Chapter 11 Algebra (Additional Questions)

The question asks to identify the term for a symbol that represents an unknown value in algebra.

Let's consider each option:

(A) Constant: A constant is a value that is fixed and does not change, such as $5$, $-10$, or $\pi$. It does not represent an unknown value.

(B) Variable: A variable is a symbol, typically a letter like $x$, $y$, $a$, $b$, etc., that is used to represent an unknown quantity or a quantity that can change its value. This perfectly matches the description in the question.

(C) Expression: An expression is a combination of variables, constants, and mathematical operations. For example, $2x + 3$ is an expression. It is not a single symbol representing an unknown value.

(D) Equation: An equation is a mathematical statement that asserts the equality of two expressions. For example, $2x + 3 = 7$ is an equation. It shows a relationship between quantities but is not the symbol for the unknown value itself.

Therefore, the symbol used to represent an unknown value in algebra is called a Variable.

The correct option is (B) Variable.

The question asks to identify which of the given options is a variable.

Recall that a variable is a symbol used to represent an unknown or changing value.

Let's examine each option:

(A) 5: This is a numerical value, which is a constant. Its value is fixed as five.

(B) +: This is a mathematical operator used for addition. It represents an operation, not a value.

(C) x: This is a symbol, commonly a letter, used in algebra to represent an unknown value. This is the definition of a variable.

(D) =: This is a mathematical symbol used to indicate equality between two expressions or values. It is part of an equation, not a symbol for an unknown value itself.

Based on the definitions, only 'x' represents a variable.

The correct option is (C) x.

The question asks for the term used for a number that has a fixed value.

Let's analyse the given options:

(A) Variable: A variable is a symbol (like $x$ or $y$) used to represent a quantity that can change or is unknown. It does not have a fixed value.

(B) Term: A term in an algebraic expression is a single number, variable, or the product of numbers and variables. For example, in the expression $3x + 5$, both $3x$ and $5$ are terms. While a term can be a fixed number (like $5$), the term itself is not exclusively defined as a fixed value; it can also contain variables.

(C) Constant: A constant is a quantity whose value does not change. Numbers like $7$, $-2$, $0$, or $\pi$ are examples of constants because their value is fixed. This matches the description in the question.

(D) Coefficient: A coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., in $5x$, $5$ is the coefficient). It is a fixed number but is specifically related to a variable term, not necessarily standing alone as just a fixed value in general algebra.

Based on the standard definitions in algebra, a number with a fixed value is called a constant.

The correct option is (C) Constant.

The question asks to identify which of the given choices is an algebraic expression.

An algebraic expression is a mathematical phrase that contains variables, constants, and algebraic operations (addition, subtraction, multiplication, division). It does not contain an equality sign ($=$).

Let's examine each option:

(A) $2x + 5 = 10$: This contains an equality sign ($=$), which equates two mathematical expressions. This is an equation, not just an expression.

(B) $3 \times 4 - 2$: This contains only numbers and mathematical operations. It is a numerical expression, not an algebraic expression because it does not contain any variables.

(C) $y - 7$: This contains a variable ($y$), a constant ($7$), and a mathematical operation (subtraction). It does not contain an equality sign. This fits the definition of an algebraic expression.

(D) $8 + 3 = 11$: This contains an equality sign ($=$) and relates two numerical values. This is a numerical equation or a statement of equality, not an algebraic expression.

Based on the analysis, $y - 7$ is the only option that is an algebraic expression.

The correct option is (C) $y - 7$.

The question asks for the relationship between two expressions in an equation.

By definition, an equation is a mathematical statement that shows that two expressions are equivalent or have the same value.

Equations use an equality sign ($=$) between the two expressions to indicate this relationship.

For example, in the equation $2x + 5 = 10$, the expression $2x + 5$ on the left side is stated to be equal to the expression $10$ on the right side.

Looking at the options:

(A) Different: An equation states that two expressions are the same in value, not different.

(B) Added: Addition is an operation performed on expressions or terms, not the defining relationship between the two sides of an equation.

(C) Subtracted: Subtraction is also an operation, not the defining relationship between the two sides of an equation.

(D) Equal: An equation explicitly states that the value of the expression on one side of the equality sign is the same as the value of the expression on the other side. Thus, the two expressions are stated to be equal.

Therefore, an equation is a statement that two expressions are Equal.

The correct option is (D) Equal.

The question asks to translate the given verbal statement into an algebraic equation using the variable 'p'.

Let's break down the statement step by step:

"A number": This unknown number is represented by the variable 'p'.

"Twice a number": This means multiplying the number 'p' by 2. This can be written as $2 \times p$, or simply $2p$.

"decreased by 3": This means subtracting 3 from the previous result ($2p$). So, we get $2p - 3$. The phrase "decreased by" implies subtraction from the preceding quantity.

"is 7": This indicates that the expression on the left side is equal to 7. So, we use the equality sign and the number 7.

Combining these parts, the algebraic equation is:

$2p - 3 = 7$

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

(A) $2p + 3 = 7$: This represents "Twice a number increased by 3 is 7". This is not correct.

(B) $2p - 3 = 7$: This matches our derived equation. This represents "Twice a number decreased by 3 is 7". This is correct.

(C) $3 - 2p = 7$: This represents "3 decreased by twice a number is 7". This is not correct.

(D) $2(p - 3) = 7$: This represents "Twice the difference of a number and 3 is 7". This is not correct.

The statement "Twice a number decreased by 3 is 7" is correctly translated to the equation $2p - 3 = 7$.

The correct option is (B) $2p - 3 = 7$.

The question asks for the algebraic expression that represents the "Sum of x and y".

In mathematics, the term "sum" refers to the result of adding two or more numbers or quantities together.

To find the sum of 'x' and 'y', we need to perform the addition operation on these two variables.

The mathematical symbol for addition is '+'.

Therefore, the sum of x and y is written as $x + y$.

Let's look at the given options:

(A) $x \times y$: This represents the product of x and y (x multiplied by y).

(B) $x + y$: This represents the sum of x and y (x added to y).

(C) $x - y$: This represents the difference between x and y (x minus y).

(D) $x / y$: This represents the quotient of x and y (x divided by y).

The expression corresponding to the "Sum of x and y" is $x + y$.

The correct option is (B) $x + y$.

The question asks us to write the algebraic expression for the verbal statement "5 times the product of a and b".

Let's break down the statement:

"the product of a and b": The product of two numbers or variables is the result of multiplying them. The product of 'a' and 'b' is written as $a \times b$ or simply $ab$.

"5 times the product of a and b": This means we need to multiply the result of "the product of a and b" by 5. So, we multiply $ab$ by 5.

Multiplying $ab$ by 5 gives us $5 \times (ab)$, which is commonly written in algebraic notation as $5ab$.

Now let's compare this with the given options:

(A) $5a + b$: This represents "5 times a plus b" or "the sum of 5 times a and b".

(B) $5(a+b)$: This represents "5 times the sum of a and b".

(C) $5ab$: This represents "5 times the product of a and b". This matches our derived expression.

(D) $a \times 5b$: This also represents "a times 5 times b". Since multiplication is commutative ($a \times b = b \times a$) and associative ($(a \times b) \times c = a \times (b \times c)$), $a \times 5b$ is equivalent to $5 \times (a \times b)$ or $5ab$. However, the standard and most simplified way to write "5 times the product of a and b" is $5ab$. Option (C) is the direct translation and the most conventional form.

The algebraic expression for "5 times the product of a and b" is $5ab$.

The correct option is (C) $5ab$.

The question gives Ram's current age as 'r' years and asks for his age in 5 years.

To find someone's age after a certain number of years, we need to add that number of years to their current age.

Given:

We want to find his age after 5 years.

Age in 5 years = Current age + 5 years

Let's check the given options:

(A) $r - 5$: This would represent Ram's age 5 years ago (current age minus 5).

(B) $r + 5$: This represents Ram's current age plus 5 years, which is his age in 5 years.

(C) $5r$: This would represent 5 times Ram's current age.

(D) $r / 5$: This would represent Ram's current age divided by 5.

The expression for Ram's age in 5 years is $r + 5$.

The correct option is (B) $r + 5$.

The question provides the cost of one pen and asks for the cost of 'p' pens.

Given:

To find the total cost of 'p' pens, we multiply the cost of one pen by the number of pens.

In algebraic form, $10 \times p$ is written as $10p$.

Let's compare this result with the given options:

(A) $10 + p$: This represents the sum of the cost of one pen and the number of pens, which is incorrect.

(B) $10 - p$: This represents the difference between the cost of one pen and the number of pens, which is incorrect.

(C) $10p$: This represents 10 times the number of pens, which is the total cost.

(D) $10/p$: This represents the cost of one pen divided by the number of pens, which is incorrect.

The cost of 'p' pens is $\textsf{₹} \ 10p$.

The correct option is (C) $10p$.

The question asks to identify the constant term in the expression $4m + 7$.

An algebraic expression is composed of one or more terms. Terms are separated by addition or subtraction signs.

In the expression $4m + 7$, there are two terms:

1. $4m$

2. $7$

A constant term is a term that does not contain any variables. Its value is fixed.

Let's look at the terms in the expression $4m + 7$:

- The term $4m$ contains the variable $m$. Therefore, this is a variable term. The number $4$ in this term is called the coefficient of $m$.

- The term $7$ is a number standing alone; it does not contain any variables. Therefore, this is a constant term.

Thus, the constant term in the expression $4m + 7$ is $7$.

Comparing with the options:

(A) 4: This is the coefficient of the variable term $4m$.

(B) m: This is the variable.

(C) 7: This is the term without a variable; it is the constant term.

(D) 4m: This is the variable term.

The correct option is (C) 7.

The question asks to identify which of the given options is an equation.

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

Let's examine each option:

(A) $x + 5$: This is an algebraic expression. It consists of a variable ($x$) and a constant ($5$) connected by an addition operation. It does not contain an equality sign.

(B) $y - 3 > 0$: This statement uses an inequality sign ($>$). It states that the expression $y - 3$ is greater than 0. This is an inequation or inequality, not an equation.

(C) $2a = 8$: This statement contains an equality sign ($=$) connecting the expression $2a$ on the left side and the expression $8$ (which is a constant, and thus an expression) on the right side. This states that $2a$ is equal to $8$. This is an equation.

(D) $10 \times 2 + 5$: This is a numerical expression. It consists of numbers and operations but does not contain any variables or an equality sign. It can be evaluated to a single value ($10 \times 2 + 5 = 20 + 5 = 25$).

Based on the definition, only option (C) is an equation because it contains an equality sign connecting two expressions.

The correct option is (C) $2a = 8$.

The question asks for the formula for the perimeter of a rectangle given its length 'l' and width 'w'.

The perimeter of a two-dimensional shape is the total distance around its boundary.

A rectangle has four sides: two sides of equal length and two sides of equal width.

Let the length of the rectangle be $l$ and the width be $w$.

The four sides of the rectangle have lengths $l$, $w$, $l$, and $w$.

To find the perimeter, we add the lengths of all four sides:

We can group the like terms:

Alternatively, we can factor out 2:

Now, let's check the given options:

(A) $l \times w$: This is the formula for the area of a rectangle.

(B) $l + w$: This is the sum of the length and width, which is half of the perimeter.

(C) $2l + 2w$: This matches the formula we derived for the perimeter.

(D) $l^2 + w^2$: This expression is not the formula for the perimeter of a rectangle. It might appear in contexts like finding the diagonal length using the Pythagorean theorem, but it's not the perimeter.

The correct formula for the perimeter of a rectangle with length $l$ and width $w$ is $2l + 2w$.

The correct option is (C) $2l + 2w$.

The question asks us to find the value of the expression $3x + 2$ when the variable $x$ is equal to 4.

Given expression: $3x + 2$

Given value of the variable: $x = 4$

To find the value of the expression, we substitute the given value of $x$ into the expression.

Substitute $x = 4$ into $3x + 2$:

First, perform the multiplication:

Now, substitute this value back into the expression:

Perform the addition:

So, the value of the expression $3x + 2$ when $x = 4$ is 14.

Comparing this result with the given options:

(A) 6: Incorrect.

(B) 12: This is $3 \times 4$, but the addition of 2 is missing.

(C) 14: This is the calculated value.

(D) 24: Incorrect.

The correct option is (C) 14.

The question asks us to write the algebraic expression for the verbal statement "5 less than a number y".

Let's break down the statement:

"a number y": This is the variable, represented by $y$.

"5 less than...": This phrase means we are subtracting 5 from the quantity that follows it. In this case, we are subtracting 5 from "a number y".

So, we start with 'y' and subtract '5'. This is written as $y - 5$.

Let's compare this with the given options:

(A) $5 - y$: This represents "y less than 5" or "5 decreased by y". This is not the same as "5 less than y".

(B) $y - 5$: This represents "5 less than y" or "y decreased by 5". This matches our derived expression.

(C) $5y$: This represents "5 times y" or "the product of 5 and y".

(D) $y/5$: This represents "y divided by 5" or "the quotient of y and 5".

The algebraic expression for "5 less than a number y" is $y - 5$.

The correct option is (B) $y - 5$.

This question is a case study requiring us to find an expression for distance based on the given speed and time.

Given:

To Find:

The distance between City A and City B.

Solution:

We know the fundamental relationship between distance, speed, and time:

Substitute the given values into this formula:

When writing the product of a number and a variable in algebra, the number is usually written before the variable. So, $v \times 5$ is written as $5v$.

Now let's compare this expression with the given options:

(A) $v + 5$: This represents the sum of speed and time, which is not the distance.

(B) $v - 5$: This represents the difference between speed and time, which is not the distance.

(C) $5v$: This represents 5 times the speed, which is the correct formula for the distance travelled.

(D) $v/5$: This represents the speed divided by the time, which is not the distance.

The expression for the distance between City A and City B is $5v$ km.

The correct option is (C) $5v$.

This question refers to the case study in Question 16 and asks for the specific distance when the speed is given as 40 km/h.

From the solution to Question 16, we found the expression for the distance between City A and City B:

where $v$ is the speed of the bus in km/h and the time taken is 5 hours.

In this question, we are given a specific speed:

To find the distance, we substitute this value of $v$ into the expression (i):

Now, we perform the multiplication:

So, the distance between City A and City B is 200 km.

Let's compare our result with the given options:

(A) 45 km: Incorrect ($40 + 5$).

(B) 35 km: Incorrect ($40 - 5$).

(C) 200 km: Correct ($5 \times 40$).

(D) 8 km: Incorrect ($40 / 5$).

The correct option is (C) 200 km.

This is an Assertion-Reason type question.

Let's analyse the Assertion (A) and the Reason (R) separately.

Assertion (A): In the expression $5x$, 5 is a constant.

In the expression $5x$, $x$ is a variable because its value can change. The number 5 is multiplied by the variable $x$. The value of 5 itself does not change. A number with a fixed value is called a constant.

Therefore, Assertion (A) is True.

Reason (R): A constant has a fixed numerical value.

This is the definition of a constant in mathematics. A constant is a quantity that does not change its value. Its value is fixed.

Therefore, Reason (R) is True.

Now, let's determine if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states that 5 is a constant. Reason (R) provides the definition of a constant as having a fixed numerical value. The reason why 5 is a constant is precisely because it has a fixed numerical value (which is 5).

Thus, Reason (R) correctly explains why Assertion (A) is true.

Based on our analysis, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).

The correct option is (A) Both A and R are true and R is the correct explanation of A.

We need to match each verbal statement with its corresponding algebraic expression.

Let's translate each verbal statement into an algebraic expression:

(i) x increased by 10: "Increased by" means addition. So, this is $x + 10$.

(ii) 5 times y: "Times" means multiplication. So, this is $5 \times y$, which can also be written as $y \times 5$ or simply $5y$.

(iii) p divided by 2: "Divided by" means division. So, this is $p \div 2$, which is written as $\frac{p}{2}$ in fractional form.

(iv) q minus 7: "Minus" means subtraction. So, this is $q - 7$.

Now let's match these expressions with the given options (a), (b), (c), and (d):

(i) $x + 10$: This matches expression (c).

(ii) $5 \times y$ (or $y \times 5$ or $5y$): This matches expression (a) $y \times 5$.

(iii) $\frac{p}{2}$: This matches expression (d) $\frac{p}{2}$.

(iv) $q - 7$: This matches expression (b) $q - 7$.

So the correct matching is:

(i) - (c)

(ii) - (a)

(iii) - (d)

(iv) - (b)

Now let's check which option corresponds to this matching:

(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)

(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)

(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)

(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)

Option (A) correctly represents the matching we found.

The correct option is (A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b).

The question asks for the expression representing the total surface area of a cube with side length 'a'.

A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

All edges of a cube are of equal length, and all faces are congruent squares.

Given that the side length of the cube is 'a'.

The shape of each face of the cube is a square.

The formula for the area of a square with side length 's' is $s^2$.

Since the side length of the cube is 'a', the side length of each square face is also 'a'.

The area of one face of the cube is $a \times a = a^2$.

A cube has a total of 6 faces.

The total surface area of the cube is the sum of the areas of all its faces.

Since all 6 faces are identical and each has an area of $a^2$, the total surface area is:

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

(A) $a^2$: This is the area of a single face, not the total surface area.

(B) $6a$: This would represent 6 times the side length, which is not the surface area.

(C) $6a^2$: This is 6 times the area of one face, which is the total surface area.

(D) $a^3$: This is the formula for the volume of the cube.

The expression that represents the total surface area of a cube with side 'a' is $6a^2$.

The correct option is (C) $6a^2$.

The question asks to identify which of the given options are examples of using variables.

Recall that a variable is a symbol (usually a letter) that represents a quantity that may change or is unknown.

Let's examine each option:

(A) Formula for perimeter of a square: $4s$

In this expression, '$s$' represents the side length of the square. The side length of a square can vary from one square to another. Therefore, '$s$' is used as a variable representing the unknown or changing side length.

This is an example of using a variable.

(B) Sum of two numbers: $a + b$

In this expression, '$a$' and '$b$' represent two numbers. These numbers are general or unknown, and their values can vary. Therefore, '$a$' and '$b$' are used as variables.

This is an example of using variables.

(C) Cost of 5 apples if one apple costs $\textsf{₹} \ 10$.

To find the cost of 5 apples, we calculate $5 \times \textsf{₹} \ 10 = \textsf{₹} \ 50$. This involves a specific calculation using fixed numerical values (5 apples and $\textsf{₹} \ 10$ per apple). No symbol is used here to represent an unknown or changing quantity within this specific calculation.

This is NOT an example of using a variable in the expression or calculation shown.

(D) Distance = Speed $\times$ Time

This is a fundamental formula in physics and mathematics. "Distance", "Speed", and "Time" represent physical quantities whose values can and usually do change depending on the scenario. Although written in words here, this formula expresses a relationship between quantities that are typically represented by variables (e.g., $d = v \times t$). The statement itself is an example of a principle that requires variables for its general algebraic representation.

This is an example of using variables (or concepts represented by variables).

Based on the analysis, options (A), (B), and (D) are examples where variables are used or implied in a general relationship/formula.

The correct options are (A), (B), and (D).

The question asks us to find the value of the variable $x$ that satisfies the given equation.

Given equation:

To find the value of $x$, we need to isolate $x$ on one side of the equation. Currently, 5 is being subtracted from $x$ on the left side.

To undo the subtraction of 5, we perform the inverse operation, which is addition of 5. We must add 5 to both sides of the equation to keep it balanced.

Add 5 to both sides of equation (i):

Simplify both sides:

On the left side, $-5 + 5 = 0$, so $x - 5 + 5 = x + 0 = x$.

On the right side, $2 + 5 = 7$.

So, the equation becomes:

The value of the variable $x$ is 7.

We can verify this by substituting $x = 7$ back into the original equation (i):

This confirms that $x = 7$ is the correct solution.

Comparing our result with the given options:

(A) 3: Incorrect ($3 - 5 = -2 \neq 2$).

(B) 7: Correct ($7 - 5 = 2$).

(C) -3: Incorrect ($-3 - 5 = -8 \neq 2$).

(D) -7: Incorrect ($-7 - 5 = -12 \neq 2$).

The correct option is (B) 7.

The question asks us to write the verbal statement that corresponds to the algebraic expression $10 + k$.

The expression is $10 + k$. It involves the number 10, the variable $k$, and the addition operation ($+$).

The addition operation can be described verbally in several ways, such as "sum", "added to", "increased by", "more than", etc.

Let's examine each option to see which verbal statement matches the expression $10 + k$:

(A) 10 times k: This translates to multiplication, $10 \times k$ or $10k$. This is not $10 + k$.

(B) 10 less than k: This translates to subtracting 10 from k, which is $k - 10$. This is not $10 + k$.

(C) k added to 10: This translates to adding k to 10, which is $10 + k$. This matches the given expression.

(D) k subtracted from 10: This translates to subtracting k from 10, which is $10 - k$. This is not $10 + k$.

The verbal statement "k added to 10" correctly represents the expression $10 + k$. Note that "10 added to k", "the sum of 10 and k", or "10 increased by k" would also be correct verbal statements for $10 + k$ because addition is commutative ($10 + k = k + 10$).

The correct option is (C) k added to 10.

The question asks for an algebraic expression representing the total cost of buying 3 books and 2 pens, given the price of one book and one pen.

Given:

To Find:

Expression for the total cost of 3 books and 2 pens.

Solution:

First, let's find the cost of 3 books. The cost of one book is $\textsf{₹} \ b$. The cost of 3 books will be 3 times the cost of one book.

Next, let's find the cost of 2 pens. The cost of one pen is $\textsf{₹} \ p$. The cost of 2 pens will be 2 times the cost of one pen.

The total cost is the sum of the cost of 3 books and the cost of 2 pens.

The expression for the total cost is $3b + 2p$.

Now let's compare this expression with the given options:

(A) $b+p+3+2$: This is the sum of the prices of one book, one pen, and the numbers 3 and 2. This is incorrect.

(B) $3b + 2p$: This matches the expression we derived for the total cost.

(C) $3+b+2+p$: This is the same as option (A) due to the commutative property of addition, just rearranged. It is incorrect.

(D) $(b+p) \times (3+2)$: This represents the product of the sum of the prices of one book and one pen and the sum of the quantities 3 and 2. This is incorrect.

The expression for the total cost of 3 books and 2 pens is $3b + 2p$.

The correct option is (B) $3b + 2p$.

The question asks us to find the value of the expression $\frac{m}{6} + 3$ when the variable $m$ is equal to 18.

Given expression: $\frac{m}{6} + 3$

Given value of the variable: $m = 18$

To find the value of the expression, we substitute the given value of $m$ into the expression.

Substitute $m = 18$ into $\frac{m}{6} + 3$:

Value = $\frac{18}{6} + 3$

Now, we evaluate the expression following the order of operations (division before addition).

First, perform the division:

$\frac{18}{6} = 3$

Next, perform the addition:

Value = $3 + 3$

$3 + 3 = 6$

So, the value of the expression $\frac{m}{6} + 3$ when $m = 18$ is 6.

Comparing this result with the given options:

(A) 3: Incorrect (This is the result of the division only).

(B) 6: Correct.

(C) 9: Incorrect ($3 \times 3$ or $18 \div 2$).

(D) 21: Incorrect ($18 + 3$).

The correct option is (B) 6.

The question asks us to find the value of the variable $y$ that satisfies the given equation $y + 8 = 12$. A solution to an equation is the value of the variable that makes the equation true.

Given equation:

To find the value of $y$, we need to isolate $y$ on one side of the equation. We can do this by performing the inverse operation of addition (which is subtraction) on both sides of the equation.

Subtract 8 from both sides of equation (i):

Simplify both sides of the equation:

On the left side, $8 - 8 = 0$, so $y + 8 - 8 = y + 0 = y$.

On the right side, $12 - 8 = 4$.

So, the equation simplifies to:

The value of the variable $y$ that solves the equation is 4.

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

This confirms that $y = 4$ is the correct solution.

Comparing our result with the given options:

(A) $y = 4$: This matches our calculated solution.

(B) $y = 20$: Incorrect ($20 + 8 = 28 \neq 12$).

(C) $y = -4$: Incorrect ($-4 + 8 = 4 \neq 12$).

(D) $y = -20$: Incorrect ($-20 + 8 = -12 \neq 12$).

The correct option is (A) $y = 4$.

The question asks to complete a sentence defining how algebra helps express relationships between quantities.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols, called variables, represent quantities that are unknown or can change.

Algebraic expressions and equations are formed using these variables, along with numbers (which are constants), and mathematical operations (like addition, subtraction, multiplication, and division).

The sentence states that algebra helps express relationships between quantities using _____ and numbers.

Let's consider the options:

(A) Constants: Numbers are already mentioned in the sentence. While constants are used, the primary symbols representing potentially changing or unknown quantities are not constants themselves (other than specific numerical constants used). If the blank was "symbols", then "variables and constants" might fit, but here it needs a single key concept alongside numbers.

(B) Operations: Operations are the actions performed on quantities (variables and numbers), not the symbols representing the quantities themselves. Algebra uses operations, but they are not the fundamental building blocks *alongside* numbers and variables for representing the quantities and their relationships.

(C) Variables: Variables are symbols that represent quantities whose values can vary. Using variables allows us to express general relationships between quantities without knowing their specific numerical values. For example, the relationship between distance, speed, and time ($d = v \times t$) uses variables ($d$, $v$, $t$) and implicit numbers (coefficients like 1) and operations ($\times$, $=$).

(D) Equations: Equations are statements of equality between expressions built using variables, numbers, and operations. An equation is a specific *type* of relationship expressed, not one of the fundamental tools used *to express* relationships in general.

Algebra's power comes from using symbols (variables) to represent quantities, allowing for general formulas, equations, and expressions that describe relationships. Alongside numbers (constants), variables are the core components for representing these relationships symbolically.

The most appropriate word to complete the sentence is "variables". Algebra uses variables and numbers (constants) to express relationships between quantities.

The completed sentence is: "Algebra helps us to express relationships between quantities using variables and numbers."

The correct option is (C) Variables.

The question asks us to identify the option that is NOT an algebraic expression.

An algebraic expression is a mathematical phrase that contains one or more variables, numbers (constants), and at least one mathematical operation (addition, subtraction, multiplication, division).

A numerical expression contains only numbers and mathematical operations, but no variables.

Let's examine each option:

(A) $p/5$: This expression contains a variable ($p$), a number (5), and a mathematical operation (division). Since it contains a variable, it is an algebraic expression.

(B) $3x + 2y$: This expression contains variables ($x$ and $y$), numbers (3 and 2, which are coefficients and constants), and mathematical operations (multiplication, implicitly $3 \times x$ and $2 \times y$, and addition). Since it contains variables, it is an algebraic expression.

(C) $7 \times 8$: This expression contains only numbers (7 and 8) and a mathematical operation (multiplication). It does not contain any variables. Therefore, it is a numerical expression.

(D) $a - b$: This expression contains variables ($a$ and $b$) and a mathematical operation (subtraction). Since it contains variables, it is an algebraic expression.

The option that is NOT an algebraic expression is the one that is purely numerical.

The expression $7 \times 8$ is a numerical expression because it contains only numbers and an operation, without any variables.

The correct option is (C) $7 \times 8$.

The question asks for the total number of chairs in 10 rows, given that each row has 'c' chairs.

Given:

To Find:

Total number of chairs in 10 rows.

Solution:

To find the total number of chairs when you have multiple rows with the same number of chairs in each row, you multiply the number of chairs per row by the number of rows.

Substitute the given values:

In algebraic notation, the product of a variable and a number is usually written with the number first:

Let's compare this result with the given options:

(A) $c + 10$: This represents the sum of the chairs in one row and the number of rows. Incorrect.

(B) $c - 10$: This represents the difference between the chairs in one row and the number of rows. Incorrect.

(C) $10c$: This represents 10 times the number of chairs in one row, which is the total number of chairs in 10 rows. Correct.

(D) $c / 10$: This represents the number of chairs in one row divided by the number of rows. Incorrect.

The expression for the total number of chairs in 10 such rows is $10c$.

The correct option is (C) $10c$.

This question is a case study asking for an expression for Priya's age based on Rahul's age.

Given:

Relationship between their ages: Priya is 5 years older than Rahul.

To Find:

An expression for Priya's age.

Solution:

The phrase "5 years older than Rahul's age" means we need to add 5 years to Rahul's age.

Rahul's age is represented by the variable $r$.

Priya's age = Rahul's age + 5 years

The expression for Priya's age is $r + 5$.

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

(A) $r - 5$: This represents "5 years younger than Rahul" or "Rahul's age decreased by 5". This is incorrect.

(B) $r + 5$: This represents "Rahul's age increased by 5" or "5 years older than Rahul". This is correct.

(C) $5r$: This represents "5 times Rahul's age". This is incorrect.

(D) $r/5$: This represents "Rahul's age divided by 5". This is incorrect.

The expression for Priya's age is $r + 5$ years.

The correct option is (B) $r + 5$.

This question refers to the Case Study in Question 30, where it was established that Priya is 5 years older than her brother Rahul. If Rahul's age is 'r' years, Priya's age is $r + 5$ years.

Given:

To Find:

Priya's age.

Solution:

From the Case Study in Question 30, we know the expression for Priya's age:

Substitute the given value of Rahul's age, $r = 8$, into the expression (i):

Perform the addition:

So, Priya's age is 13 years.

Let's compare our result with the given options:

(A) 3 years: Incorrect ($8 - 5$).

(B) 13 years: Correct ($8 + 5$).

(C) 40 years: Incorrect ($5 \times 8$).

(D) 1.6 years: Incorrect ($8 / 5$).

The correct option is (B) 13 years.

The question asks us to identify which of the given quantities are variables. A variable is a quantity whose value can change.

Let's examine each option:

(A) Number of days in a week: The number of days in a standard week is always 7. This value is fixed and does not change. Therefore, this is a constant.

(B) Temperature of a city: The temperature of a city changes throughout the day, across different days, and seasons. For example, the temperature might be $15^\circ\text{C}$ in the morning and $25^\circ\text{C}$ in the afternoon. This value is not fixed. Therefore, this is a variable.

(C) Number of wheels in a bicycle: A standard bicycle has 2 wheels. This is a fixed value. While there might be variations like unicycles (1 wheel) or tricycles (3 wheels), the number of wheels in *a* bicycle (singular, typically implying a standard one) is conventionally understood as 2, which is constant. Even if considering different types, for *a* specific bicycle, the number of wheels is fixed. Therefore, this is generally considered a constant in this context.

(D) Height of a growing plant: The height of a growing plant increases over time. Its height at different points in time will be different. This value changes. Therefore, this is a variable.

The quantities that represent variables are those whose values can change: the temperature of a city and the height of a growing plant.

The correct options are (B) Temperature of a city and (D) Height of a growing plant.

The question asks for the value of the variable $x$ in the equation $2x = 12$.

Given equation:

The equation $2x = 12$ means that 2 times the value of $x$ is equal to 12.

To find the value of $x$, we need to isolate $x$ on one side of the equation. Currently, $x$ is being multiplied by 2 on the left side.

To undo the multiplication by 2, we perform the inverse operation, which is division by 2. We must divide both sides of the equation by 2 to maintain the equality.

Divide both sides of equation (i) by 2:

Simplify both sides:

On the left side, $\frac{\cancel{2}x}{\cancel{2}} = x$.

On the right side, $\frac{12}{2} = 6$.

So, the equation simplifies to:

The value of the variable $x$ is 6.

We can verify this by substituting $x = 6$ back into the original equation (i):

This confirms that $x = 6$ is the correct solution.

Comparing our result with the given options:

(A) 6: This matches our calculated value.

(B) 10: Incorrect ($2 \times 10 = 20 \neq 12$).

(C) 14: Incorrect ($2 \times 14 = 28 \neq 12$).

(D) 24: Incorrect ($2 \times 24 = 48 \neq 12$).

The value of the variable $x$ is 6.

The correct option is (A) 6.

The question asks us to write the algebraic expression that represents the verbal statement "Half of the sum of a and b".

Let's break down the statement into parts:

"the sum of a and b": The sum of two quantities is found by adding them. So, the sum of 'a' and 'b' is written as $a + b$.

"Half of the sum of a and b": This means taking the result of the sum $(a+b)$ and dividing it by 2 or multiplying it by $\frac{1}{2}$.

Dividing the sum $(a+b)$ by 2 is written as $\frac{a+b}{2}$.

Multiplying the sum $(a+b)$ by $\frac{1}{2}$ is written as $\frac{1}{2}(a+b)$ or $\frac{a+b}{2}$.

Thus, the algebraic expression for "Half of the sum of a and b" is $\frac{a+b}{2}$ or $\frac{1}{2}(a+b)$.

Now let's compare this with the given options:

(A) $\frac{1}{2}a + b$: This represents "Half of a plus b". The addition is performed after taking half of a, not after taking the sum of a and b.

(B) $a + \frac{1}{2}b$: This represents "a plus half of b". Similar to (A), the multiplication by $\frac{1}{2}$ is applied only to b, not the sum of a and b.

(C) $\frac{a+b}{2}$: This represents the sum of a and b, divided by 2. This matches our derived expression.

(D) $2(a+b)$: This represents "Twice the sum of a and b", which is the opposite of taking half of the sum.

The correct expression is $\frac{a+b}{2}$.

The correct option is (C) $\frac{a+b}{2}$.

The question asks to identify the coefficient of the variable $y$ in the expression $5y - 3$.

An algebraic expression is made up of one or more terms. In the expression $5y - 3$, the terms are $5y$ and $-3$.

A term containing a variable consists of a numerical factor and a variable part.

In the term $5y$, the variable part is $y$. The numerical factor that is multiplied by the variable is called the coefficient of that variable in that term.

In the term $5y$, the number multiplying the variable $y$ is 5.

Therefore, the coefficient of $y$ in the term $5y$ is 5.

The expression $5y - 3$ can be thought of as $5y + (-3)$.

The terms are $5y$ and $-3$.

The coefficient of the variable $y$ is the numerical part of the term containing $y$, which is 5.

The term $-3$ is a constant term.

Let's compare this with the given options:

(A) y: This is the variable itself.

(B) 5: This is the numerical factor multiplied by $y$, which is the coefficient of $y$.

(C) -3: This is the constant term in the expression.

(D) 5y: This is the entire variable term, not just the coefficient.

The coefficient of $y$ in the expression $5y - 3$ is 5.

The correct option is (B) 5.

In algebra, a variable is a symbol, typically a letter (such as $x, y, z, a, b, c$), that represents a quantity that can change or take on different values.

It is used to represent an unknown quantity, a quantity that is not fixed, or to express a general relationship between different quantities. Unlike a constant, which has a fixed value (like the number 5 or $\pi$), the value of a variable can vary depending on the context of the problem or equation.

Variables are fundamental building blocks in algebraic expressions, equations, and inequalities. For example, in the expression $x + 5$, $x$ is the variable. In the equation $2y = 10$, $y$ is the variable. In the formula for the area of a circle, $A = \pi r^2$, $A$ and $r$ are variables (representing area and radius, respectively), while $\pi$ is a constant.

In algebra, a constant is a quantity or value that is fixed and does not change. It is the opposite of a variable, which can take on different values.

Constants are typically represented by specific numbers (like $3, -7, \frac{1}{2}, \sqrt{2}, \pi$) or sometimes by letters that are understood to represent a specific, unchanging value within a particular context (like $a$ or $b$ in $ax + b = 0$, where $a$ and $b$ are coefficients and typically treated as constants for a given equation).

For example, in the expression $y = 2x + 5$, the numbers $2$ and $5$ are constants. The number $2$ is the coefficient of $x$, and $5$ is the constant term. Their values remain the same regardless of the value of the variable $x$.

Consider the algebraic expression: $3x + 7$

In this expression, $x$ is the variable. Its value is not fixed and can change. For example, if $x=2$, the expression becomes $3(2) + 7 = 6 + 7 = 13$. If $x=5$, it becomes $3(5) + 7 = 15 + 7 = 22$. The value of $x$ varies.

The numbers $3$ and $7$ are constants. Their values are fixed and do not change regardless of the value of $x$. The number $3$ is the coefficient of the variable $x$, and $7$ is the constant term.

An algebraic expression is a mathematical phrase that combines variables and constants using mathematical operations such as addition, subtraction, multiplication, and division.

Unlike an algebraic equation, an expression does not contain an equals sign ($=$) and therefore does not state a relationship between two quantities or solve for the value of a variable. It is simply a combination of terms.

Examples of algebraic expressions include:

$5x + 3$

$y^2 - 4z$

$\frac{a+b}{2}$

$m \times n$

An example of an algebraic expression is:

$2x + 5$

In this expression:

$x$ is the variable.

$2$ is a constant (specifically, the coefficient of $x$).

$5$ is a constant (the constant term).

The operation used is multiplication (between $2$ and $x$) and addition (between $2x$ and $5$).

Other examples include $y - 3$, $\frac{p}{4}$, $a^2$, $7m^2 - 3n + 1$.

The phrase '5 added to $x$' means we are adding the number 5 to the variable $x$.

The operation is addition.

The algebraic expression is:

$x + 5$

Alternatively, it can also be written as $5 + x$, as addition is commutative.

The phrase '$y$ decreased by $7$' means we are subtracting the number 7 from the variable $y$.

The operation is subtraction.

The algebraic expression is:

$y - 7$

The phrase '$3$ times $m$' means we are multiplying the number 3 by the variable $m$.

The operation is multiplication.

The algebraic expression is:

$3 \times m$

In algebraic notation, multiplication between a number and a variable is usually written without the multiplication symbol. So, the expression is commonly written as:

$3m$

The phrase 'the product of $p$ and $q$' means we are multiplying the variable $p$ by the variable $q$.

The operation is multiplication.

The algebraic expression can be written as:

$p \times q$

In standard algebraic notation, the multiplication between two variables is written by placing them next to each other. So, the expression is commonly written as:

$pq$

The phrase 'divide $10$ by $k$' means we are performing the operation of division, with 10 as the dividend and $k$ as the divisor.

The operation is division.

The algebraic expression is written as a fraction:

$\frac{10}{k}$

The phrase 'thrice of $n$' means multiplying $n$ by 3.

This gives us $3 \times n$, which is written as $3n$.

Now, we need to 'add 8 to' this result ($3n$).

The operation is addition.

Combining the parts, the algebraic expression is:

$3n + 8$

In the expression $2x + 5$, we need to identify the symbol that represents a quantity whose value can change.

The number $2$ is a constant (the coefficient of $x$). Its value is fixed.

The number $5$ is also a constant (the constant term). Its value is fixed.

The symbol $x$ is a letter used to represent a quantity that can take on different values.

Therefore, the variable in the expression $2x + 5$ is $x$.

In the expression $3y - 10$, we need to identify the numerical quantities that have a fixed value.

The symbol $y$ is the variable, as its value can change.

The number $3$ is multiplying the variable $y$. Its value is fixed.

The number $-10$ is being subtracted. Its value is fixed.

Therefore, the constants in the expression $3y - 10$ are $3$ (the coefficient of $y$) and $-10$ (the constant term).

An equation is a mathematical statement that asserts that two expressions are equal.

It is characterized by the presence of an equals sign ($=$) between the two expressions.

Equations are used to show the relationship between quantities or to find the value(s) of a variable that make the statement true.

Examples of equations include:

$x + 5 = 10$

$2y - 3 = 7$

$a^2 + b^2 = c^2$

$5 = 5$

No, $x + 3$ is not an equation.

An equation is a mathematical statement that shows two expressions are equal, and it must contain an equals sign ($=$).

$x + 3$ is an algebraic expression. It is a combination of a variable ($x$) and a constant ($3$) connected by an operation (addition), but it does not state that this expression is equal to anything else.

For example, $x + 3 = 7$ would be an equation because it has an equals sign relating the expression $x+3$ to the value $7$.

Yes, $2p = 10$ is an equation.

An equation is defined as a mathematical statement that shows that two expressions are equal.

The statement $2p = 10$ contains an equals sign ($=$), which connects the expression $2p$ on the left side to the expression $10$ on the right side, asserting their equality.

Therefore, it fits the definition of an equation.

The main difference between an algebraic expression and an algebraic equation lies in what they represent and the presence of the equals sign.

An algebraic expression is a mathematical phrase that combines variables and constants using mathematical operations (addition, subtraction, multiplication, division). It represents a single value or a quantity. It does not contain an equals sign ($=$). Examples: $3x + 5$, $y - 7$, $\frac{m}{2}$.

An algebraic equation is a mathematical statement that asserts that two expressions are equal. It always contains an equals sign ($=$) separating two expressions. Equations are used to show a relationship between quantities or to find the values of variables that make the statement true. Examples: $3x + 5 = 11$, $y - 7 = 10$, $\frac{m}{2} = 4$.

In short, an expression is a mathematical phrase, while an equation is a mathematical sentence stating equality between two expressions.

Given that the cost of one pen is $\textsf{₹}p$.

To find the cost of $5$ pens, we need to multiply the cost of one pen by the number of pens.

Cost of $5$ pens = (Cost of one pen) $\times$ (Number of pens)

Cost of $5$ pens = $\textsf{₹}p \times 5$

Writing this in standard algebraic form, the expression for the cost of $5$ pens is:

$\textsf{₹}5p$

Given that Radha's current age is $y$ years.

To find her age after $5$ years, we need to add $5$ years to her current age.

Age after $5$ years = Current age + Number of years added

Age after $5$ years = $y$ years + $5$ years

The algebraic expression for Radha's age after $5$ years is:

$y + 5$ years

Given:

Speed of the bus = $v$ km/hr

Time taken = $t$ hours

The formula relating distance, speed, and time is:

Distance = Speed $\times$ Time

Substituting the given values into the formula:

Distance = $v$ km/hr $\times$ $t$ hours

The algebraic expression for the distance covered in $t$ hours is:

$vt$ km

In algebra, a variable is a symbol, usually a letter, that represents a quantity whose value can change or vary. Think of it as a placeholder for a number that isn't fixed. In real life, things that change or vary can be represented by variables.

Real-life example of a Variable: Consider the time it takes for you to get to school each morning. This time can change depending on traffic, the weather, or if you stop to talk to a friend. So, the 'time taken to get to school' is a variable. We could represent it with a letter like $t$. Another example is the temperature outside; it changes throughout the day, so 'temperature' is a variable.

A constant, on the other hand, is a quantity or value that is fixed and does not change. It has a specific, unchanging numerical value.

Real-life example of a Constant: The number of days in a week is always 7. This number is fixed and doesn't change, so 'the number of days in a week' is a constant (the value 7). Another example is the number of wheels on a standard bicycle, which is always 2.

Difference between Variables and Constants: The key difference is that a variable's value can change, while a constant's value is fixed. Variables represent unknown or varying quantities, while constants represent specific, unchanging values.

Daily life examples of Variables:

1. The number of students present in your class on any given day (it can vary due to absence).

2. The amount of money you spend each day (it changes depending on what you buy).

Daily life examples of Constants:

1. The number of months in a year (always 12).

2. The number of centimeters in a meter (always 100).

Variables are used in geometry to write general rules or formulas that apply to any shape of a particular type, regardless of its specific size. Instead of stating a rule only for a rectangle with specific dimensions (like length 5 cm and width 3 cm), variables allow us to express the rule for a rectangle of *any* length and *any* width.

By using variables, we create a concise and universal way to calculate properties like area, perimeter, or volume for an entire class of shapes. This avoids the need to write out the calculation process in words every time or to provide separate rules for every possible size.

For example, the formula for the perimeter of a rectangle uses variables:

The perimeter ($P$) of a rectangle is found by adding the lengths of all four sides. A rectangle has two pairs of equal sides: two lengths and two widths.

The formula using variables is:

$P = 2l + 2w$

or

$P = 2(l + w)$

In this formula:

$P$ is a variable representing the Perimeter of the rectangle. Its value changes depending on the dimensions of the rectangle.

$l$ is a variable representing the Length of the rectangle. Its value can be any positive number.

$w$ is a variable representing the Width of the rectangle. Its value can be any positive number.

The number $2$ is a constant because there are always two sides of length $l$ and two sides of length $w$ in a rectangle.

An algebraic expression is a mathematical phrase that combines variables and constants using basic mathematical operations such as addition, subtraction, multiplication, and division. It does not contain an equals sign ($=$).

Forming the algebraic expressions for the given statements:

(a) The sum of $a$ and $b$.

The operation is addition. The quantities are $a$ and $b$.

Expression: $a + b$

(b) $x$ multiplied by $8$.

The operation is multiplication. The quantities are $x$ and $8$.

Expression: $8x$

(c) $12$ subtracted from $y$.

The operation is subtraction. $12$ is taken away from $y$.

Expression: $y - 12$

(d) $p$ divided by $4$.

The operation is division. $p$ is divided by $4$.

Expression: $\frac{p}{4}$

(e) Five times $m$ minus $3$.

'Five times $m$' is $5m$. 'minus $3$' means subtracting $3$.

Expression: $5m - 3$

The fundamental difference between an algebraic expression and an algebraic equation lies in their purpose and structure, specifically the presence of an equals sign.

An algebraic expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, etc.). It represents a value or quantity. It does not contain an equals sign ($=$). It's like a phrase in mathematics.

Examples of expressions:

1. $5x + 2$

2. $y^2 - 3y + 6$

An algebraic equation is a mathematical statement that asserts that two expressions are equal. It always contains an equals sign ($=$) separating two expressions. Equations are used to show a relationship between two quantities or to find the value(s) of a variable that make the statement true. It's like a complete sentence in mathematics.

Examples of equations:

1. $5x + 2 = 12$

2. $y^2 - 3y + 6 = 0$

Yes, an equation can be formed by equating an expression to a number or another expression. This is precisely how equations are constructed.

Explanation with an example:

Consider the algebraic expression $3z - 4$. This expression represents some value depending on the value of $z$.

To form an equation, we can state that this expression is equal to a specific number, say $11$.

Equating the expression to the number gives us the equation:

$3z - 4 = 11$

Here, we have an expression ($3z - 4$) on the left side and a constant (which is also a simple expression) ($11$) on the right side, connected by an equals sign. This statement asserts that the value of the expression $3z - 4$ is equal to $11$.

Alternatively, we can equate two different expressions. For example, if we have another expression like $z + 8$, we can form an equation by stating that the first expression is equal to the second expression:

$3z - 4 = z + 8$

This equation equates the expression $3z - 4$ to the expression $z + 8$. Both sides of the equals sign are expressions.

(a) Rakesh's age is given as $r$ years.

His father is $25$ years older than Rakesh, which means we add $25$ to Rakesh's age.

Father's age = Rakesh's age + $25$ years

Father's age = $r + 25$ years

The expression for his father's age is $r + 25$.

(b) The side of a square is given as $s$ cm.

The perimeter of a square is the sum of the lengths of its four sides. Since all sides of a square are equal, the perimeter is $4$ times the length of one side.

Perimeter = $4 \times$ side length

Perimeter = $4 \times s$ cm

The expression for the perimeter is $4s$ cm.

(c) The cost of one book is given as $\textsf{₹}b$.

To find the cost of $7$ such books, we multiply the cost of one book by the number of books.

Cost of $7$ books = (Cost of one book) $\times 7$

Cost of $7$ books = $\textsf{₹}b \times 7$

The expression for the cost of $7$ books is $\textsf{₹}7b$.

(d) The total number of sweets in the box is given as $k$.

If $10$ sweets are taken out, the number of sweets left is the total number minus the number taken out.

Sweets left = Total sweets - $10$ sweets

Sweets left = $k - 10$

The expression for the number of sweets left is $k - 10$.

To solve an equation means to find the value or values of the variable (or variables) in the equation that make the statement of equality true. This value is called the solution or root of the equation.

The concept of finding the value that satisfies the equation means finding the specific number that, when substituted in place of the variable, makes the left side of the equation equal to the right side. It's like finding the correct key that fits a lock.

We will use the trial and error method to solve the equation $x + 5 = 12$. The goal is to find a value for $x$ such that when we add 5 to it, the result is 12.

Trial and Error Method:

We will try different values for $x$ and check if the equation $x + 5 = 12$ holds true (i.e., if the Left-Hand Side (LHS) equals the Right-Hand Side (RHS)).

Trial 1: Let's try $x = 1$.

LHS = $x + 5 = 1 + 5 = 6$.

RHS = $12$.

Since $6 \neq 12$, $x = 1$ is not the solution.

Trial 2: Let's try $x = 5$.

LHS = $x + 5 = 5 + 5 = 10$.

RHS = $12$.

Since $10 \neq 12$, $x = 5$ is not the solution.

Trial 3: Let's try $x = 7$.

LHS = $x + 5 = 7 + 5 = 12$.

RHS = $12$.

Since $12 = 12$, LHS = RHS. This means $x = 7$ satisfies the equation.

Therefore, the solution to the equation $x + 5 = 12$ is $x = 7$.

To frame an equation, we represent the unknown quantity with a variable, identify the operations involved, and use the equals sign ($=$) to show that the result of the operations is equal to a given value.

(a) "The sum of a number $p$ and $9$ is $20$."

The sum of $p$ and $9$ is written as $p + 9$. This sum is equal to $20$.

Equation: $p + 9 = 20$

(b) "$3$ times a number $y$ is $21$."

$3$ times $y$ is written as $3 \times y$ or $3y$. This product is equal to $21$.

Equation: $3y = 21$

(c) "When $7$ is subtracted from a number $m$, the result is $15$."

$7$ subtracted from $m$ is written as $m - 7$. The result is $15$.

Equation: $m - 7 = 15$

(d) "A number $q$ divided by $4$ gives $5$."

$q$ divided by $4$ is written as $\frac{q}{4}$. This result is equal to $5$.

Equation: $\frac{q}{4} = 5$

To identify whether a mathematical statement is an equation, we look for the presence of an equals sign ($=$) which states that two expressions are equal.

(a) $2x + 3$

This is not an equation. It is an algebraic expression. It combines a variable ($x$) and constants ($2$ and $3$) using multiplication and addition. There is no equals sign asserting that this expression is equal to something else.

(b) $y - 8 = 5$

This is an equation. It contains an equals sign ($=$), which states that the expression $y - 8$ on the left side is equal to the constant $5$ on the right side. It is a statement of equality between two quantities.

(c) $5p > 10$

This is not an equation. It is an inequality. It uses the symbol $>$ (greater than) instead of an equals sign. Inequalities show that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

(d) $a + b + c$

This is not an equation. It is an algebraic expression. It combines variables ($a, b, c$) using addition. There is no equals sign asserting that this expression is equal to something else.

The algebraic expression $2x + 10$ involves a variable $x$, a constant $10$, and the operations of multiplication (by 2) and addition.

Here are two real-life situations that can be represented by the expression $2x + 10$:

Situation 1: Cost of purchasing items with a fixed fee

Suppose you are buying multiple copies of the same item and there is a fixed additional cost, such as a delivery fee.

Let $x$ represent the cost of one item (e.g., the cost of one book in $\textsf{₹}$).

Then $2x$ represents the cost of buying 2 of these items.

If there is a fixed delivery fee of $\textsf{₹}10$ for the entire order, then the total cost would be the cost of the items plus the delivery fee.

The expression $2x + 10$ represents the total cost (in $\textsf{₹}$) of buying 2 items at $\textsf{₹}x$ each plus a fixed $\textsf{₹}10$ delivery fee.

Situation 2: Relationship between ages

Suppose the age of one person is related to the age of another person.

Let $x$ represent the current age of a child (in years).

Then $2x$ represents twice the age of the child.

If an adult's age is 10 years more than twice the child's age, then the adult's age can be represented.

The expression $2x + 10$ represents the age of an adult (in years) who is 10 years older than twice the age of the child ($x$).

(a) The perimeter of a rectangle is the total length of its boundary. A rectangle has two sides of length $l$ and two sides of length $b$.

Perimeter = length + breadth + length + breadth

Perimeter = $l + b + l + b$

Perimeter = $2l + 2b$

Alternatively, Perimeter = $2 \times (\text{length} + \text{breadth})$

Perimeter = $2(l + b)$

The algebraic expression for the perimeter is $2(l + b)$ or $2l + 2b$ units.

(b) The area of a rectangle is the measure of the surface enclosed by its sides. It is calculated by multiplying the length by the breadth.

Area = length $\times$ breadth

Area = $l \times b$

The algebraic expression for the area is $lb$ square units.

(c) We are given that the perimeter of the rectangle is $30$ units and the length $l$ is $8$ units. We need to frame an equation to find the breadth $b$.

We use the formula for the perimeter: $P = 2(l + b)$.

Substitute the given values into the formula:

$30 = 2(8 + b)$

This is an equation that can be used to find the breadth $b$ of the rectangle.

The equation is $2(8 + b) = 30$.