 is –9.

Given:

The statement "x exceeds y by 7".

To Find:

Express the given statement as an algebraic equation.

Solution:

The phrase "x exceeds y by 7" means that the value of $x$ is greater than the value of $y$ by 7. In other words, if we add 7 to $y$, we get $x$.

This can be written as an equation in a few ways:

1. $x = y + 7$ (x is equal to y plus 7)

2. $x - y = 7$ (The difference between x and y is 7, with x being larger)

3. $x - 7 = y$ (If you subtract 7 from x, you get y)

Any of these equations accurately represents the statement "x exceeds y by 7". The simplest and most direct translation is often $x = y + 7$ or $x - y = 7$. If the blank requires a single expression, $x - y$ (which equals 7) or $y + 7$ (which equals x) could be considered depending on context, but typically, the expression refers to the comparison or the relationship itself, which forms an equation.

Let's write the most common forms:

or

Either of these can be considered a correct way to express the statement as an equation. Without specific constraints on the format of the answer (like whether it should be an equation or an expression), both are mathematically equivalent representations of the relationship.

Assuming the blank requires an equation, either $x = y + 7$ or $x - y = 7$ would fit. The choice often depends on the conventions being followed or the format expected.

Common ways to express "x exceeds y by 7" as an equation are $x = y + 7$ or $x - y = 7$.

Let's provide both common forms as possible fillings for the blank, noting that either is acceptable.

‘x exceeds y by 7’ can be expressed as $x = y + 7$ or $x - y = 7$.

Given:

The phrase "8 more than three times the number x".

To Find:

Write the given phrase as an algebraic expression.

Solution:

Let the number be $x$.

"three times the number x" means multiplying the number $x$ by 3. This is written as $3 \times x$, or simply $3x$.

"8 more than three times the number x" means adding 8 to the result of "three times the number x".

So, we add 8 to $3x$. This is written as $3x + 8$.

The algebraic expression for "8 more than three times the number x" is $3x + 8$.

The blank should be filled with the expression $3x + 8$.

‘8 more than three times the number x’ can be written as $3x + 8$.

Given:

Total amount spent = $\textsf{₹} x$

Rate per pencil = $\textsf{₹} 2$ per pencil

To Find:

The number of pencils bought.

Solution:

To find the number of items bought when the total cost and the cost per item are known, we divide the total cost by the cost per item.

Number of pencils = $\frac{\text{Total amount spent}}{\text{Cost per pencil}}$

Number of pencils = $\frac{\textsf{₹} x}{\textsf{₹} 2 \text{ per pencil}}$

Number of pencils = $\frac{x}{2}$

The number of pencils bought is $\frac{x}{2}$.

The blank should be filled with the expression $\frac{x}{2}$.

Number of pencils bought for Rs x at the rate of Rs 2 per pencil is $\frac{x}{2}$.

Given:

Number of weeks = w

To Find:

The number of days in w weeks.

Solution:

We know that there are 7 days in one week.

To find the number of days in w weeks, we multiply the number of weeks (w) by the number of days in a single week (7).

Number of days = (Number of weeks) $\times$ (Number of days per week)

Number of days = w $\times$ 7

In algebraic form, $w \times 7$ is written as $7w$.

Number of days = $7w$

The number of days in w weeks is $7w$.

The blank should be filled with the expression $7w$.

The number of days in w weeks is $7w$.

Given:

Monthly salary = $\textsf{₹} r$

Festival bonus = $\textsf{₹} 2000$

To Find:

The annual salary including the festival bonus.

Solution:

First, we need to calculate the total salary earned from monthly payments over a year. There are 12 months in a year.

Annual salary from monthly payments = (Monthly salary) $\times$ (Number of months in a year)

Annual salary from monthly payments = $\textsf{₹} r \times 12$

Annual salary from monthly payments = $\textsf{₹} 12r$

The total annual salary is the sum of the annual salary from monthly payments and the festival bonus.

Total annual salary = (Annual salary from monthly payments) + (Festival bonus)

Total annual salary = $\textsf{₹} 12r + \textsf{₹} 2000$

Total annual salary = $\textsf{₹} (12r + 2000)$

The annual salary including the festival bonus is $\textsf{₹} (12r + 2000)$.

The blank should be filled with the expression $12r + 2000$.

Annual salary at r rupees per month alongwith a festival bonus of Rs 2000 is $12r + 2000$.

Given:

Ten's digit of a two-digit number = t

Unit's digit of a two-digit number = u

To Find:

Express the two-digit number in terms of t and u.

Solution:

In a two-digit number, the ten's digit represents the number of tens, and the unit's digit represents the number of ones.

For example, in the number 35, the ten's digit is 3 and the unit's digit is 5. The value of the number is $3 \times 10 + 5 \times 1 = 30 + 5 = 35$.

Following this pattern, if the ten's digit is 't' and the unit's digit is 'u', the value of the number is:

(Ten's digit $\times$ Value of ten's place) + (Unit's digit $\times$ Value of unit's place)

Value of the number = $t \times 10 + u \times 1$

Value of the number = $10t + u$

The two-digit number is $10t + u$.

The blank should be filled with the expression $10t + u$.

The two digit number whose ten’s digit is ‘t’ and units’s digit is ‘u’ is $10t + u$.

Given:

The equation $2p + 8 = 18$.

To Find:

Identify the variable used in the given equation.

Solution:

In an algebraic equation, a variable is a letter or symbol that represents an unknown quantity or a quantity that can change its value. Constants are fixed numerical values.

In the equation $2p + 8 = 18$:

• The numbers 2, 8, and 18 are constants.
• The letter 'p' represents a quantity whose value can be determined by solving the equation, or it can represent a quantity that can take on different values in a more general context. Here, in the context of solving an equation, 'p' is the unknown we are trying to find.

Therefore, the variable used in the equation is 'p'.

The variable used in the equation is p.

The blank should be filled with the letter 'p'.

The variable used in the equation 2p + 8 = 18 is p.

Given:

A length expressed in metres: x metres.

To Find:

Express the given length in terms of centimetres.

Solution:

We need to know the conversion factor between metres and centimetres.

We know that:

To convert a length from metres to centimetres, we multiply the number of metres by 100.

Given length = x metres

Length in centimetres = x $\times$ (Number of centimetres in 1 metre)

Length in centimetres = x $\times$ 100

Length in centimetres = $100x$ centimetres

So, x metres is equal to $100x$ centimetres.

The blank should be filled with the expression $100x$.

x metres = $100x$ centimetres

Given:

A volume expressed in litres: p litres.

To Find:

Express the given volume in terms of millilitres.

Solution:

We need to know the conversion factor between litres and millilitres.

We know that:

To convert a volume from litres to millilitres, we multiply the number of litres by 1000.

Given volume = p litres

Volume in millilitres = p $\times$ (Number of millilitres in 1 litre)

Volume in millilitres = p $\times$ 1000

Volume in millilitres = $1000p$ millilitres

So, p litres is equal to $1000p$ millilitres.

The blank should be filled with the expression $1000p$.

p litres = $1000p$ millilitres

Given:

An amount of money expressed in rupees: r rupees.

To Find:

Express the given amount in terms of paise.

Solution:

We need to know the conversion factor between rupees and paise.

We know that:

To convert an amount from rupees to paise, we multiply the number of rupees by 100.

Given amount = r rupees

Amount in paise = r $\times$ (Number of paise in 1 rupee)

Amount in paise = r $\times$ 100

Amount in paise = $100r$ paise

So, r rupees is equal to $100r$ paise.

The blank should be filled with the expression $100r$.

r rupees = $100r$ paise

Given:

Ramandeep's present age = n years

To Find:

Ramandeep's age after 7 years.

Solution:

To find Ramandeep's age after 7 years, we add 7 to her present age.

Age after 7 years = (Present age) + 7 years

Age after 7 years = n years + 7 years

Age after 7 years = $(n + 7)$ years

So, Ramandeep's age after 7 years will be $(n + 7)$ years.

The blank should be filled with the expression $n + 7$.

If the present age of Ramandeep is n years, then her age after 7 years will be $n + 7$.

Given:

Initial amount of money = $\textsf{₹} 100$

Amount spent = $\textsf{₹} f$

To Find:

The amount of money left.

Solution:

To find the amount of money left after spending, we subtract the amount spent from the initial amount.

Money left = (Initial amount) - (Amount spent)

Money left = $\textsf{₹} 100 - \textsf{₹} f$

Money left = $\textsf{₹} (100 - f)$

The amount of money left is $\textsf{₹} (100 - f)$.

The blank should be filled with the expression $100 - f$.

If I spend f rupees from 100 rupees, the money left with me is $100 - f$ rupees.

Given:

The statement: "0 is a solution of the equation $x + 1 = 0$".

The equation is $x + 1 = 0$.

The proposed solution is $x = 0$.

To Verify:

Determine if the given statement is true or false.

Solution:

To check if a value is a solution to an equation, we substitute the value into the equation and see if the equality holds true.

Substitute $x = 0$ into the equation $x + 1 = 0$:

LHS = $0 + 1 = 1$

RHS = 0

Since the LHS ($1$) is not equal to the RHS ($0$), the value $x = 0$ does not satisfy the equation $x + 1 = 0$.

Therefore, 0 is not a solution of the equation $x + 1 = 0$.

The statement "0 is a solution of the equation $x + 1 = 0$" is false.

The statement is False.

Given:

Two equations: Equation 1: $x + 1 = 0$ and Equation 2: $2x + 2 = 0$.

The statement: "The equations $x + 1 = 0$ and $2x + 2 = 0$ have the same solution."

To Verify:

Determine if the given statement is true or false by finding the solution of each equation.

Solution:

Let's find the solution for each equation separately.

Equation 1: $x + 1 = 0$

Subtract 1 from both sides:

$x + 1 - 1 = 0 - 1$

The solution for Equation 1 is $x = -1$.

Equation 2: $2x + 2 = 0$

Subtract 2 from both sides:

$2x + 2 - 2 = 0 - 2$

Divide both sides by 2:

The solution for Equation 2 is $x = -1$.

Comparing the solutions from (i) and (ii):

Solution of Equation 1 is $x = -1$.

Solution of Equation 2 is $x = -1$.

Since the solutions are the same ($x=-1$), the statement "The equations $x + 1 = 0$ and $2x + 2 = 0$ have the same solution" is true.

The statement is True.

Given:

m is a whole number.

The expression is 2m.

The statement: "If m is a whole number, then 2m denotes a multiple of 2."

To Verify:

Determine if the given statement is true or false based on the definition of a multiple of 2 and whole numbers.

Solution:

A whole number is any non-negative integer (0, 1, 2, 3, ...).

A multiple of 2 is any number that can be obtained by multiplying 2 by an integer. In other words, a multiple of 2 is any even integer.

The expression $2m$ represents the product of 2 and the whole number $m$.

Let's consider some examples by taking different whole number values for $m$:

• If $m = 0$, then $2m = 2 \times 0 = 0$. $0$ is a multiple of 2 ($0 = 2 \times 0$).
• If $m = 1$, then $2m = 2 \times 1 = 2$. $2$ is a multiple of 2 ($2 = 2 \times 1$).
• If $m = 2$, then $2m = 2 \times 2 = 4$. $4$ is a multiple of 2 ($4 = 2 \times 2$).
• If $m = 3$, then $2m = 2 \times 3 = 6$. $6$ is a multiple of 2 ($6 = 2 \times 3$).
• If $m = k$ (any whole number), then $2m = 2 \times k$. By definition, any number that can be expressed as 2 multiplied by an integer is a multiple of 2. Since $m$ is a whole number (which is a subset of integers), $2m$ is always a multiple of 2.

The expression $2m$ generates all non-negative even numbers (multiples of 2) when $m$ is a whole number.

The statement "If m is a whole number, then 2m denotes a multiple of 2" is true.

The statement is True.

Given:

An integer x.

The statement: "The additive inverse of an integer x is 2x."

To Verify:

Determine if the given statement is true or false based on the definition of additive inverse.

Solution:

The additive inverse of an integer (or any number) is the number that, when added to the original number, results in zero.

If y is the additive inverse of x, then $x + y = 0$.

To find y, we subtract x from both sides of the equation:

$y = 0 - x$

$y = -x$

So, the additive inverse of an integer x is $-x$.

The statement claims that the additive inverse of x is $2x$. We need to check if $2x$ is equal to $-x$ for any integer x.

Is $2x = -x$?

Add x to both sides:

$2x + x = -x + x$

$3x = 0$

Divide by 3:

$x = 0$

The equality $2x = -x$ is only true when $x=0$. The statement claims that this is true for an integer x, implying it should hold for any integer x. Since it is not true for all integers (e.g., if $x=5$, $2x = 10$ and $-x = -5$, and $10 \neq -5$), the statement is false.

The additive inverse of x is always $-x$, not $2x$ (unless $x=0$).

The statement is False.

Given:

x is a negative integer.

The statement: "If x is a negative integer, – x is a positive integer."

To Verify:

Determine if the given statement is true or false based on the properties of integers.

Solution:

A negative integer is an integer that is less than zero, e.g., -1, -2, -3, ...

A positive integer is an integer that is greater than zero, e.g., 1, 2, 3, ...

The expression – x represents the additive inverse of x. If x is a negative integer, it means x has a value like $-k$, where $k$ is a positive integer.

Let's consider some examples:

• If $x = -1$ (a negative integer), then $-x = -(-1) = 1$. $1$ is a positive integer.
• If $x = -5$ (a negative integer), then $-x = -(-5) = 5$. $5$ is a positive integer.
• If $x = -100$ (a negative integer), then $-x = -(-100) = 100$. $100$ is a positive integer.

In general, if $x$ is a negative integer, it means $x < 0$. Multiplying both sides of the inequality by -1 reverses the inequality sign:

$-1 \times x > -1 \times 0$

$-x > 0$

Since $-x$ is greater than 0, it is a positive number. If x is an integer, then -x is also an integer. Therefore, -x is a positive integer.

The statement "If x is a negative integer, – x is a positive integer" is true.

The statement is True.

Given:

The mathematical statement $2x – 5 > 11$.

The statement: "$2x – 5 > 11$ is an equation."

To Verify:

Determine if the given statement is true or false based on the definition of an equation.

Solution:

An equation is a mathematical statement that shows that two mathematical expressions are equal. It is characterised by the presence of an equality sign (=).

The given statement is $2x – 5 > 11$. This statement uses the symbol $>$ which means "greater than". This symbol indicates a relationship of inequality, not equality.

Statements involving inequality signs such as $>, <, \ge, \le, \neq$ are called inequalities.

Since the statement $2x – 5 > 11$ uses the inequality sign $>$, it is an inequality, not an equation.

The statement "$2x – 5 > 11$ is an equation" is false.

The statement is False.

Given:

The statement: "In an equation, the LHS is equal to the RHS."

To Verify:

Determine if the given statement is true or false based on the definition of an equation.

Solution:

An equation is a mathematical statement that asserts the equality of two expressions. The equality sign (=) is used to indicate that the expression on the left side of the equality sign (Left Hand Side, LHS) has the same value as the expression on the right side of the equality sign (Right Hand Side, RHS).

For example, in the equation $x + 5 = 12$, the LHS is $x + 5$ and the RHS is $12$. The equation states that the value of the expression $x + 5$ is equal to the value of the expression $12$. This equality is the defining characteristic of an equation.

The statement "In an equation, the LHS is equal to the RHS" is directly in line with the definition of an equation.

The statement is true.

The statement is True.

Given:

The equation $7k – 7 = 7$.

The statement: "In the equation $7k – 7 = 7$, the variable is 7."

To Verify:

Determine if the given statement is true or false by identifying the variable in the equation.

Solution:

In an algebraic equation, a variable is a letter or symbol that represents an unknown quantity or a quantity that can change its value. Constants are fixed numerical values.

In the equation $7k – 7 = 7$:

• The number 7 appears multiple times. It is a fixed numerical value. Therefore, 7 is a constant.
• The letter 'k' is present. It represents an unknown value that we can find by solving the equation. Therefore, 'k' is the variable in this equation.

The statement claims that the variable is 7. However, 7 is a constant.

The statement "In the equation $7k – 7 = 7$, the variable is 7" is false.

The statement is False.

Given:

The equation is $2a – 1 = 5$.

The proposed solution is $a = 3$.

The statement: "a = 3 is a solution of the equation $2a – 1 = 5$".

To Verify:

Determine if the given statement is true or false by checking if $a=3$ satisfies the equation.

Solution:

To check if $a=3$ is a solution, we substitute $a=3$ into the given equation and evaluate both sides.

Equation: $2a – 1 = 5$

Substitute $a=3$ into the Left Hand Side (LHS):

LHS = $2 \times a – 1$

LHS = $2 \times 3 – 1$

LHS = $6 – 1$

LHS = $5$

The Right Hand Side (RHS) of the equation is 5.

Since LHS = 5 and RHS = 5, we have LHS = RHS.

This means that when $a=3$ is substituted into the equation $2a – 1 = 5$, the equation is true.

Therefore, $a=3$ is a solution of the equation $2a – 1 = 5$.

The statement "a = 3 is a solution of the equation $2a – 1 = 5$" is true.

The statement is True.

Given:

The statement: "The distance between New Delhi and Bhopal is not a variable."

To Verify:

Determine if the given statement is true or false based on the definition of a variable.

Solution:

In mathematics, a variable is a quantity that may change its value or represents an unknown value. A constant is a quantity that has a fixed value.

The distance between two fixed geographical points on the Earth's surface, such as New Delhi and Bhopal, is a specific, determined value. While there might be slight variations due to measurement methods or specific routes, the inherent geographical distance between these two cities is considered a fixed property.

This distance does not change arbitrarily; it is a defined value (approximately 780 km by road, or a specific straight-line distance). Since the distance is a fixed value (within the context of typical mathematical problems), it is considered a constant, not a variable.

The statement says that this distance is *not* a variable. This aligns with our understanding that the distance between two fixed points is a constant.

The statement "The distance between New Delhi and Bhopal is not a variable" is true.

The statement is True.

Given:

A quantity of time expressed as 't minutes'.

The statement: "t minutes are equal to 60t seconds."

To Verify:

Determine if the given statement is true or false based on the conversion between minutes and seconds.

Solution:

We need to know the conversion factor between minutes and seconds.

We know that:

To convert a duration from minutes to seconds, we multiply the number of minutes by 60.

Given duration = t minutes

Duration in seconds = t $\times$ (Number of seconds in 1 minute)

Duration in seconds = t $\times$ 60

Duration in seconds = $60t$ seconds

So, t minutes is equal to $60t$ seconds.

The statement "t minutes are equal to 60t seconds" is consistent with the conversion factor.

The statement is true.

The statement is True.

Given:

The equation is $3x + 2 = 20$.

The proposed solution is $x = 5$.

The statement: "$x = 5$ is the solution of the equation $3x + 2 = 20$".

To Verify:

Determine if the given statement is true or false by checking if $x=5$ satisfies the equation.

Solution:

To check if $x=5$ is a solution, we substitute $x=5$ into the given equation and evaluate both sides.

Equation: $3x + 2 = 20$

Substitute $x=5$ into the Left Hand Side (LHS):

LHS = $3 \times x + 2$

LHS = $3 \times 5 + 2$

LHS = $15 + 2$

LHS = $17$

The Right Hand Side (RHS) of the equation is 20.

Since LHS = 17 and RHS = 20, we have LHS $\neq$ RHS.

This means that when $x=5$ is substituted into the equation $3x + 2 = 20$, the equation is not true.

Therefore, $x=5$ is not a solution of the equation $3x + 2 = 20$.

The statement "$x = 5$ is the solution of the equation $3x + 2 = 20$" is false.

Alternatively, we can solve the equation $3x + 2 = 20$ to find its correct solution:

Subtract 2 from both sides:

Divide both sides by 3:

The solution of the equation $3x + 2 = 20$ is $x = 6$, not $x = 5$. This confirms that the statement is false.

The statement is False.

Given:

The verbal statement: "One third of a number added to itself gives 8".

The proposed equation: $\frac{x}{3} + 8 = x$.

The statement: "‘One third of a number added to itself gives 8’, can be expressed as $\frac{x}{3} + 8 = x$".

To Verify:

Determine if the given statement is true or false by translating the verbal statement into an algebraic equation and comparing it with the proposed equation.

Solution:

Let the unknown number be represented by the variable $x$.

"One third of a number" means $\frac{1}{3}$ of $x$, which is written as $\frac{x}{3}$.

"added to itself" means adding the original number ($x$) to "one third of the number" ($\frac{x}{3}$). This translates to $\frac{x}{3} + x$.

"gives 8" means the result of the operation is equal to 8. So, the equation is $\frac{x}{3} + x = 8$.

Let's compare the equation derived from the verbal statement with the proposed equation:

Derived equation: $\frac{x}{3} + x = 8$

Proposed equation: $\frac{x}{3} + 8 = x$

These two equations are different. The proposed equation means "one third of a number plus 8 gives the number itself". The verbal statement means "one third of a number plus the number itself gives 8".

Therefore, the proposed equation $\frac{x}{3} + 8 = x$ does not correctly represent the verbal statement "One third of a number added to itself gives 8".

The statement "‘One third of a number added to itself gives 8’, can be expressed as $\frac{x}{3} + 8 = x$" is false.

The statement is False.

Given:

Two sisters, Leela and Yamini.

The statement: "The difference between the ages of two sisters Leela and Yamini is a variable."

To Verify:

Determine if the given statement is true or false based on the nature of the age difference between two individuals.

Solution:

Let Leela's current age be $L$ years and Yamini's current age be $Y$ years.

The difference between their ages is $|L - Y|$ or $(L - Y)$ if we assume one is older than the other, or simply the absolute difference.

Assuming Leela is older, the difference in age is $D = L - Y$.

Consider their ages in the future, say after $t$ years.

Leela's age after $t$ years will be $L + t$.

Yamini's age after $t$ years will be $Y + t$.

The difference in their ages after $t$ years will be $(L + t) - (Y + t) = L + t - Y - t = L - Y$.

The difference in their ages remains constant over time. It is the difference between their birth years, which is a fixed value.

Since the difference between their ages is a fixed value (a constant) and does not change as time passes, it is not a variable.

The statement says the difference is a variable. This is false.

The statement is False.

Given:

The statement: "The number of lines that can be drawn through a point is a variable."

To Verify:

Determine if the given statement is true or false based on a geometric property.

Solution:

In geometry, a point is a location in space that has no size (no length, width, or depth). A line is a straight one-dimensional figure that extends infinitely in both directions.

Consider a single point in a plane or in space. We can draw a line that passes through this point. We can draw another line that passes through the same point but in a different direction. We can continue to draw lines through this point in infinitely many directions.

There is no limit to the number of distinct lines that can pass through a single given point. Any line passing through that point, regardless of its orientation, satisfies the condition.

The number of lines that can be drawn through a single point is not a specific, countable number like 1, 2, 100, or any fixed value. It is infinite.

A variable is a quantity whose value can change or is unknown. A quantity that has a fixed value is called a constant.

The number of lines through a point is always the same (infinitely many); it does not vary depending on the point chosen. Although the number is infinite, it is a fixed, determined quantity in this geometric context, unlike a quantity that could take on different finite values.

However, the statement asks if it is a variable. Since the number of lines through a point is always the same fixed quantity (infinite), it is not a variable in the sense that its value can differ from case to case or within an algebraic problem (where variables typically take finite values). A variable is something that is unknown or whose value can vary in an expression, equation, or formula.

In the context of elementary algebra where the term 'variable' is introduced for quantities that can take on different numerical values or represent unknowns in equations, the number of lines through a point (which is always infinite) is not a variable.

The statement claims that this number is a variable. This is false because the number is a constant (specifically, an infinite constant).

The statement is False.

Given:

The phrase "One more than twice the number".

To Express:

Write the corresponding algebraic expression using a variable for the unknown number.

Solution:

Let the unknown number be represented by the variable $x$.

"twice the number" means multiplying the number $x$ by 2. This is written as $2 \times x$, or simply $2x$.

"One more than twice the number" means adding 1 to the result of "twice the number".

So, we add 1 to $2x$. This is written as $2x + 1$.

The algebraic expression for "One more than twice the number" is $2x + 1$.

Let the number be x.

Twice the number is $2x$.

One more than twice the number is $2x + 1$.

The expression is $2x + 1$.

Given:

The phrase "20$^\circ$C less than the present temperature".

To Express:

Write the corresponding algebraic expression using a variable for the present temperature.

Solution:

Let the present temperature (in $^\circ$C) be represented by the variable $t$.

"20$^\circ$C less than the present temperature" means subtracting 20 from the present temperature.

So, we subtract 20 from $t$. This is written as $t - 20$.

The resulting temperature will be in $^\circ$C.

Let the present temperature be $t^\circ$C.

20$^\circ$C less than the present temperature is $t - 20$ $^\circ$C.

The expression is $t - 20$.

Given:

The phrase "The successor of an integer".

To Express:

Write the corresponding algebraic expression using a variable for the integer.

Solution:

Let the integer be represented by the variable $n$.

The successor of an integer is the next integer in the counting sequence. It is obtained by adding 1 to the integer.

For example, the successor of 5 is $5 + 1 = 6$. The successor of -2 is $-2 + 1 = -1$.

The successor of the integer $n$ is $n + 1$.

Let the integer be $n$.

The successor of the integer is $n + 1$.

The expression is $n + 1$.

Given:

The figure is an equilateral triangle.

The side length of the equilateral triangle is m.

To Express:

Write the expression for the perimeter of the triangle.

Solution:

An equilateral triangle is a triangle with all three sides equal in length.

The perimeter of a triangle is the sum of the lengths of its three sides.

Since the triangle is equilateral and its side length is m, all three sides have length m.

Perimeter = Side 1 length + Side 2 length + Side 3 length

Perimeter = m + m + m

Combining the terms involving m:

Perimeter = 3m

The perimeter of the equilateral triangle is $3m$.

Side length of the equilateral triangle = m.

Number of sides in a triangle = 3.

Perimeter = Sum of lengths of sides = m + m + m = $3m$.

The expression is $3m$.

Given:

The figure is a rectangle.

Length of the rectangle = k units

Breadth of the rectangle = n units

To Express:

Write the expression for the area of the rectangle.

Solution:

A rectangle is a quadrilateral with four right angles. The opposite sides are equal in length.

The area of a rectangle is calculated by multiplying its length by its breadth.

Area of a rectangle = Length $\times$ Breadth

Substitute the given values for length and breadth:

Area = k units $\times$ n units

Area = $(k \times n)$ square units

In algebraic form, $k \times n$ is written as $kn$.

Area = $kn$ square units

The area of the rectangle is $kn$ square units.

Length of the rectangle = k units.

Breadth of the rectangle = n units.

Area = Length $\times$ Breadth = $k \times n = kn$ square units.

The expression is $kn$.

Given:

The phrase "Omar helps his mother 1 hour more than his sister does".

To Express:

Write the corresponding algebraic expression relating the time Omar helps to the time his sister helps.

Solution:

Let the amount of time Omar's sister helps (in hours) be represented by the variable $s$.

The phrase "Omar helps his mother 1 hour more than his sister does" means that the time Omar helps is equal to the time his sister helps plus 1 hour.

Let the amount of time Omar helps (in hours) be represented by the variable $o$.

According to the statement:

Omar's time = Sister's time + 1 hour

$o = s + 1$

The expression representing the time Omar helps in terms of the time his sister helps is $s + 1$. The statement itself describes the relationship between their helping times.

Let the time Omar's sister helps be $s$ hours.

Omar helps 1 hour more than his sister.

Time Omar helps = $s + 1$ hours.

The statement implies a comparison. If we want to express Omar's helping time using the sister's helping time ($s$), the expression is $s+1$. If we are asked to express the statement as a relationship or equation, it would be (Time Omar helps) = (Time sister helps) + 1 hour.

Let's assume the question asks for an expression related to the time Omar helps, given the sister's helping time is represented by a variable.

Let the time Omar's sister helps be $x$ hours.

Omar helps 1 hour more than his sister.

Time Omar helps = $x + 1$ hours.

The expression is $x + 1$ (where x is the time Omar's sister helps in hours).

Given:

The phrase "Two consecutive odd integers".

To Express:

Write algebraic expressions for two consecutive odd integers using a variable.

Solution:

Consecutive odd integers are odd integers that follow each other in sequence, with a difference of 2 between them (e.g., 3 and 5, -1 and 1, 11 and 13).

Let the first odd integer be represented by a variable. A common way to represent any odd integer using a variable is $2n + 1$ or $2n - 1$, where $n$ is an integer.

Let's use the variable $n$ to represent the first odd integer. So, the first odd integer is $n$.

The next consecutive odd integer is obtained by adding 2 to the first odd integer.

The second consecutive odd integer = (First odd integer) + 2

The second consecutive odd integer = $n + 2$

So, two consecutive odd integers can be represented as $n$ and $n+2$, where $n$ is an odd integer.

Alternatively, if we represent the first odd integer in the form $2k+1$ (where $k$ is an integer), the next consecutive odd integer is:

$(2k + 1) + 2 = 2k + 3$

So, two consecutive odd integers can also be represented as $2k+1$ and $2k+3$, where $k$ is an integer.

The question asks for expressions, and using a single variable to define the first term is standard.

Let the first consecutive odd integer be $x$.

The next consecutive odd integer is $x + 2$.

The expressions are $x$ and $x + 2$ (where x is an odd integer).

Given:

The phrase "Two consecutive even integers".

To Express:

Write algebraic expressions for two consecutive even integers using a variable.

Solution:

Consecutive even integers are even integers that follow each other in sequence, with a difference of 2 between them (e.g., 2 and 4, -6 and -4, 10 and 12).

Let the first even integer be represented by a variable. A common way to represent any even integer using a variable is $2n$, where $n$ is an integer.

Let's use the variable $n$ to represent the first even integer. So, the first even integer is $n$.

The next consecutive even integer is obtained by adding 2 to the first even integer.

The second consecutive even integer = (First even integer) + 2

The second consecutive even integer = $n + 2$

So, two consecutive even integers can be represented as $n$ and $n+2$, where $n$ is an even integer.

Alternatively, if we represent the first even integer in the form $2k$ (where $k$ is an integer), the next consecutive even integer is:

$2k + 2$

So, two consecutive even integers can also be represented as $2k$ and $2k+2$, where $k$ is an integer.

The question asks for expressions, and using a single variable to define the first term is standard.

Let the first consecutive even integer be $x$.

The next consecutive even integer is $x + 2$.

The expressions are $x$ and $x + 2$ (where x is an even integer).

A multiple of a number is the result obtained by multiplying that number by an integer.

To find the multiples of a number, we multiply it by any integer (positive, negative, or zero).

For the number 5, the multiples are obtained by multiplying 5 by various integers.

Let's list some examples:

If we multiply 5 by positive integers ($1, 2, 3, ...$), we get positive multiples:

$5 \times 1 = 5$

$5 \times 2 = 10$

$5 \times 3 = 15$

$5 \times 4 = 20$

and so on.

If we multiply 5 by zero ($0$), we get:

$5 \times 0 = 0$

If we multiply 5 by negative integers ($-1, -2, -3, ...$), we get negative multiples:

$5 \times (-1) = -5$

$5 \times (-2) = -10$

$5 \times (-3) = -15$

and so on.

Combining these, the multiples of 5 are a sequence of numbers that extend infinitely in both positive and negative directions, including zero.

The set of multiples of 5 can be written as: $..., -15, -10, -5, 0, 5, 10, 15, 20, ...$

In general, a multiple of 5 can be represented in the form $5k$, where $k$ is any integer.

We can write this using set notation as $\{5k \mid k \in \mathbb{Z}\}$, where $\mathbb{Z}$ represents the set of all integers ($..., -3, -2, -1, 0, 1, 2, 3, ...$).

A simple way to identify if a number is a multiple of 5 is to check if its last digit is either 0 or 5.

A fraction is a number representing a part of a whole. It is written in the form $\frac{p}{q}$, where $p$ is the numerator and $q$ is the denominator, and $q \neq 0$.

In this problem, we are given a relationship between the denominator and the numerator of a fraction.

Let the numerator of the fraction be represented by the variable $n$.

The problem states that the denominator is 1 more than its numerator.

Therefore, the denominator can be represented as $n + 1$.

Using these representations, the fraction can be written as:

$\text{Fraction} = \frac{\text{Numerator}}{\text{Denominator}}$

$\text{Fraction} = \frac{n}{n+1}$

Here, $n$ represents the numerator and $n+1$ represents the denominator, satisfying the condition that the denominator is 1 more than the numerator. Note that for this to be a valid fraction, the denominator cannot be zero, so $n+1 \neq 0$, which means $n \neq -1$. Also, typically, fractions involve integers, so $n$ would be an integer (often a non-negative integer in many contexts, but the problem doesn't specify).

We are given a relationship between the heights of two landmarks: Mount Everest and the Empire State Building.

Let $H_{Everest}$ represent the height of Mount Everest.

Let $H_{ESB}$ represent the height of the Empire State Building.

The problem states that the height of Mount Everest is 20 times the height of the Empire State Building.

This can be expressed mathematically as:

$H_{Everest} = 20 \times H_{ESB}$

Or simply:

$H_{Everest} = 20 H_{ESB}$

This equation shows that to find the height of Mount Everest, you would multiply the height of the Empire State Building by 20.

We are given the individual costs of a notebook and a pencil and need to find the total cost of a specific quantity of each.

Given:

Cost of one notebook = $\textsf{₹} p$

Cost of one pencil = $\textsf{₹} 3$

We need to find the total cost of two notebooks and one pencil.

The cost of two notebooks will be the cost of one notebook multiplied by 2.

Cost of two notebooks = $2 \times (\textsf{₹} p) = \textsf{₹} 2p$

The total cost is the sum of the cost of two notebooks and the cost of one pencil.

Total cost = (Cost of two notebooks) + (Cost of one pencil)

Total cost = $\textsf{₹} 2p + \textsf{₹} 3$

Thus, the total cost (in Rs) of two notebooks and one pencil is $2p + 3$.

We need to translate the given word statement into a mathematical expression.

The statement involves a variable z and two operations: multiplication and subtraction.

First, "z is multiplied by –3". This operation can be written as:

$z \times (-3)$ or $-3z$

The result of this operation is $-3z$.

Next, "the result is subtracted from 13". This means we take 13 and subtract the result of the first operation from it.

This can be written as:

$13 - (\text{result})$

Substitute the result $(-3z)$ into this expression:

$13 - (-3z)$

Subtracting a negative number is the same as adding the corresponding positive number. Therefore, the expression simplifies to:

$13 + 3z$

So, the algebraic expression for the given statement is $13 + 3z$.

We are asked to translate the given word statement into an algebraic expression.

The statement involves a variable p and two operations: division and addition.

First, the statement says "p is divided by 11". This can be written mathematically as:

$\frac{p}{11}$

This is the result of the first step.

Next, the statement says "the result is added to 10". This means we take the result from the first step ($\frac{p}{11}$) and add 10 to it.

This can be written as:

$10 + \frac{p}{11}$

Thus, the algebraic expression for the given statement is $10 + \frac{p}{11}$.

We need to translate the given word statement into an algebraic expression.

The statement involves a variable x, multiplication, the smallest natural number, and addition.

First, let's identify the smallest natural number.

The set of natural numbers is $\{1, 2, 3, ...\}$.

The smallest natural number is 1.

Next, consider the phrase "x times of 3".

This means multiplying x by 3.

Mathematically, this is $3 \times x$, which is written as $3x$.

Finally, the statement says "the result is added to the smallest natural number".

This means we add the result from the previous step ($3x$) to the smallest natural number (1).

This can be written as:

$3x + 1$

Therefore, the algebraic expression for the given statement is $3x + 1$.

We need to translate the given word statement into an algebraic expression.

The statement involves a variable q, multiplication, the smallest two-digit number, and subtraction.

First, let's identify the smallest two-digit number.

Two-digit numbers range from 10 to 99.

The smallest number in this range is 10.

Next, consider the phrase "6 times q".

This means multiplying q by 6.

Mathematically, this is $6 \times q$, which is written as $6q$.

Finally, the statement says "is subtracted from the smallest two digit number".

This means we take the result from the previous step ($6q$) and subtract it from the smallest two-digit number (10).

This can be written as:

$10 - 6q$

Therefore, the algebraic expression for the given statement is $10 - 6q$.

An equation is a mathematical statement that shows two expressions are equal. A solution to an equation is a value for the variable that makes the equation true.

To write an equation for which 2 is the solution, we can start with the statement that the variable (let's use $x$) is equal to 2.

$x = 2$

Then, we perform the same operation on both sides of this equation to create a more complex equation.

Equation 1:

Start with $x = 2$.

Multiply both sides by 3:

$3 \times x = 3 \times 2$

This gives us the equation:

$3x = 6$

Let's verify if $x=2$ is the solution:

Left side = $3(2) = 6$

Right side = $6$

Since Left side = Right side, $x=2$ is indeed the solution.

Equation 2:

Start again with $x = 2$.

Add 4 to both sides:

$x + 4 = 2 + 4$

This gives us the equation:

$x + 4 = 6$

Let's verify if $x=2$ is the solution:

Left side = $2 + 4 = 6$

Right side = $6$

Since Left side = Right side, $x=2$ is indeed the solution.

Therefore, two equations for which 2 is the solution are:

$\mathbf{3x = 6}$

and

$\mathbf{x + 4 = 6}$

To write an equation for which 0 is the solution, we can start with the statement that the variable (let's use $y$) is equal to 0.

$y = 0$

Then, we perform the same operation on both sides of this equation to create another equation.

Start with $y = 0$.

Multiply both sides by any non-zero number, for example, 5:

$5 \times y = 5 \times 0$

This gives us the equation:

$5y = 0$

Let's verify if $y=0$ is the solution:

Left side = $5(0) = 0$

Right side = $0$

Since Left side = Right side, $y=0$ is indeed the solution.

Alternatively, we could add or subtract a number from both sides.

Start with $y = 0$.

Add 7 to both sides:

$y + 7 = 0 + 7$

This gives the equation:

$y + 7 = 7$

Let's verify if $y=0$ is the solution:

Left side = $0 + 7 = 7$

Right side = $7$

Since Left side = Right side, $y=0$ is indeed the solution.

Therefore, one equation for which 0 is a solution is $5y = 0$ (or $y + 7 = 7$, or many others).

A whole number is a non-negative integer, i.e., a number from the set $\{0, 1, 2, 3, ...\}$.

We need to create an equation such that the value of the variable that satisfies the equation is not in this set.

We can achieve this by constructing an equation whose solution is a fraction, a negative number, or a decimal that is not an integer.

Let's choose a solution that is not a whole number, for example, $\frac{1}{2}$.

We start with a simple equation where the variable (say, $x$) is equal to this value:

$x = \frac{1}{2}$

Now, we can perform the same operation on both sides to form a slightly more complex equation.

Multiply both sides by 2:

$2 \times x = 2 \times \frac{1}{2}$

This simplifies to:

$2x = 1$

Let's verify if the solution to $2x = 1$ is indeed $\frac{1}{2}$ and if it's not a whole number.

To solve $2x = 1$, divide both sides by 2:

$\frac{2x}{2} = \frac{1}{2}$

$x = \frac{1}{2}$

The solution is $\frac{1}{2}$.

The set of whole numbers is $\{0, 1, 2, 3, ...\}$. The number $\frac{1}{2}$ (or 0.5) is not in this set.

Therefore, an equation whose solution is not a whole number is $2x = 1$.

Alternate Equation:

Let's choose a negative integer as the solution, for example, -3.

Start with $y = -3$.

Add 5 to both sides:

$y + 5 = -3 + 5$

$y + 5 = 2$

The solution to $y+5=2$ is $y=-3$. Since -3 is not a whole number, this is another valid equation.

So, another equation is $y + 5 = 2$.

The given statement involves the costs of a pencil and a pen, represented by algebraic expressions.

Let $p$ be the cost of a pencil in Rupees ($\textsf{₹}$).

The cost of the pen is given as $\textsf{₹} 5p$.

Comparing the cost of the pen ($\textsf{₹} 5p$) to the cost of the pencil ($\textsf{₹} p$), we see that the cost of the pen is 5 times the cost of the pencil.

Converting this relationship into an ordinary language statement:

The cost of a pen is five times the cost of a pencil.

We are given information about Leela's contribution and the amount of money she has left, expressed using an algebraic expression.

Leela's contribution is given as $\textsf{₹} y$.

The amount of money Leela is left with is given as $\textsf{₹} (y + 10000)$.

The expression $(y + 10000)$ means that the amount left is the contribution amount ($y$) plus $\textsf{₹} 10000$.

In ordinary language, this means the amount of money Leela is left with is $\textsf{₹} 10000$ more than the amount she contributed.

Converting the expression into a statement in ordinary language:

Leela is now left with $\textsf{₹} 10000$ more than the amount she contributed to the Prime Minister’s Relief Fund.

We are given the ages of Kartik and his father using an algebraic expression.

Kartik's age is given as $n$ years.

His father's age is given as $7n$ years.

The expression $7n$ means that the father's age is 7 times Kartik's age ($n$).

Converting this relationship into an ordinary language statement:

Kartik's father is seven times as old as Kartik.

We are given the maximum and minimum temperatures on a day in Delhi using an algebraic expression.

Maximum temperature = $p^\circ\text{C}$.

Minimum temperature = $(p - 10)^\circ\text{C}$.

The expression $(p - 10)$ means that the minimum temperature is 10 less than the maximum temperature ($p$).

In ordinary language, this means the minimum temperature was $10^\circ\text{C}$ lower than the maximum temperature.

Converting the expression into a statement in ordinary language:

The minimum temperature on a day in Delhi was $10^\circ\text{C}$ less than the maximum temperature.

We are given the number of plants planted by John and his friend Jay using an algebraic expression.

Number of plants John planted = $t$.

Number of plants Jay planted = $2t + 10$.

The expression $2t + 10$ means that the number of plants Jay planted is 2 times the number John planted ($t$) plus 10.

In ordinary language, this means Jay planted twice the number of plants John planted, plus 10 more plants.

Converting the expression into a statement in ordinary language:

Jay planted 10 more than twice the number of plants that John planted.

We are given the number of cups of tea Sharad used to drink and the number he drinks now, using an algebraic expression.

Original number of cups of tea per day = $p$.

New number of cups of tea per day = $p - 5$.

The expression $p - 5$ means that the new number of cups is 5 less than the original number of cups ($p$).

In ordinary language, this means Sharad now drinks 5 cups of tea less than he used to.

Converting the expression into a statement in ordinary language:

Sharad now takes 5 cups less than the number of cups of tea he used to take a day.

We are given the number of students dropping out of school in two different years using an algebraic expression.

Number of students dropping out last year = $m$.

Number of students dropping out this year = $m - 30$.

The expression $m - 30$ means that the number of students dropping out this year is 30 less than the number who dropped out last year ($m$).

In ordinary language, this means the number of students dropping out this year is 30 fewer than last year.

Converting the expression into a statement in ordinary language:

The number of students dropping out of school this year is 30 less than the number of students who dropped out last year.

We are given the price of petrol last month and the price now, using an algebraic expression.

Price of petrol last month = $\textsf{₹} p$ per litre.

Price of petrol now = $\textsf{₹} (p - 5)$ per litre.

The expression $(p - 5)$ means that the current price is $\textsf{₹} 5$ less than the price last month ($p$).

In ordinary language, this means the price of petrol has decreased by $\textsf{₹} 5$ per litre compared to last month.

Converting the expression into a statement in ordinary language:

The price of petrol now is $\textsf{₹} 5$ less than the price last month.

We are given Khader's monthly salary in two different years using an algebraic expression.

Khader's monthly salary in 2005 = $\textsf{₹} P$.

Khader's monthly salary in 2006 = $\textsf{₹} (P + 1000)$.

The expression $(P + 1000)$ means that his salary in 2006 was $\textsf{₹} 1000$ more than his salary in 2005 ($P$).

In ordinary language, this means Khader's salary increased by $\textsf{₹} 1000$ from 2005 to 2006.

Converting the expression into a statement in ordinary language:

Khader’s monthly salary in 2006 was $\textsf{₹} 1000$ more than his salary in 2005.

We are given the number of girls enrolled in a school last year and this year using an algebraic expression.

Number of girls enrolled last year = $g$.

Number of girls enrolled this year = $3g - 10$.

The expression $3g - 10$ means that the number of girls enrolled this year is 3 times the number of girls enrolled last year ($g$), minus 10.

In ordinary language, this means the number of girls enrolled this year is 10 less than three times the number enrolled last year.

Converting the expression into a statement in ordinary language:

The number of girls enrolled in the school this year is 10 less than three times the number of girls enrolled last year.

We need to translate each given statement into an algebraic equation using the variable $x$.

(a) 13 subtracted from twice a number gives 3.

Let the number be $x$.

"Twice a number" translates to $2x$.

"13 subtracted from twice a number" translates to $2x - 13$.

"gives 3" means the result is equal to 3.

The equation is:

$\mathbf{2x - 13 = 3}$

(b) One fifth of a number is 5 less than that number.

Let the number be $x$.

"One fifth of a number" translates to $\frac{1}{5}x$ or $\frac{x}{5}$.

"5 less than that number" translates to $x - 5$.

"is" means the two expressions are equal.

The equation is:

$\mathbf{\frac{x}{5} = x - 5}$

(c) Two-third of number is 12.

Let the number be $x$.

"Two-third of a number" translates to $\frac{2}{3}x$.

"is 12" means the result is equal to 12.

The equation is:

$\mathbf{\frac{2}{3}x = 12}$

(d) 9 added to twice a number gives 13.

Let the number be $x$.

"Twice a number" translates to $2x$.

"9 added to twice a number" translates to $2x + 9$.

"gives 13" means the result is equal to 13.

The equation is:

$\mathbf{2x + 9 = 13}$

(e) 1 subtracted from one-third of a number gives 1.

Let the number be $x$.

"One-third of a number" translates to $\frac{1}{3}x$ or $\frac{x}{3}$.

"1 subtracted from one-third of a number" translates to $\frac{x}{3} - 1$.

"gives 1" means the result is equal to 1.

The equation is:

$\mathbf{\frac{x}{3} - 1 = 1}$

We need to translate each statement into an algebraic equation using the variables provided.

(a) The perimeter (p) of an equilateral triangle is three times of its side (a).

The variables are $p$ (perimeter) and $a$ (side length).

The relationship is that the perimeter is three times the side.

Equation:

$\mathbf{p = 3a}$

(b) The diameter (d) of a circle is twice its radius (r).

The variables are $d$ (diameter) and $r$ (radius).

The relationship is that the diameter is twice the radius.

Equation:

$\mathbf{d = 2r}$

(c) The selling price (s) of an item is equal to the sum of the cost price (c) of an item and the profit (p) earned.

The variables are $s$ (selling price), $c$ (cost price), and $p$ (profit).

The relationship is that the selling price is the sum of the cost price and the profit.

Equation:

$\mathbf{s = c + p}$

(d) Amount (a) is equal to the sum of principal (p) and interest (i).

The variables are $a$ (amount), $p$ (principal), and $i$ (interest).

The relationship is that the amount is the sum of the principal and the interest.

Equation:

$\mathbf{a = p + i}$

We are given that Kanika's present age is $x$ years. We need to write algebraic expressions for the ages of her relatives based on the given descriptions.

Kanika's age = $x$ years.

(i) Her brother is 2 years younger

This means the brother's age is Kanika's age minus 2 years.

Brother's age $= x - 2$ years.

(ii) Her father's age exceed her age by 35 years.

This means the father's age is Kanika's age plus 35 years.

Father's age $= x + 35$ years.

(iii) Mother's age is 3 years less than that of her father.

From (ii), the father's age is $(x + 35)$ years.

The mother's age is the father's age minus 3 years.

Mother's age $= (x + 35) - 3 = x + 32$ years.

(iv) Her grand father's age is 8 times of her age.

This means the grandfather's age is 8 multiplied by Kanika's age.

Grandfather's age $= 8 \times x = 8x$ years.

Now, we complete the table with these expressions:

We are given that $m$ is a whole number less than 5. The whole numbers are $\{0, 1, 2, 3, ...\}$.

The whole numbers less than 5 are 0, 1, 2, 3, and 4.

We need to complete the table by evaluating the expression $2m - 5$ for each of these values of $m$.

• When $m = 0$, $2m - 5 = 2(0) - 5 = 0 - 5 = -5$
• When $m = 1$, $2m - 5 = 2(1) - 5 = 2 - 5 = -3$
• When $m = 2$, $2m - 5 = 2(2) - 5 = 4 - 5 = -1$
• When $m = 3$, $2m - 5 = 2(3) - 5 = 6 - 5 = 1$
• When $m = 4$, $2m - 5 = 2(4) - 5 = 8 - 5 = 3$

Now we fill these values into the table:

We need to find the solution to the equation $2m - 5 = -1$ by inspection of the table.

We look for the row where the value of the expression $2m - 5$ is equal to -1.

From the table, we see that when $2m - 5$ is -1, the corresponding value of $m$ is 2.

Therefore, the solution to the equation $2m - 5 = -1$ is $\mathbf{m = 2}$.

We need to find the amount of money remaining after paying the advance for transport.

Given:

Number of students in the class = $p$

Amount collected per student = $\textsf{₹} 50$

Advance paid for transport = $\textsf{₹} 1800$

First, let's calculate the total amount of money collected from all the students.

Total money collected = (Number of students) $\times$ (Amount collected per student)

Total money collected = $p \times \textsf{₹} 50 = \textsf{₹} 50p$

Next, we need to find the money left after paying the advance for transport.

Money left = (Total money collected) - (Advance paid for transport)

Money left = $\textsf{₹} 50p - \textsf{₹} 1800$

The amount of money left with them to spend on other items is $\textsf{₹} (50p - 1800)$.

We need to calculate the total amount of water in all the tanks on a particular day, considering the rain collected and the initial amount in one tank.

Given:

Number of water tanks = 8

Amount of rain water collected per tank on that day = $x$ litres

Initial amount of water in one specific tank = 100 litres

The amount of rain water collected in 8 tanks on that day is the amount collected per tank multiplied by the number of tanks.

Total rain water collected in 8 tanks = 8 $\times x$ litres = $8x$ litres.

However, the problem states that 100 litres was already in *one* of the tanks.

Let's consider the distribution of water:

The amount of water in each of the 8 tanks after collecting $x$ litres of rain is $(x + \text{initial amount})$.

One tank had 100 litres initially, and collected $x$ litres of rain. So, this tank has $(100 + x)$ litres.

The other 7 tanks likely started empty (or had a negligible amount not mentioned) and collected $x$ litres each. So, these 7 tanks each have $x$ litres.

Total amount of water in all tanks = (Water in the tank with initial 100 litres) + (Water in the other 7 tanks)

Total amount of water = $(100 + x) + 7 \times x$

Total amount of water = $100 + x + 7x$

Combine the terms with $x$:

$x + 7x = 8x$

Total amount of water = $100 + 8x$ litres.

Alternatively, consider the total rain collected ($8x$) and add the initial 100 litres which is present in *one* of those 8 tanks.

Total amount of water = (Total rain collected in all 8 tanks) + (Initial amount in one tank)

Total amount of water = $8x + 100$ litres.

Both approaches lead to the same expression. The total amount of water in the tanks on that day is $(8x + 100)$ litres.

We need to find the area of a square given the length of its side.

Given:

The side length of the square is $m$ cm.

The formula for the area of a square is:

Area = $(\text{side length})^2$

In this case, the side length is $m$ cm.

Area $= (m \text{ cm})^2$

Area $= m^2 \text{ cm}^2$

The area of the square is $m^2$ square cm (or $\mathbf{m^2 \text{ cm}^2}$).

The given formula for the perimeter ($P$) of a triangle is $P = a + b + c$, where $a$, $b$, and $c$ are the lengths of the three sides of the triangle.

The formula tells us how to calculate the perimeter of any triangle.

The perimeter is the total distance around the outside of the shape.

The formula states that this perimeter ($P$) is equal to the sum of the lengths of its three sides ($a$, $b$, and $c$).

Expressing the rule in words:

The rule expressed by the formula $P = a + b + c$ is: The perimeter of a triangle is the sum of the lengths of its three sides.

The given formula for the perimeter ($P$) of a rectangle is $P = 2 (l + w)$, where $l$ is the length and $w$ is the breadth (or width) of the rectangle.

The formula involves two steps:

1. Find the sum of the length and the breadth ($l + w$).

2. Multiply the sum by 2 ($2 \times (l + w)$).

This corresponds to the fact that a rectangle has two sides of length $l$ and two sides of length $w$. The perimeter is $l + w + l + w = 2l + 2w = 2(l + w)$.

Expressing the rule in words:

The rule expressed by the formula $P = 2 (l + w)$ is: The perimeter of a rectangle is twice the sum of its length and breadth.

Alternate wording:

The perimeter of a rectangle is the sum of twice its length and twice its breadth.

We need to find the present weight by adding the weight gained to the weight from last year.

Given:

Weight on last birthday = 40 kg

Weight gained after a year = $m$ kg

The present weight is the sum of the weight on the last birthday and the weight gained over the year.

Present weight = (Weight on last birthday) + (Weight gained after a year)

Present weight = $40 \text{ kg} + m \text{ kg}$

Present weight = $(40 + m) \text{ kg}$

Therefore, the present weight is $(40 + m)$ kg.

We are given the dimensions of a bulletin board and asked to solve several problems related to it.

Given:

Length of the bulletin board = $r$ cm

Breadth of the bulletin board = $t$ cm

(i) Length of the aluminium strip required to frame the board, if 10cm extra strip is required to fix it properly.

The aluminium strip is required to frame the board, which means it goes around the perimeter of the board.

The perimeter of a rectangle is given by the formula $P = 2(l + w)$.

Here, length $l = r$ cm and breadth $w = t$ cm.

Perimeter of the board = $2(r + t)$ cm.

An extra 10 cm strip is required.

Total length of aluminium strip required = (Perimeter of the board) + (Extra strip)

Total length $= 2(r + t) \text{ cm} + 10 \text{ cm}$

Total length $= \mathbf{(2(r + t) + 10) \text{ cm}}$

(ii) If x nails are used to repair one board, how many nails will be required to repair 15 such boards?

Given:

Number of nails used to repair one board = $x$ nails.

We need to find the number of nails required for 15 such boards.

Total nails required = (Number of boards) $\times$ (Nails per board)

Total nails required $= 15 \times x$ nails

Total nails required $= \mathbf{15x \text{ nails}}$

(iii) If 500sqcm extra cloth per board is required to cover the edges, what will be the total area of the cloth required to cover 8 such boards?

To cover the board, we need cloth equal to the area of the board plus the extra area for edges.

The area of a rectangle is given by the formula $A = l \times w$.

Area of one board = $r \times t$ cm$^2$ = $rt$ cm$^2$.

Extra cloth required per board for edges = 500 cm$^2$.

Total cloth required per board = (Area of the board) + (Extra area for edges)

Total cloth per board $= rt \text{ cm}^2 + 500 \text{ cm}^2 = (rt + 500) \text{ cm}^2$.

We need to find the total area of cloth required for 8 such boards.

Total area of cloth for 8 boards = (Number of boards) $\times$ (Total cloth per board)

Total area of cloth $= 8 \times (rt + 500) \text{ cm}^2$

Total area of cloth $= \mathbf{8(rt + 500) \text{ cm}^2}$ or $\mathbf{(8rt + 4000) \text{ cm}^2}$

(iv) What will be the expenditure for making 23 boards, if the carpenter charges Rs x per board.

Given:

Cost charged by the carpenter per board = $\textsf{₹} x$.

We need to find the total expenditure for making 23 boards.

Total expenditure = (Number of boards) $\times$ (Cost per board)

Total expenditure $= 23 \times \textsf{₹} x$

Total expenditure $= \mathbf{\textsf{₹} 23x}$

Let Sunita's present age be $S$ years.

Let Geeta's present age be $G$ years.

The problem states that Sunita is half the age of her mother Geeta.

This means $S = \frac{1}{2}G$ or $G = 2S$.

Let's express both ages in terms of one variable, say Sunita's present age, $S$.

Sunita's present age = $S$ years.

Geeta's present age = $2S$ years.

(i) Ages after 4 years:

After 4 years, both Sunita and Geeta will be 4 years older.

Sunita's age after 4 years = (Sunita's present age) + 4

Sunita's age after 4 years $= S + 4$ years.

Geeta's age after 4 years = (Geeta's present age) + 4

Geeta's age after 4 years $= 2S + 4$ years.

So, their ages after 4 years will be $(S + 4)$ years and $(2S + 4)$ years respectively, where $S$ is Sunita's present age.

(ii) Ages before 3 years:

Before 3 years, both Sunita and Geeta were 3 years younger.

Sunita's age before 3 years = (Sunita's present age) - 3

Sunita's age before 3 years $= S - 3$ years.

Geeta's age before 3 years = (Geeta's present age) - 3

Geeta's age before 3 years $= 2S - 3$ years.

So, their ages before 3 years were $(S - 3)$ years and $(2S - 3)$ years respectively, where $S$ is Sunita's present age.

Let's evaluate each item in Column I and find its corresponding match in Column II.

(i) The number of corners of a quadrilateral

A quadrilateral is a polygon with four sides. It has 4 vertices, which are its corners.

The number of corners of a quadrilateral is 4.

Looking at Column II, there is no option that corresponds to the number 4. Based on the provided options, this item does not have a direct match.

(ii) The variable in the equation $2p + 3 = 5$

In the equation $2p + 3 = 5$, the letter representing the unknown quantity is $p$. This is the variable.

Looking at Column II, option (E) is 'p'.

Match: (ii) - (E)

(iii) The solution of the equation $x + 2 = 3$

To find the solution, we solve the equation:

$x + 2 = 3$

Subtract 2 from both sides:

$x = 3 - 2$

$x = 1$

The solution is 1.

Looking at Column II, option (C) is '+1'.

Match: (iii) - (C)

(iv) solution of the equation $2p + 3 = 5$

To find the solution, we solve the equation:

$2p + 3 = 5$

Subtract 3 from both sides:

$2p = 5 - 3$

$2p = 2$

Divide both sides by 2:

$p = \frac{2}{2}$

$p = 1$

The solution is 1.

Looking at Column II, option (C) is '+1'.

Match: (iv) - (C)

(v) A sign used in an equation

An equation uses the equality sign to show that two expressions are equal.

Looking at Column II, option (A) is '='.

Match: (v) - (A)

Summary of matches based on provided options:

(i) The number of corners of a quadrilateral (4) - No match in Column II.

(ii) The variable in the equation $2p + 3 = 5$ - (E) p

(iii) The solution of the equation $x + 2 = 3$ - (C) +1

(iv) solution of the equation $2p + 3 = 5$ - (C) +1

(v) A sign used in an equation - (A) =