Chapter 12 Ratio And Proportion (Additional Questions)

To find the ratio of two quantities, they must be in the same units.

We are given 15 minutes and 1 hour.

Convert 1 hour into minutes. There are 60 minutes in 1 hour.

$1 \text{ hour} = 60 \text{ minutes}$.

Now, the ratio of 15 minutes to 1 hour is the ratio of 15 minutes to 60 minutes.

Ratio = $15 : 60$

To simplify the ratio, we find the greatest common divisor (GCD) of 15 and 60, which is 15. Divide both parts of the ratio by 15.

$15 \div 15 = 1$

$60 \div 15 = 4$

The simplified ratio is $1 : 4$.

Comparing this ratio with the given options, we find that it matches option (C).

Thus, the ratio of 15 minutes to 1 hour is $1:4$.

The correct option is (C) $1:4$.

To write the ratio $36:48$ in its simplest form, we need to divide both numbers by their greatest common divisor (GCD).

We find the GCD of 36 and 48.

Factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The greatest common divisor of 36 and 48 is 12.

Now, divide both parts of the ratio by 12:

$36 \div 12 = 3$

$48 \div 12 = 4$

The simplified ratio is $3:4$.

Comparing this ratio with the given options, we see that option (A) matches our result.

The correct option is (A) $3:4$.

We are given:

Number of boys = 25

Number of girls = 15

Total number of students = 40

We need to find the ratio of girls to boys.

Ratio of girls to boys = (Number of girls) : (Number of boys)

Ratio = $15 : 25$

To write the ratio in its simplest form, we find the greatest common divisor (GCD) of 15 and 25.

The common factors of 15 and 25 are 1 and 5.

The GCD of 15 and 25 is 5.

Divide both parts of the ratio by 5:

$15 \div 5 = 3$

$25 \div 5 = 5$

The simplified ratio of girls to boys is $3:5$.

Comparing this ratio with the given options, we find that it matches option (D).

The correct option is (D) $3:5$.

Given that the cost of 5 pens is $\textsf{₹}$ 50.

To find the cost of 1 pen, we divide the total cost by the number of pens.

Cost of 1 pen = $\frac{\text{Total cost}}{\text{Number of pens}}$

Cost of 1 pen = $\frac{\textsf{₹}50}{5}$

Dividing 50 by 5:

$\frac{50}{5} = 10$

So, the cost of 1 pen is $\textsf{₹}$ 10.

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

The correct option is (B) $\textsf{₹}$ 10.

The correct option is (B).

In a proportion $a:b :: c:d$, the terms are named as follows:

First term: $a$

Second term: $b$

Third term: $c$

Fourth term: $d$

The first term ($a$) and the fourth term ($d$) are called the extremes.

The second term ($b$) and the third term ($c$) are called the means.

Therefore, the first and fourth terms of a proportion are called Extremes.

The correct option is (A).

For four numbers, say $a, b, c, d$, to be in proportion, the ratio of the first two numbers must be equal to the ratio of the last two numbers. That is, $a:b :: c:d$, which can be written as:

$\frac{a}{b} = \frac{c}{d}$

This equality is true if and only if the product of the extremes (first and fourth terms) is equal to the product of the means (second and third terms).

$a \times d = b \times c$

Given the numbers $2, 3, 4, 6$. Let $a=2$, $b=3$, $c=4$, and $d=6$.

We check if the product of the extremes equals the product of the means:

Product of extremes: $a \times d = 2 \times 6 = 12$

Product of means: $b \times c = 3 \times 4 = 12$

Since $2 \times 6 = 3 \times 4$, which is $12 = 12$, the numbers $2, 3, 4, 6$ are in proportion.

Option (A) correctly states that the numbers are in proportion because $2 \times 6 = 3 \times 4$.

The correct option is (B).

A proportion states that two ratios are equal. The given proportion is $3:5 :: x:15$. This can be written as an equation:

$\frac{3}{5} = \frac{x}{15}$

To solve for $x$, we can use the property that the product of the extremes is equal to the product of the means. In this proportion, the extremes are $3$ and $15$, and the means are $5$ and $x$.

Product of extremes $= 3 \times 15 = 45$

Product of means $= 5 \times x = 5x$

Equating the product of extremes and means:

Now, divide both sides of the equation by 5 to find the value of $x$:

Thus, the missing term in the proportion is 9.

The correct option is (B).

Given:

Cost of 10 kg of rice = $\textsf{₹}$ 400

To Find:

Cost of 15 kg of rice

Solution (Using Unitary Method):

The unitary method involves first finding the value of a single unit (in this case, the cost of 1 kg of rice) and then using that value to find the value of the required number of units (the cost of 15 kg of rice).

We are given the cost of 10 kg of rice is $\textsf{₹}$ 400.

Cost of 10 kg rice = $\textsf{₹}$ 400

To find the cost of 1 kg of rice, we divide the total cost by the quantity (10 kg):

Cost of 1 kg rice $= \frac{\textsf{₹} 400}{10}$

Cost of 1 kg rice $= \textsf{₹} 40$

Now, we need to find the cost of 15 kg of rice. We multiply the cost of 1 kg by the required quantity (15 kg):

Cost of 15 kg rice = Cost of 1 kg rice $\times$ 15

Cost of 15 kg rice = $\textsf{₹} 40 \times 15$

Cost of 15 kg rice = $\textsf{₹} 600$

Thus, the cost of 15 kg of rice is $\textsf{₹}$ 600.

Alternate Solution (Using Proportion):

Let the cost of 15 kg of rice be $\textsf{₹} x$. The relationship between the quantity of rice and its cost is a direct proportion. This means that if the quantity of rice increases, the cost also increases proportionally. We can set up a proportion:

Quantity of rice (kg) : Cost ($\textsf{₹}$) :: Quantity of rice (kg) : Cost ($\textsf{₹}$)

$10 : 400 :: 15 : x$

Using the property that the product of the extremes equals the product of the means:

Product of extremes $= 10 \times x = 10x$

Product of means $= 400 \times 15 = 6000$

Equating the products:

Divide both sides by 10:

So, the cost of 15 kg of rice is $\textsf{₹} 600$.

The correct option is (C).

To find the ratio of two quantities, they must be in the same unit.

We are asked to find the ratio of 50 cm to 2 metres.

First, convert 2 metres to centimetres. We know that 1 metre = 100 centimetres.

So, 2 metres $= 2 \times 100$ cm $= 200$ cm.

Now, the ratio of 50 cm to 2 metres is the ratio of 50 cm to 200 cm.

Ratio $= \frac{50 \text{ cm}}{200 \text{ cm}}$

To simplify the ratio, we divide both the numerator and the denominator by their greatest common divisor, which is 50.

$\frac{50}{200} = \frac{50 \div 50}{200 \div 50} = \frac{1}{4}$

The ratio is $1:4$.

Therefore, the ratio of 50 cm to 2 metres is $1:4$.

The correct option is (A).

Two ratios are equivalent if their values are the same when simplified to their lowest terms.

The given ratio is $2:5$, which can be written as a fraction $\frac{2}{5}$. This fraction is already in its simplest form because the greatest common divisor of 2 and 5 is 1.

Now, let's examine each option:

(A) The ratio is $4:10$. As a fraction, this is $\frac{4}{10}$.

To simplify $\frac{4}{10}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{\cancel{4}^{2}}{\cancel{10}_{5}} = \frac{2}{5}$

Since $\frac{4}{10}$ simplifies to $\frac{2}{5}$, the ratio $4:10$ is equivalent to $2:5$.

(B) The ratio is $10:4$. As a fraction, this is $\frac{10}{4}$.

To simplify $\frac{10}{4}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{\cancel{10}^{5}}{\cancel{4}_{2}} = \frac{5}{2}$

The ratio $10:4$ is equivalent to $5:2$, which is not equivalent to $2:5$.

(C) The ratio is $6:12$. As a fraction, this is $\frac{6}{12}$.

To simplify $\frac{6}{12}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 6.

$\frac{\cancel{6}^{1}}{\cancel{12}_{2}} = \frac{1}{2}$

The ratio $6:12$ is equivalent to $1:2$, which is not equivalent to $2:5$.

(D) The ratio is $8:15$. As a fraction, this is $\frac{8}{15}$.

The greatest common divisor of 8 and 15 is 1, so the fraction $\frac{8}{15}$ is already in its simplest form.

The ratio $8:15$ is not equivalent to $2:5$.

Therefore, the only ratio equivalent to $2:5$ is $4:10$.

The correct option is (B).

A proportion is an equality between two ratios. If four quantities $a, b, c, d$ are in proportion, we write this as $a:b :: c:d$.

This means that the ratio of the first two quantities is equal to the ratio of the last two quantities:

$\frac{a}{b} = \frac{c}{d}$

In the proportion $a:b :: c:d$:

The extremes are the first term ($a$) and the fourth term ($d$).

The means are the second term ($b$) and the third term ($c$).

The fundamental property of a proportion states that the product of the extremes is equal to the product of the means.

Therefore, in a proportion, the product of the extremes is equal to the product of the Means.

The correct option is (B).

A proportion is a statement that two ratios are equal. If four quantities $a, b, c, d$ are in proportion, it is written as $a:b :: c:d$. This is equivalent to the equation:

$\frac{a}{b} = \frac{c}{d}$

In the proportion $a:b :: c:d$:

The first term ($a$) and the fourth term ($d$) are called the extremes.

The second term ($b$) and the third term ($c$) are called the means.

The property derived from the equality $\frac{a}{b} = \frac{c}{d}$ by cross-multiplication is $a \times d = b \times c$.

This property, which states that the product of the extremes is equal to the product of the means, is formally known as the Property of extremes and means.

The correct option is (C).

Given:

Total amount to be divided = $\textsf{₹}$ 60

Ratio of division between Rahul and Deepak = $2:3$

To Find:

Deepak's share

Solution:

The given ratio is $2:3$. The sum of the parts of the ratio represents the total number of equal parts into which the amount is divided.

Sum of the ratio parts $= 2 + 3 = 5$

This means the total amount $\textsf{₹}$ 60 is divided into 5 equal parts.

Value of one ratio part $= \frac{\text{Total amount}}{\text{Sum of ratio parts}}$

Rahul's share corresponds to 2 parts of the ratio.

Rahul's share $= 2 \times \text{Value of one ratio part}$

Rahul's share $= 2 \times \textsf{₹} 12 = \textsf{₹} 24$

Deepak's share corresponds to 3 parts of the ratio.

Deepak's share $= 3 \times \text{Value of one ratio part}$

Deepak's share $= 3 \times \textsf{₹} 12 = \textsf{₹} 36$

To verify, the sum of their shares should be equal to the total amount:

Rahul's share + Deepak's share $= \textsf{₹} 24 + \textsf{₹} 36 = \textsf{₹} 60$

This matches the total amount given.

Therefore, Deepak gets $\textsf{₹}$ 36.

The correct option is (B).

Given:

Distance traveled = 120 km

Time taken = 3 hours

To Find:

Time taken to travel 200 km at the same speed.

Solution (Using Unitary Method):

First, we find the speed of the car. Speed is calculated as distance divided by time.

Now that we know the speed is 40 km/hour, we can find the time taken to travel 200 km using the formula: Time = Distance / Speed.

Thus, it will take 5 hours to travel 200 km at the same speed.

Alternate Solution (Using Proportion):

Since the speed is constant, the distance traveled is directly proportional to the time taken. We can set up a proportion relating the two situations.

Let $t$ be the time (in hours) taken to travel 200 km.

The proportion is: Distance$_1$ : Time$_1$ :: Distance$_2$ : Time$_2$

$120 \text{ km} : 3 \text{ hours} :: 200 \text{ km} : t \text{ hours}$

This can be written as an equation:

Using the cross-multiplication property (Product of extremes = Product of means):

Now, solve for $t$ by dividing both sides by 120:

So, the time taken is 5 hours.

The correct option is (B).

To check which pair of numbers is in the ratio $1:2$, we need to find the ratio of the numbers in each pair and simplify it to its lowest terms. The ratio $1:2$ can be represented as the fraction $\frac{1}{2}$.

Let's check each option:

(A) The pair is 10, 12. The ratio is $10:12$.

As a fraction, this is $\frac{10}{12}$. Simplifying by dividing the numerator and denominator by their greatest common divisor (GCD), which is 2:

$\frac{\cancel{10}^{5}}{\cancel{12}_{6}} = \frac{5}{6}$

The ratio $10:12$ simplifies to $5:6$. This is not equal to $1:2$.

(B) The pair is 15, 30. The ratio is $15:30$.

As a fraction, this is $\frac{15}{30}$. Simplifying by dividing the numerator and denominator by their GCD, which is 15:

$\frac{\cancel{15}^{1}}{\cancel{30}_{2}} = \frac{1}{2}$

The ratio $15:30$ simplifies to $1:2$. This matches the required ratio.

(C) The pair is 20, 25. The ratio is $20:25$.

As a fraction, this is $\frac{20}{25}$. Simplifying by dividing the numerator and denominator by their GCD, which is 5:

$\frac{\cancel{20}^{4}}{\cancel{25}_{5}} = \frac{4}{5}$

The ratio $20:25$ simplifies to $4:5$. This is not equal to $1:2$.

(D) The pair is 5, 15. The ratio is $5:15$.

As a fraction, this is $\frac{5}{15}$. Simplifying by dividing the numerator and denominator by their GCD, which is 5:

$\frac{\cancel{5}^{1}}{\cancel{15}_{3}} = \frac{1}{3}$

The ratio $5:15$ simplifies to $1:3$. This is not equal to $1:2$.

Therefore, the pair of numbers that are in the ratio $1:2$ is 15, 30.

The equivalent ratios to $3:7$ are (A) $6:14$, (B) $9:21$, and (C) $12:28$.

Two ratios are equivalent if they have the same value when simplified to their lowest terms. The given ratio is $3:7$, which can be written as the fraction $\frac{3}{7}$. This fraction is already in its simplest form.

Let's check each option by simplifying the given ratio to its lowest terms:

(A) Ratio is $6:14$. As a fraction, this is $\frac{6}{14}$.

Simplify by dividing the numerator and denominator by their greatest common divisor (GCD), which is 2:

$\frac{\cancel{6}^{3}}{\cancel{14}_{7}} = \frac{3}{7}$

Since $\frac{6}{14}$ simplifies to $\frac{3}{7}$, the ratio $6:14$ is equivalent to $3:7$.

(B) Ratio is $9:21$. As a fraction, this is $\frac{9}{21}$.

Simplify by dividing the numerator and denominator by their GCD, which is 3:

$\frac{\cancel{9}^{3}}{\cancel{21}_{7}} = \frac{3}{7}$

Since $\frac{9}{21}$ simplifies to $\frac{3}{7}$, the ratio $9:21$ is equivalent to $3:7$.

(C) Ratio is $12:28$. As a fraction, this is $\frac{12}{28}$.

Simplify by dividing the numerator and denominator by their GCD, which is 4:

$\frac{\cancel{12}^{3}}{\cancel{28}_{7}} = \frac{3}{7}$

Since $\frac{12}{28}$ simplifies to $\frac{3}{7}$, the ratio $12:28$ is equivalent to $3:7$.

(D) Ratio is $15:30$. As a fraction, this is $\frac{15}{30}$.

Simplify by dividing the numerator and denominator by their GCD, which is 15:

$\frac{\cancel{15}^{1}}{\cancel{30}_{2}} = \frac{1}{2}$

The ratio $15:30$ simplifies to $1:2$, which is not equivalent to $3:7$.

Therefore, the ratios equivalent to $3:7$ are $6:14$, $9:21$, and $12:28$.

The correct option is (D).

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

Reason (R):

The Reason states: "To find a ratio, the quantities must be in the same unit, and 1 metre = 100 cm."

This statement is accurate. It is a fundamental rule that quantities must have the same unit before their ratio can be calculated. Also, the conversion 1 metre = 100 cm is correct.

Therefore, Reason (R) is True.

Assertion (A):

The Assertion states: "The ratio of 1 metre to 100 cm is $1:100$."

To find the ratio of 1 metre to 100 cm, we must first convert them to the same unit. Using the conversion from Reason (R):

1 metre = 100 cm

Now, we find the ratio of 1 metre to 100 cm, which is the ratio of 100 cm to 100 cm.

Ratio $= \frac{100 \text{ cm}}{100 \text{ cm}}$

Simplifying this fraction:

$\frac{\cancel{100}^{1}}{\cancel{100}_{1}} = \frac{1}{1}$

The ratio is $1:1$.

The Assertion claims the ratio is $1:100$, which is incorrect.

Therefore, Assertion (A) is False.

Based on our analysis, Assertion (A) is false, and Reason (R) is true.

Comparing this with the given options:

(A) Both A and R are true (False)

(B) Both A and R are true (False)

(C) A is true but R is false (False)

(D) A is false but R is true (True)

The correct option is (D).

The correct option is (C).

Given:

Ratio of sugar to flour = $1:3$

Amount of sugar used = 2 cups

To Find:

Amount of flour to be used.

Solution:

The ratio of sugar to flour is $1:3$. This means for every 1 part of sugar, 3 parts of flour are needed.

We are using 2 cups of sugar.

Let the amount of flour needed be $x$ cups.

We can set up a proportion based on the given ratio:

$\frac{\text{Sugar}}{\text{Flour}} = \frac{1}{3}$

Using the amounts in the recipe:

Now, we can solve for $x$ using cross-multiplication (product of extremes equals product of means):

Therefore, you should use 6 cups of flour.

Alternate Solution (Using Unitary Method):

The ratio of sugar to flour is $1:3$. This means the amount of flour is 3 times the amount of sugar.

Given that 2 cups of sugar are used:

So, you should use 6 cups of flour.

The correct option is (B).

Given:

Ratio of sugar to flour = $1:3$

Total amount of sugar and flour = 8 cups

To Find:

Amount of sugar used.

Solution:

The ratio of sugar to flour is $1:3$. This means that for every 1 part of sugar, there are 3 parts of flour in the mixture.

The total number of parts in the mixture is the sum of the ratio parts:

Sum of ratio parts $= 1 + 3 = 4$

The total amount of 8 cups is divided into 4 equal parts.

To find the quantity represented by one ratio part, we divide the total amount by the sum of the ratio parts:

The ratio of sugar is 1 part.

Amount of sugar used $= 1 \times \text{Quantity per ratio part}$

Amount of sugar used $= 1 \times 2 \text{ cups} = 2 \text{ cups}$

We can also find the amount of flour: The ratio of flour is 3 parts.

Amount of flour used $= 3 \times \text{Quantity per ratio part}$

Amount of flour used $= 3 \times 2 \text{ cups} = 6 \text{ cups}$

Check: Total amount = Sugar + Flour = 2 cups + 6 cups = 8 cups. This matches the given total amount.

Therefore, 2 cups of sugar were used.

The correct option is (A).

Let's analyze each ratio problem and identify the most appropriate method for solving it:

(i) Find the cost of 10 kg of apples if 4 kg cost $\textsf{₹}$ 400

This problem can be solved by first finding the cost of 1 kg of apples ($\frac{\textsf{₹} 400}{4} = \textsf{₹} 100$) and then multiplying that cost by 10 to find the cost of 10 kg ($\textsf{₹} 100 \times 10 = \textsf{₹} 1000$). This is the **Unitary Method**. It corresponds to option (b) Unitary Method.

(ii) Check if $2:3$ and $10:15$ are equivalent

To check if two ratios are equivalent, we can either simplify both ratios to their lowest terms ( $2:3$ is already in lowest terms; $10:15 = \frac{\cancel{10}^{2}}{\cancel{15}_{3}} = 2:3$ ) or use cross-multiplication on the corresponding fractions ($\frac{2}{3}$ and $\frac{10}{15}$, checking if $2 \times 15 = 3 \times 10$, which is $30=30$). These methods fall under **Ratio simplification or cross-multiplication**. It corresponds to option (c) Ratio simplification or cross-multiplication.

(iii) Divide $\textsf{₹}$ 100 in the ratio $1:4$

This involves distributing a total quantity based on a given ratio. The method is to find the sum of the ratio parts ($1+4=5$), determine the value of one part ($\frac{\textsf{₹} 100}{5} = \textsf{₹} 20$), and then calculate the shares for each part ($1 \times \textsf{₹} 20 = \textsf{₹} 20$ and $4 \times \textsf{₹} 20 = \textsf{₹} 80$). This is the method of **Dividing a quantity in a given ratio**. It corresponds to option (d) Dividing a quantity in a given ratio.

(iv) Find the missing term in $5:10 :: x:20$

This is a problem involving a proportion. We can write the proportion as $\frac{5}{10} = \frac{x}{20}$ and solve for $x$ using the property that the product of the extremes equals the product of the means ($5 \times 20 = 10 \times x$). This method is solving a **Proportion**. It corresponds to option (a) Proportion.

Based on the analysis, the correct matching is:

(i) - (b)

(ii) - (c)

(iii) - (d)

(iv) - (a)

This corresponds to option (A).

The correct option is (B).

A ratio is a comparison of two quantities of the same kind and in the same unit. Quantities of different kinds cannot be compared in a ratio.

Let's consider each option:

(A) Ratio of heights of two students: Height is a measure of length. Comparing the heights of two students (both quantities of length) is a valid ratio comparison, assuming the units are the same (e.g., cm to cm, or inches to inches).

(B) Ratio of weight of a box to the length of a road: Weight is a measure of mass, while length is a measure of distance. Mass and length are quantities of different kinds. You cannot form a ratio between weight and length.

(C) Ratio of speeds of two cars: Speed is a rate (distance per unit time). Comparing the speeds of two cars (both quantities of speed) is a valid ratio comparison, assuming the units are the same (e.g., km/h to km/h, or mph to mph).

(D) Ratio of money in two wallets: Money is a measure of value. Comparing the amount of money in two wallets (both quantities of the same currency value) is a valid ratio comparison, assuming the units are the same (e.g., Rupees to Rupees, or Dollars to Dollars).

Based on the principle that a ratio compares quantities of the same kind, the ratio of the weight of a box to the length of a road is not a valid ratio comparison because weight and length are different types of quantities.

The correct option is (B).

Given:

Ratio of teachers to students = $1:20$

Number of teachers = 10

To Find:

Number of students

Solution:

The ratio $1:20$ means that for every 1 teacher, there are 20 students.

We are given that there are 10 teachers. Comparing this to the ratio, the number of teachers is 10 times the 'teacher' part of the ratio (since $1 \times 10 = 10$).

Since the ratio must be maintained, the number of students must also be 10 times the 'student' part of the ratio.

Number of students $= 20 \times 10$

$= 200$

Thus, there are 200 students in the school.

The correct option is (B).

Given:

The proportion is $4:12 :: \text{_____}:9$.

To Find:

The missing term in the proportion.

Solution:

Let the missing term be $x$. The proportion can be written as $4:12 :: x:9$.

In a proportion, the product of the extremes is equal to the product of the means.

The extremes are the first and fourth terms (4 and 9).

The means are the second and third terms (12 and $x$).

Product of extremes $= 4 \times 9 = 36$

Product of means $= 12 \times x = 12x$

Equating the product of extremes and means:

To find $x$, divide both sides of the equation by 12:

Simplify the fraction:

The missing term is 3.

The complete proportion is $4:12 :: 3:9$.

We can check this by simplifying the ratios: $\frac{4}{12} = \frac{1}{3}$ and $\frac{3}{9} = \frac{1}{3}$. Since the ratios are equal, the proportion is valid.

The correct option is (C).

Given:

Cost of 6 books = $\textsf{₹}$ 360

To Find:

Cost of 8 books

Solution (Using Unitary Method):

First, we need to find the cost of a single book. We divide the total cost by the number of books.

Now that we know the cost of 1 book is $\textsf{₹}$ 60, we can find the cost of 8 books by multiplying the cost of one book by 8.

Thus, the cost of 8 books is $\textsf{₹}$ 480.

Alternate Solution (Using Proportion):

The cost of books is directly proportional to the number of books. We can set up a proportion.

Let the cost of 8 books be $\textsf{₹} x$.

Number of books : Cost :: Number of books : Cost

$6 : 360 :: 8 : x$

Using the property that the product of the extremes equals the product of the means:

Equating the products:

Now, solve for $x$ by dividing both sides by 6:

So, the cost of 8 books is $\textsf{₹} 480$.

The correct option is (B).

We are asked to find the ratio of the number of sides of a square to the number of sides of a triangle.

A square is a quadrilateral with four equal sides and four right angles. Therefore, the number of sides of a square is 4.

A triangle is a polygon with three sides. Therefore, the number of sides of a triangle is 3.

The ratio of the number of sides of a square to the number of sides of a triangle is the ratio of 4 to 3.

Ratio = Number of sides of square : Number of sides of triangle

Ratio = $4 : 3$

The numbers 4 and 3 have no common factors other than 1, so the ratio $4:3$ is in its simplest form.

Thus, the ratio of the number of sides of a square to the number of sides of a triangle is $4:3$.

The correct option is (B).

Given:

Earning in 5 months = $\textsf{₹}$ 15000

Earning is uniform (constant per month).

To Find:

Earning in 7 months.

Solution (Using Unitary Method):

First, we find the earning per month. We divide the total earning by the number of months.

Now that we know the earning per month is $\textsf{₹}$ 3000, we can find the earning in 7 months by multiplying the earning per month by 7.

Thus, the man will earn $\textsf{₹}$ 21000 in 7 months.

Alternate Solution (Using Proportion):

Assuming uniform earning, the earning is directly proportional to the number of months. We can set up a proportion.

Let the earning in 7 months be $\textsf{₹} x$.

Months : Earning :: Months : Earning

$5 \text{ months} : \textsf{₹} 15000 :: 7 \text{ months} : \textsf{₹} x$

This can be written as an equation:

Using the cross-multiplication property (Product of extremes = Product of means):

Now, solve for $x$ by dividing both sides by 5:

So, the earning in 7 months is $\textsf{₹} 21000$.

The sets of numbers in proportion are (A) $1, 2, 3, 6$, (B) $4, 8, 5, 10$, and (D) $10, 20, 15, 30$.

Four numbers $a, b, c, d$ are in proportion if the ratio of the first two numbers is equal to the ratio of the last two numbers, i.e., $a:b :: c:d$. This is equivalent to the condition that the product of the extremes ($a \times d$) is equal to the product of the means ($b \times c$). We will check this condition for each set of numbers.

(A) Numbers: 1, 2, 3, 6

Here, $a=1$, $b=2$, $c=3$, $d=6$.

Product of extremes $= a \times d = 1 \times 6 = 6$

Product of means $= b \times c = 2 \times 3 = 6$

Since $1 \times 6 = 2 \times 3$ ($6=6$), the numbers $1, 2, 3, 6$ are in proportion ($1:2 :: 3:6$ or $\frac{1}{2} = \frac{3}{6}$).

(B) Numbers: 4, 8, 5, 10

Here, $a=4$, $b=8$, $c=5$, $d=10$.

Product of extremes $= a \times d = 4 \times 10 = 40$

Product of means $= b \times c = 8 \times 5 = 40$

Since $4 \times 10 = 8 \times 5$ ($40=40$), the numbers $4, 8, 5, 10$ are in proportion ($4:8 :: 5:10$ or $\frac{4}{8} = \frac{5}{10}$).

(C) Numbers: 3, 6, 9, 12

Here, $a=3$, $b=6$, $c=9$, $d=12$.

Product of extremes $= a \times d = 3 \times 12 = 36$

Product of means $= b \times c = 6 \times 9 = 54$

Since $3 \times 12 \neq 6 \times 9$ ($36 \neq 54$), the numbers $3, 6, 9, 12$ are NOT in proportion.

(D) Numbers: 10, 20, 15, 30

Here, $a=10$, $b=20$, $c=15$, $d=30$.

Product of extremes $= a \times d = 10 \times 30 = 300$

Product of means $= b \times c = 20 \times 15 = 300$

Since $10 \times 30 = 20 \times 15$ ($300=300$), the numbers $10, 20, 15, 30$ are in proportion ($10:20 :: 15:30$ or $\frac{10}{20} = \frac{15}{30}$).

Therefore, the sets of numbers that are in proportion are (A), (B), and (D).

The correct option is (A).

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

Assertion (A): The unitary method involves first finding the value of a single unit.

This statement accurately describes the initial step in the unitary method. For example, if you know the cost of 5 pens, the first step using the unitary method is to find the cost of 1 pen.

Therefore, Assertion (A) is True.

Reason (R): Once the value of a single unit is known, the value of any required number of units can be found.

This statement explains why finding the value of a single unit is useful. If you know the cost of 1 pen (say $\textsf{₹}$ 10), you can easily find the cost of any number of pens (e.g., the cost of 7 pens would be $7 \times \textsf{₹} 10 = \textsf{₹} 70$). This step directly follows from knowing the value of the single unit and is the purpose of finding it.

Therefore, Reason (R) is True and it is the correct explanation for Assertion (A).

Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains why Assertion (A) is the first step of the unitary method.

This matches option (A).

The correct option is (D).

Given:

Quantity 1: 1 litre

Quantity 2: 500 ml

To Find:

The ratio of 1 litre to 500 ml.

Solution:

To find the ratio of two quantities, they must be expressed in the same unit.

We know the conversion between litres and millilitres:

$1 \text{ litre} = 1000 \text{ ml}$

Now, we can express both quantities in millilitres:

Quantity 1 = 1 litre = 1000 ml

Quantity 2 = 500 ml

The ratio of 1 litre to 500 ml is the ratio of 1000 ml to 500 ml.

To simplify the ratio, we divide both the numerator and the denominator by their greatest common divisor, which is 500.

The ratio is $2:1$.

Therefore, the ratio of 1 litre to 500 ml is $2:1$.

The correct option is (C).

Given:

Ratio of stamps collected by Alia to Kabir = $4:5$

Number of stamps collected by Alia = 24

To Find:

Number of stamps collected by Kabir.

Solution (Using Unitary Method):

The ratio $4:5$ means that for every 4 stamps Alia collected, Kabir collected 5 stamps.

The number of stamps collected by Alia corresponds to the first part of the ratio, which is 4.

So, 4 parts of the ratio are equal to 24 stamps.

To find the number of stamps represented by one ratio part, we divide Alia's stamps by her ratio part:

Kabir's stamps correspond to the second part of the ratio, which is 5. To find the number of stamps Kabir collected, we multiply his ratio part by the number of stamps per ratio part.

Thus, Kabir collected 30 stamps.

Alternate Solution (Using Proportion):

Let the number of stamps collected by Kabir be $x$. The ratio of Alia's stamps to Kabir's stamps is proportional to the ratio of their shares in the given ratio.

Alia's stamps : Kabir's stamps :: Alia's ratio part : Kabir's ratio part

$24 : x :: 4 : 5$

This can be written as a proportion:

Using the cross-multiplication property (Product of extremes = Product of means), we get:

Now, solve for $x$ by dividing both sides by 4:

So, Kabir collected 30 stamps.

The correct option is (C).

Given:

Ratio of stamps collected by Alia to Kabir = $4:5$

Number of stamps collected by Alia = 24

To Find:

Total number of stamps collected by Alia and Kabir together.

Solution:

The ratio of stamps collected by Alia to Kabir is $4:5$. This means Alia collected 4 parts of stamps, and Kabir collected 5 parts of stamps, for a total of $4 + 5 = 9$ parts.

We are given that Alia collected 24 stamps, which corresponds to the 4 parts in the ratio for Alia.

So, 4 parts $= 24$ stamps.

To find the number of stamps in one ratio part, we divide Alia's stamps by her ratio part:

Now we can find the number of stamps collected by Kabir, whose ratio part is 5:

The total number of stamps collected by them together is the sum of stamps collected by Alia and Kabir.

Alternatively, we know the total number of ratio parts is 9 and each part is worth 6 stamps. So, the total number of stamps is:

Thus, the total number of stamps they collected together is 54.

The correct option is (B).

Given:

Cost of 5 notebooks = $\textsf{₹}$ 100

Amount of money available = $\textsf{₹}$ 240

To Find:

Number of notebooks that can be bought for $\textsf{₹}$ 240.

Solution (Using Unitary Method):

First, we find the cost of a single notebook. We divide the total cost by the number of notebooks.

Now that we know the cost of 1 notebook is $\textsf{₹}$ 20, we can find the number of notebooks that can be bought for $\textsf{₹}$ 240. We divide the total amount of money by the cost of one notebook.

Thus, you can buy 12 notebooks for $\textsf{₹}$ 240.

Alternate Solution (Using Proportion):

The number of notebooks is directly proportional to the cost. We can set up a proportion.

Let the number of notebooks that can be bought for $\textsf{₹}$ 240 be $x$.

Number of notebooks : Cost :: Number of notebooks : Cost

$5 \text{ notebooks} : \textsf{₹} 100 :: x \text{ notebooks} : \textsf{₹} 240$

This can be written as an equation:

Using the cross-multiplication property (Product of extremes = Product of means):

Now, solve for $x$ by dividing both sides by 100:

So, you can buy 12 notebooks for $\textsf{₹}$ 240.

Number of notebooks = 12

The correct option is (D).

A proportion is an equality between two ratios. Four numbers $a, b, c, d$ are in proportion, written as $a:b :: c:d$, if the ratio $\frac{a}{b}$ is equal to the ratio $\frac{c}{d}$. This is equivalent to the property that the product of the extremes ($a$ and $d$) is equal to the product of the means ($b$ and $c$), i.e., $a \times d = b \times c$.

We will check this property for each given option to determine which one is NOT a proportion.

(A) $1:2 :: 4:8$

Check if the product of extremes equals the product of means:

Extremes: 1 and 8

Means: 2 and 4

$1 \times 8 = 8$

$2 \times 4 = 8$

Since $1 \times 8 = 2 \times 4$ ($8=8$), this is a proportion.

(B) $3:4 :: 6:8$

Check if the product of extremes equals the product of means:

Extremes: 3 and 8

Means: 4 and 6

$3 \times 8 = 24$

$4 \times 6 = 24$

Since $3 \times 8 = 4 \times 6$ ($24=24$), this is a proportion.

(C) $2:5 :: 4:10$

Check if the product of extremes equals the product of means:

Extremes: 2 and 10

Means: 5 and 4

$2 \times 10 = 20$

$5 \times 4 = 20$

Since $2 \times 10 = 5 \times 4$ ($20=20$), this is a proportion.

(D) $1:3 :: 2:5$

Check if the product of extremes equals the product of means:

Extremes: 1 and 5

Means: 3 and 2

$1 \times 5 = 5$

$3 \times 2 = 6$

Since $1 \times 5 \neq 3 \times 2$ ($5 \neq 6$), this is NOT a proportion.

Therefore, the set of numbers that is NOT a proportion is $1:3 :: 2:5$.

The correct option is (A).

Given Word: INDIA

To Find:

The ratio of the number of vowels to the number of consonants in the word 'INDIA'.

Solution:

First, let's identify the letters in the word 'INDIA': I, N, D, I, A.

Next, we identify the vowels and consonants among these letters.

Vowels: The vowels are A, E, I, O, U.

In the word 'INDIA', the vowels are I, I, A.

Number of vowels = 3

Consonants: The consonants are all letters that are not vowels.

In the word 'INDIA', the consonants are N, D.

Number of consonants = 2

We need to find the ratio of the number of vowels to the number of consonants.

Ratio = Number of vowels : Number of consonants

Ratio = $3 : 2$

The ratio $3:2$ is already in its simplest form as there are no common factors for 3 and 2 other than 1.

Therefore, the ratio of the number of vowels to the number of consonants in the word 'INDIA' is $3:2$.

The correct option is (C).

Given:

Ratio of blue to red marbles = $3:5$

Number of blue marbles = 15

To Find:

Number of red marbles.

Solution:

The given ratio of blue marbles to red marbles is $3:5$. This means that for every 3 blue marbles, there are 5 red marbles.

We are given that the number of blue marbles is 15.

The 'blue' part of the ratio is 3, which corresponds to 15 blue marbles.

So, 3 parts of the ratio represent 15 marbles.

To find the number of marbles represented by one ratio part, we divide the number of blue marbles by their corresponding ratio part:

The 'red' part of the ratio is 5. To find the number of red marbles, we multiply the number of marbles per ratio part by the red ratio part:

Thus, there are 25 red marbles.

Alternate Solution (Using Proportion):

Let the number of red marbles be $x$. The ratio of blue marbles to red marbles is $3:5$. We can set up a proportion comparing the ratio parts to the actual numbers of marbles:

Blue ratio part : Red ratio part :: Number of blue marbles : Number of red marbles

$3 : 5 :: 15 : x$

This proportion can be written as an equation:

Using the cross-multiplication property (Product of extremes = Product of means):

Now, solve for $x$ by dividing both sides by 3:

So, the number of red marbles is 25.

A ratio is a comparison of two quantities of the same kind, expressed as a fraction or using a colon.

A ratio between two quantities, say $a$ and $b$ (where $b \neq 0$), is represented in the following ways:

1. Using a colon symbol (:): It is written as $a : b$. This is read as "a is to b".

2. As a fraction: It is written as $\frac{a}{b}$.

In the ratio $a:b$ or $\frac{a}{b}$, the quantity $a$ is called the antecedent and the quantity $b$ is called the consequent.

To find the ratio of $10$ kg to $25$ kg, we compare the two quantities. Since the units are the same (kg), we can directly write the ratio.

The ratio is given by:

$10$ kg : $25$ kg

We can write this ratio as a fraction:

$\frac{10}{25}$

To simplify the ratio, we find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of $10$ and $25$ is $5$.

We divide both the numerator and the denominator by $5$:

$\frac{\cancel{10}^{2}}{\cancel{25}_{5}} = \frac{2}{5}$

So, the simplified ratio of $10$ kg to $25$ kg is $\frac{2}{5}$ or $2:5$.

We need to find the ratio of two quantities: $30$ minutes and $1$ hour.

To find the ratio of two quantities, they must be in the same units.

We know that $1$ hour is equal to $60$ minutes.

Now we can find the ratio of $30$ minutes to $60$ minutes.

Ratio = $\frac{\text{Quantity 1}}{\text{Quantity 2}} = \frac{30 \text{ minutes}}{60 \text{ minutes}}$

Writing the ratio as a fraction:

$\frac{30}{60}$

To simplify the fraction, we find the greatest common divisor (GCD) of $30$ and $60$. The GCD is $30$.

Divide the numerator and denominator by $30$:

$\frac{\cancel{30}^{1}}{\cancel{60}_{2}} = \frac{1}{2}$

The ratio of $30$ minutes to $1$ hour is $\frac{1}{2}$, which can also be written as $1:2$.

No, the ratio $2:3$ is not the same as the ratio $3:2$.

A ratio compares two quantities in a specific order.

The ratio $2:3$ can be written as the fraction $\frac{2}{3}$. It represents a situation where the first quantity is 2 parts and the second quantity is 3 parts.

The ratio $3:2$ can be written as the fraction $\frac{3}{2}$. It represents a situation where the first quantity is 3 parts and the second quantity is 2 parts.

Since $\frac{2}{3} \neq \frac{3}{2}$, the ratios $2:3$ and $3:2$ are different.

The order of the numbers in a ratio is important. $a:b$ is generally not the same as $b:a$ unless $a=b$.

To write the ratio $12:18$ in its simplest form, we need to divide both parts of the ratio by their greatest common divisor (GCD).

The two numbers in the ratio are $12$ and $18$.

Let's find the GCD of $12$ and $18$.

Factors of $12$: $1, 2, 3, 4, 6, 12$

Factors of $18$: $1, 2, 3, 6, 9, 18$

The common factors are $1, 2, 3, 6$.

The greatest common divisor (GCD) of $12$ and $18$ is $6$.

Now, we divide both parts of the ratio $12:18$ by the GCD, which is $6$.

$12 \div 6 = 2$

$18 \div 6 = 3$

So, the ratio $12:18$ in its simplest form is $2:3$.

Yes, the ratios $1:2$ and $3:6$ are equivalent.

Equivalent ratios are ratios that express the same relationship between two quantities.

Here are ways to check if the ratios are equivalent:

Method 1: Simplify both ratios to their simplest form.

The first ratio is $1:2$. This ratio is already in its simplest form as the GCD of $1$ and $2$ is $1$. The fraction is $\frac{1}{2}$.

The second ratio is $3:6$. We find the GCD of $3$ and $6$. The factors of $3$ are $1, 3$. The factors of $6$ are $1, 2, 3, 6$. The GCD is $3$.

Divide both parts of the ratio $3:6$ by $3$: $3 \div 3 = 1$ and $6 \div 3 = 2$.

The simplest form of the ratio $3:6$ is $1:2$. The fraction is $\frac{3}{6} = \frac{\cancel{3}^{1}}{\cancel{6}_{2}} = \frac{1}{2}$.

Since the simplest forms of both ratios are the same ($1:2$), the ratios $1:2$ and $3:6$ are equivalent.

Method 2: Convert ratios to fractions and check for equivalence of fractions (Cross-multiplication).

Ratio $1:2$ is equivalent to the fraction $\frac{1}{2}$.

Ratio $3:6$ is equivalent to the fraction $\frac{3}{6}$.

To check if $\frac{1}{2}$ and $\frac{3}{6}$ are equivalent fractions, we can use cross-multiplication.

Multiply the numerator of the first fraction by the denominator of the second fraction: $1 \times 6 = 6$.

Multiply the denominator of the first fraction by the numerator of the second fraction: $2 \times 3 = 6$.

Since the cross-products are equal ($6 = 6$), the fractions $\frac{1}{2}$ and $\frac{3}{6}$ are equivalent. Therefore, the ratios $1:2$ and $3:6$ are equivalent.

Method 3: Check if one ratio can be obtained by multiplying or dividing the parts of the other ratio by the same non-zero number.

Consider the ratio $1:2$. If we multiply both parts by $3$, we get $1 \times 3 : 2 \times 3 = 3:6$.

Since we obtained the ratio $3:6$ by multiplying both parts of the ratio $1:2$ by the same number ($3$), the ratios are equivalent.

Alternatively, consider the ratio $3:6$. If we divide both parts by $3$, we get $3 \div 3 : 6 \div 3 = 1:2$.

Since we obtained the ratio $1:2$ by dividing both parts of the ratio $3:6$ by the same number ($3$), the ratios are equivalent.

A proportion is a statement that two ratios are equal.

If the ratio $a:b$ is equal to the ratio $c:d$, then we say that $a, b, c, d$ are in proportion. This is written as:

$a:b :: c:d$

or in fractional form as:

$\frac{a}{b} = \frac{c}{d}$

In a proportion $a:b :: c:d$, the terms $a$ and $d$ are called the extremes, and the terms $b$ and $c$ are called the means.

A key property of a proportion is that the product of the extremes is equal to the product of the means.

This means:

$a \times d = b \times c$

or

$ad = bc$

A proportion is a statement that two ratios are equal.

If two ratios, say $a:b$ and $c:d$, are equal, they form a proportion.

A proportion is represented using symbols in the following ways:

1. Using the double colon symbol (::) between the two ratios:

$a:b :: c:d$

This is read as "a is to b as c is to d". The double colon symbol (::) signifies "is in proportion to" or "is equal to the ratio".

2. Using the equals sign (=) between the two ratios expressed as fractions:

$\frac{a}{b} = \frac{c}{d}$

This representation clearly shows the equality of the two ratios.

In both representations, it is understood that $b \neq 0$ and $d \neq 0$, as division by zero is undefined.

For four numbers $a, b, c, d$ to be in proportion, the ratio $a:b$ must be equal to the ratio $c:d$. This can be written as $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$.

In this case, the four numbers are $2, 4, 6, 12$. We check if the ratio $2:4$ is equal to the ratio $6:12$.

Method 1: Comparing the ratios as fractions in simplest form.

The first ratio is $2:4$, which can be written as the fraction $\frac{2}{4}$. Simplifying this fraction:

$\frac{2}{4} = \frac{\cancel{2}^{1}}{\cancel{4}_{2}} = \frac{1}{2}$

The second ratio is $6:12$, which can be written as the fraction $\frac{6}{12}$. Simplifying this fraction:

$\frac{6}{12} = \frac{\cancel{6}^{1}}{\cancel{12}_{2}} = \frac{1}{2}$

Since both ratios simplify to the same value ($\frac{1}{2}$), the ratios $2:4$ and $6:12$ are equal.

Method 2: Using the property that the product of extremes equals the product of means.

If $2, 4, 6, 12$ are in proportion ($2:4 :: 6:12$), then the product of the extremes must be equal to the product of the means.

Extremes are the first and fourth terms: $2$ and $12$.

Means are the second and third terms: $4$ and $6$.

Product of extremes = $2 \times 12 = 24$

Product of means = $4 \times 6 = 24$

Since the product of the extremes ($24$) is equal to the product of the means ($24$), the numbers are in proportion.

Therefore, yes, the numbers $2, 4, 6, 12$ are in proportion because the ratio of the first two numbers ($2:4$) is equal to the ratio of the last two numbers ($6:12$).

If four numbers $a, b, c, d$ are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers.

This is written as $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$.

In a proportion, the first term ($a$) and the fourth term ($d$) are called the extremes.

The second term ($b$) and the third term ($c$) are called the means.

A fundamental property of proportion states that:

The product of the extremes is equal to the product of the means.

This can be expressed algebraically as:

$a \times d = b \times c$

Given that the numbers $3, x, 12, 16$ are in proportion, we have $a=3$, $b=x$, $c=12$, and $d=16$.

The proportion is $3:x :: 12:16$.

Applying the property of proportion:

Product of extremes = $3 \times 16$

Product of means = $x \times 12$

So, the relationship between the product of extremes and the product of means for the proportion $3:x :: 12:16$ is:

$(3 \times 16) = (x \times 12)$

This simplifies to:

Thus, the relationship is that the product of the first and fourth terms ($3 \times 16 = 48$) is equal to the product of the second and third terms ($x \times 12 = 12x$).

The unitary method is a technique used to solve problems involving proportions or direct/inverse variation.

The core idea of the unitary method is to first find the value of a single unit (the "unit" part of "unitary") and then use that value to find the value of the required number of units.

The steps typically involved are:

1. Find the value of a single unit from the given information.

2. Calculate the value for the required number of units by multiplying the value of the single unit by the desired quantity.

For example, if you know the cost of $5$ pens, you first find the cost of $1$ pen (by dividing the total cost by $5$). Then, to find the cost of $10$ pens, you multiply the cost of $1$ pen by $10$.

This method is particularly useful for solving problems related to cost, time, speed, work, etc., where a direct relationship exists between quantities.

Given: The cost of $5$ pens is $\textsf{₹}50$.

We need to find the cost of $1$ pen.

Using the unitary method, we can find the cost of a single unit (1 pen).

Cost of $5$ pens = $\textsf{₹}50$

Cost of $1$ pen = $\frac{\text{Total cost}}{\text{Number of pens}}$

Cost of $1$ pen = $\frac{\textsf{₹}50}{5}$

Cost of $1$ pen = $\textsf{₹}10$

Therefore, the cost of $1$ pen is $\textsf{₹}10$.

Given: The cost of $1$ meter of cloth is $\textsf{₹}200$.

We need to find the cost of $3$ meters of cloth.

Using the unitary method, since we know the cost of one unit (1 meter), we can find the cost of $3$ units by multiplying the cost of one unit by the number of units required.

Cost of $1$ meter of cloth = $\textsf{₹}200$

Cost of $3$ meters of cloth = Cost of $1$ meter $\times$ $3$

Cost of $3$ meters of cloth = $\textsf{₹}200 \times 3$

Cost of $3$ meters of cloth = $\textsf{₹}600$

Therefore, the cost of $3$ meters of cloth is $\textsf{₹}600$.

To find the ratio of $50$ paise to $\textsf{₹}5$, we must express both quantities in the same units.

We know that $1$ Rupee is equal to $100$ paise.

So, $\textsf{₹}5$ can be converted to paise by multiplying by $100$.

Now we need to find the ratio of $50$ paise to $500$ paise.

Ratio = $\frac{\text{Quantity 1}}{\text{Quantity 2}} = \frac{50 \text{ paise}}{500 \text{ paise}}$

Writing the ratio as a fraction:

$\frac{50}{500}$

To simplify the fraction, we find the greatest common divisor (GCD) of $50$ and $500$. The GCD is $50$.

Divide the numerator and denominator by $50$:

$\frac{\cancel{50}^{1}}{\cancel{500}_{10}} = \frac{1}{10}$

The ratio of $50$ paise to $\textsf{₹}5$ is $\frac{1}{10}$, which can also be written as $1:10$.

Given:

Number of girls in the class = $20$

Number of boys in the class = $15$

We need to find the ratio of the number of girls to the number of boys.

Ratio = Number of girls : Number of boys

Ratio = $20 : 15$}

To express the ratio in its simplest form, we need to divide both numbers by their greatest common divisor (GCD).

The numbers are $20$ and $15$.

Let's find the GCD of $20$ and $15$.

Factors of $20$: $1, 2, 4, 5, 10, 20$

Factors of $15$: $1, 3, 5, 15$

The common factors are $1$ and $5$.

The greatest common divisor (GCD) is $5$.

Divide both parts of the ratio $20:15$ by the GCD, $5$:

$20 \div 5 = 4$

$15 \div 5 = 3$

So, the ratio of the number of girls to the number of boys is $4:3$.}

As a fraction, this is $\frac{20}{15} = \frac{\cancel{20}^{4}}{\cancel{15}_{3}} = \frac{4}{3}$.

In a proportion $a:b :: c:d$, the four terms are referred to by specific names based on their position.

The first term ($a$) and the fourth term ($d$) are called the extremes.

The second term ($b$) and the third term ($c$) are called the means.

The given proportion is $4:5 :: 12:15$.

Here, the terms are:

First term = $4$

Second term = $5$

Third term = $12$

Fourth term = $15$}

Identifying the extremes and means:

The extremes are the first and fourth terms.

Extremes: $4$ and $15$

The means are the second and third terms.

Means: $5$ and $12$

Given:

Distance traveled by the car = $100$ km

Time taken = $2$ hours

To Find:

Distance traveled in $1$ hour at the same speed.

Solution:

We can solve this problem using the unitary method.

We are given the distance traveled in $2$ hours, and we want to find the distance traveled in $1$ hour (a single unit of time).

In $2$ hours, the car travels $100$ km.

To find the distance traveled in $1$ hour, we divide the total distance by the total time:

Distance in $1$ hour = $\frac{\text{Distance traveled}}{\text{Time taken}}$

Therefore, the car travels $50$ km in $1$ hour at the same speed.

Given:

Ratio of red balls to blue balls = $3:5$

Number of red balls = $12$

To Find:

Number of blue balls.

Solution:

Let the number of blue balls be $x$.

The ratio of red balls to blue balls is given as $3:5$.

Using the given information, the ratio of red balls to blue balls is also $12:x$.

Since these two ratios represent the same relationship, they are equivalent and form a proportion:

$3:5 :: 12:x$

This can be written as a fraction:

Using the property of proportion (product of extremes equals product of means), we multiply diagonally:

Now, we solve for $x$ by dividing both sides of the equation by $3$:

So, the number of blue balls is $20$.

Alternate Solution (Unitary Method):

The ratio $3:5$ means that for every $3$ red balls, there are $5$ blue balls.

The number of red balls is $12$, which corresponds to $3$ parts of the ratio.

To find the value of one 'part', we divide the number of red balls by the corresponding ratio part:

This means each 'part' in the ratio represents $4$ balls.

The number of blue balls corresponds to $5$ parts of the ratio.

Number of blue balls = Value of 1 part $\times$ Ratio part for blue balls

Using both methods, we find that there are $20$ blue balls.

Two ratios form a proportion if they are equivalent, meaning they represent the same relationship.

We can check if the ratios $8:12$ and $10:15$ form a proportion by comparing their values or by using the property of proportion.

Method 1: Simplifying the ratios.

First ratio: $8:12$. As a fraction: $\frac{8}{12}$.

Find the greatest common divisor (GCD) of $8$ and $12$. Factors of $8$: $1, 2, 4, 8$. Factors of $12$: $1, 2, 3, 4, 6, 12$. The GCD is $4$.

Simplify the fraction by dividing numerator and denominator by $4$:

$\frac{8}{12} = \frac{\cancel{8}^{2}}{\cancel{12}_{3}} = \frac{2}{3}$

The simplest form of the ratio $8:12$ is $2:3$.

Second ratio: $10:15$. As a fraction: $\frac{10}{15}$.

Find the greatest common divisor (GCD) of $10$ and $15$. Factors of $10$: $1, 2, 5, 10$. Factors of $15$: $1, 3, 5, 15$. The GCD is $5$.

Simplify the fraction by dividing numerator and denominator by $5$:

$\frac{10}{15} = \frac{\cancel{10}^{2}}{\cancel{15}_{3}} = \frac{2}{3}$

The simplest form of the ratio $10:15$ is $2:3$.

Since the simplest forms of both ratios are the same ($2:3$), the ratios $8:12$ and $10:15$ are equivalent and form a proportion.

Method 2: Using the property of proportion (Product of Extremes = Product of Means).

If the numbers $8, 12, 10, 15$ are in proportion, then $8:12 :: 10:15$.

The extremes are the first and fourth terms: $8$ and $15$.

The means are the second and third terms: $12$ and $10$.

Product of extremes = $8 \times 15$

$8 \times 15 = 120$

Product of means = $12 \times 10$

$12 \times 10 = 120$

Since the product of the extremes ($120$) is equal to the product of the means ($120$), the ratios $8:12$ and $10:15$ form a proportion.

Conclusion: Yes, the ratios $8:12$ and $10:15$ do form a proportion.

Given:

The cost of $6$ notebooks is $\textsf{₹}90$.

To Find:

The cost of $1$ notebook.

Solution:

We can use the unitary method to find the cost of a single notebook.

We are given the cost of $6$ notebooks and want to find the cost of $1$ notebook.

Cost of $6$ notebooks = $\textsf{₹}90$

Cost of $1$ notebook = $\frac{\text{Total cost}}{\text{Number of notebooks}}$

Perform the division:

Therefore, the cost of $1$ notebook is $\textsf{₹}15$.

Given:

Total amount to be shared = $\textsf{₹}200$

Ratio of shares between A and B = $2:3$

To Find:

The share of A and the share of B.

Solution:

The ratio $2:3$ means that the total amount is divided into parts, where A gets $2$ parts and B gets $3$ parts.

The total number of parts in the ratio is the sum of the ratio terms:

Now, we find the value of one ratio part by dividing the total amount by the total number of parts.

Now, we calculate the share for A and B by multiplying their respective ratio parts by the value of one part.

Share of A = A's ratio part $\times$ Value of one part

Share of B = B's ratio part $\times$ Value of one part

So, A receives $\textsf{₹}80$ and B receives $\textsf{₹}120$.

Check: $\textsf{₹}80 + \textsf{₹}120 = \textsf{₹}200$, which is the total amount.

Given:

Ratio of length to breadth of a rectangle = $5:2$

Breadth of the rectangle = $10$ cm

To Find:

Length of the rectangle.

Solution:

Let the length of the rectangle be $L$ and the breadth be $B$.

We are given the ratio $\text{Length} : \text{Breadth} = 5:2$.

This can be written as a fraction: $\frac{\text{Length}}{\text{Breadth}} = \frac{5}{2}$.

We are given that the breadth ($B$) is $10$ cm. We need to find the length ($L$).

We can set up a proportion using the given ratio and the actual measurements:

$L : 10 :: 5 : 2$

Writing this as an equation of fractions:

To solve for $L$, we can cross-multiply (using the property that the product of the extremes equals the product of the means).

Now, divide both sides by $2$ to find $L$:

The unit for the length is the same as the unit for the breadth, which is cm.

Therefore, the length of the rectangle is $25$ cm.

Alternate Solution (Unitary Method based on ratio parts):

The ratio of length to breadth is $5:2$. This means that the length is divided into $5$ parts and the breadth is divided into $2$ parts.

The breadth is given as $10$ cm, which corresponds to $2$ parts of the ratio.

Value of 2 parts = $10$ cm

To find the value of one part, divide the breadth by the number of parts it represents:

The length corresponds to $5$ parts of the ratio.

Length = Value of 1 part $\times$ Number of parts for length

Both methods show that the length of the rectangle is $25$ cm.

Given:

Map scale = $1$ cm : $10$ km

Distance between two cities on the map = $5$ cm

To Find:

The actual distance between the two cities.

Solution:

The map scale $1$ cm : $10$ km means that $1$ centimeter on the map represents an actual distance of $10$ kilometers on the ground.

We are given that the distance on the map is $5$ cm.

To find the actual distance, we can multiply the map distance by the scale factor (the actual distance represented by $1$ cm on the map).

From the scale, $1$ cm on the map corresponds to $10$ km in reality.

Actual distance = Distance on map $\times$ Value of $1$ cm on the map

Therefore, the actual distance between the two cities is $50$ km.

Given:

The proportion $6:10 :: x:15$.

To Find:

The value of the missing term, $x$.

Solution:

A proportion is a statement that two ratios are equal. The given proportion $6:10 :: x:15$ means that the ratio $6:10$ is equal to the ratio $x:15$.

This can be written in fractional form as:

In a proportion $a:b :: c:d$, the product of the extremes ($a$ and $d$) is equal to the product of the means ($b$ and $c$).

In our proportion $6:10 :: x:15$:

Extremes are $6$ and $15$.

Means are $10$ and $x$.

Applying the property:

To find $x$, divide both sides of the equation by $10$:

The missing term in the proportion is $9$.

We can verify this by checking if the ratios are equal: $6:10 = 6/10 = 3/5$. And $9:15 = 9/15 = 3/5$. Since $3/5 = 3/5$, the proportion is true with $x=9$.

Given:

The cost of $8$ bananas is $\textsf{₹}48$.

To Find:

The cost of $1$ banana.

Solution:

We can use the unitary method to find the cost of a single banana.

We are given the cost of $8$ bananas and need to find the cost of $1$ banana.

Cost of $8$ bananas = $\textsf{₹}48$

To find the cost of $1$ banana, we divide the total cost by the number of bananas:

Cost of $1$ banana = $\frac{\text{Total cost}}{\text{Number of bananas}}$

Perform the division:

Therefore, the cost of $1$ banana is $\textsf{₹}6$.

Concept of Ratio:

A ratio is a comparison of two quantities of the same kind by division. It shows how many times one quantity is relative to another quantity.

For example, if there are $5$ apples and $10$ oranges, the ratio of apples to oranges is $5:10$, which simplifies to $1:2$. This means for every $1$ apple, there are $2$ oranges.

Terms of a Ratio:

In a ratio $a:b$, where $b \neq 0$, the two terms are:

The first term ($a$) is called the antecedent.

The second term ($b$) is called the consequent.

Important Consideration for Ratios:

When comparing two quantities using a ratio, it is crucial that both quantities are in the same units. If they are not in the same units, you must convert one quantity so that both have the same unit before finding the ratio. Ratios are unitless.

Finding the Ratio of $2$ meters to $75$ cm:

Given:

Quantity 1 = $2$ meters

Quantity 2 = $75$ cm

To Find:

The ratio of $2$ meters to $75$ cm in its simplest form.

Solution:

The given quantities are in different units (meters and centimeters). We need to convert them to the same unit. We know that $1$ meter is equal to $100$ centimeters.

So, we convert $2$ meters to centimeters:

Now both quantities are in the same unit (cm): $200$ cm and $75$ cm.

The ratio of $2$ meters to $75$ cm is the ratio of $200$ cm to $75$ cm.

Ratio = $\frac{\text{First Quantity}}{\text{Second Quantity}} = \frac{200 \text{ cm}}{75 \text{ cm}}$

Writing the ratio as a fraction:

$\frac{200}{75}$

To write the ratio in its simplest form, we find the greatest common divisor (GCD) of $200$ and $75$.

We can find the GCD by prime factorization:

Prime factors of $200 = 2 \times 2 \times 2 \times 5 \times 5 = 2^3 \times 5^2$

Prime factors of $75 = 3 \times 5 \times 5 = 3^1 \times 5^2$

The common prime factors are $5^2 = 25$. So, the GCD of $200$ and $75$ is $25$.

Now, we divide the numerator and the denominator of the fraction $\frac{200}{75}$ by their GCD, $25$:

$\frac{\cancel{200}^{8}}{\cancel{75}_{3}} = \frac{8}{3}$

The ratio in its simplest form is $\frac{8}{3}$, which can also be written as $8:3$.

Equivalent Ratios:

Equivalent ratios are ratios that express the same relationship between two quantities. They represent the same proportion, even though the numbers themselves may be different.

For instance, the ratio $1:2$ and $2:4$ are equivalent because they both describe a situation where the second quantity is twice the first quantity.

How to Find Equivalent Ratios:

To find equivalent ratios, you multiply or divide both terms of the ratio by the same non-zero number.

If you have a ratio $a:b$, you can get an equivalent ratio by:

Multiplying: $(a \times k) : (b \times k)$, where $k$ is any non-zero number.

Dividing: $(a \div k) : (b \div k)$, where $k$ is any non-zero common divisor of $a$ and $b$.

Two Equivalent Ratios for $4:5$:

Given ratio: $4:5$.

1. Multiply both terms by $2$:

$(4 \times 2) : (5 \times 2) = 8:10$

So, $8:10$ is an equivalent ratio to $4:5$.

2. Multiply both terms by $3$:

$(4 \times 3) : (5 \times 3) = 12:15$

So, $12:15$ is another equivalent ratio to $4:5$.

Other examples include $40:50$ (multiplying by 10), $20:25$ (multiplying by 5), etc.

Similarity to Equivalent Fractions:

Equivalent ratios are very similar to equivalent fractions because a ratio $a:b$ can be represented as a fraction $\frac{a}{b}$.

Two fractions are equivalent if they represent the same value on the number line. Similarly, two ratios are equivalent if they represent the same proportional relationship.

Just as you find equivalent fractions by multiplying or dividing the numerator and denominator by the same non-zero number (e.g., $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$), you find equivalent ratios by performing the same operation on both terms of the ratio.

The ratio $4:5$ corresponds to the fraction $\frac{4}{5}$.

The equivalent ratio $8:10$ corresponds to the fraction $\frac{8}{10}$, which simplifies to $\frac{4}{5}$.

The equivalent ratio $12:15$ corresponds to the fraction $\frac{12}{15}$, which simplifies to $\frac{4}{5}$.

Since equivalent ratios correspond to equivalent fractions (fractions that simplify to the same value), the methods for finding them are essentially the same.

Definition of Proportion:

A proportion is a statement that two ratios are equal. If the ratio $a:b$ is equal to the ratio $c:d$, then we say that $a, b, c, d$ are in proportion.

This can be written as $a:b :: c:d$ or as $\frac{a}{b} = \frac{c}{d}$ (where $b \neq 0$ and $d \neq 0$).

Relationship between the four terms in a Proportion:

In a proportion $a:b :: c:d$:

The first term ($a$) and the fourth term ($d$) are called the extremes.

The second term ($b$) and the third term ($c$) are called the means.

The fundamental property of a proportion states that the product of the extremes is equal to the product of the means.

Mathematically, this means:

Checking if the numbers $5, 15, 10,$ and $30$ are in proportion:

We need to check if the ratio of the first two numbers ($5:15$) is equal to the ratio of the last two numbers ($10:30$). If they are in proportion, then $5:15 :: 10:30$.

According to the property of proportion, the product of the extremes must equal the product of the means.

The extremes are the first and fourth terms: $5$ and $30$.

The means are the second and third terms: $15$ and $10$.

Calculate the product of the extremes:

Product of Extremes = $5 \times 30$

Calculate the product of the means:

Product of Means = $15 \times 10$}

Compare the two products:

Product of Extremes ($150$) = Product of Means ($150$).

Since the product of the extremes is equal to the product of the means, the numbers $5, 15, 10,$ and $30$ are in proportion.

Explanation of the Unitary Method:

The unitary method is a technique used to solve problems where a relationship exists between two quantities, often directly proportional. The method involves finding the value of a single unit of one quantity, and then using that value to determine the value of the other quantity for a different number of units.

Why it is called the Unitary Method:

It is called the "unitary" method because the crucial first step is to find the value of a unit quantity (i.e., a single item, one kilogram, one hour, one meter, etc.). The word "unitary" comes from the word "unit". Once the value of one unit is known, finding the value for any number of units becomes straightforward.

Example Problem Solution:

Given:

Cost of $10$ kgs of rice = $\textsf{₹}650$

To Find:

Cost of $17$ kgs of rice.

Solution:

We will use the unitary method. First, we find the cost of $1$ kg of rice.

Cost of $10$ kgs of rice = $\textsf{₹}650$

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

Now that we know the cost of $1$ kg of rice ($\textsf{₹}65$), we can find the cost of $17$ kgs of rice by multiplying the cost of $1$ kg by $17$.

Cost of $17$ kgs of rice = Cost of $1$ kg $\times 17$

Let's perform the multiplication:

Therefore, the cost of $17$ kgs of rice is $\textsf{₹}1105$.

Given:

Ratio of students who passed to students who failed = $5:2$

Total number of students who appeared for the test = $40$

To Find:

1. The number of students who passed.

2. The number of students who failed.

Solution:

The ratio of passed students to failed students is $5:2$. This means that for every $5$ parts representing passed students, there are $2$ parts representing failed students.

The total number of parts in the ratio is the sum of the parts for passed and failed students.

The total number of students ($40$) corresponds to these $7$ total parts.

We can find the number of students that each ratio part represents by dividing the total number of students by the total number of ratio parts.

The number of students who passed corresponds to $5$ parts of the ratio.

The number of students who failed corresponds to $2$ parts of the ratio.

Checking the problem statement: It seems there might be an issue with the numbers provided in the question as the number of students should be a whole number. If we assume the total number of students was meant to be a multiple of 7, like 42 or 35, the calculation would yield whole numbers. Let's proceed with the given numbers, acknowledging the non-whole number result.

Number of students who passed = $\frac{200}{7} \approx 28.57$

Number of students who failed = $\frac{80}{7} \approx 11.43$

Total students = $\frac{200}{7} + \frac{80}{7} = \frac{280}{7} = 40$. This matches the total number of students.

Assuming the numbers in the question are precisely as given:

The number of students who passed is $\frac{200}{7}$.

The number of students who failed is $\frac{80}{7}$.

However, if the question intended for a whole number of students, and the total number of students was a multiple of 7 (like 35 or 42), the steps would be the same but result in integer values. Let's assume for illustration that the total students were 35.

Total students = 35

Total ratio parts = 7

Value of 1 ratio part = $\frac{35}{7} = 5$ students

Number of passed students = $5 \times 5 = 25$ students

Number of failed students = $2 \times 5 = 10$ students

Total = $25 + 10 = 35$.

Given the exact numbers in the question:

Number of students who passed = $\frac{200}{7}$

Number of students who failed = $\frac{80}{7}$

Given:

Distance traveled by car = $180$ km

Time taken to travel $180$ km = $3$ hours

Required distance to travel = $300$ km

Speed is constant.

To Find:

Time taken to travel $300$ km.

Solution (Using Unitary Method):

The unitary method involves finding the value for a single unit. In this case, we can find the time taken to travel $1$ km.

Time taken to travel $180$ km = $3$ hours

To find the time taken to travel $1$ km, we divide the total time by the total distance:

Now that we know the time taken to travel $1$ km ($\frac{1}{60}$ hours), we can find the time taken to travel $300$ km by multiplying the time for $1$ km by $300$.

Therefore, it will take $5$ hours for the car to travel $300$ km at the same speed.

Given:

Length of the rectangular field = $60$ meters

Breadth of the rectangular field = $40$ meters

To Find:

1. The ratio of its length to its breadth.

2. The ratio of its breadth to its perimeter.

Solution:

First, let's find the ratio of the length to the breadth.

Ratio (Length : Breadth) = $60 \text{ m} : 40 \text{ m}$

To simplify this ratio, we find the greatest common divisor (GCD) of $60$ and $40$. The GCD of $60$ and $40$ is $20$.

Divide both terms by $20$:

$60 \div 20 = 3$

$40 \div 20 = 2$

The ratio of length to breadth is $3:2$.

As a fraction: $\frac{60}{40} = \frac{\cancel{60}^{3}}{\cancel{40}_{2}} = \frac{3}{2}$.

Next, we need to find the ratio of the breadth to the perimeter. First, we must calculate the perimeter of the rectangle.

The formula for the perimeter of a rectangle is $P = 2 \times (\text{Length} + \text{Breadth})$.

Now we can find the ratio of the breadth to the perimeter.

Breadth = $40$ meters

Perimeter = $200$ meters

Ratio (Breadth : Perimeter) = $40 \text{ m} : 200 \text{ m}$

To simplify this ratio, we find the greatest common divisor (GCD) of $40$ and $200$. The GCD of $40$ and $200$ is $40$.

Divide both terms by $40$:

$40 \div 40 = 1$

$200 \div 40 = 5$

The ratio of breadth to perimeter is $1:5$.

As a fraction: $\frac{40}{200} = \frac{\cancel{40}^{1}}{\cancel{200}_{5}} = \frac{1}{5}$.

The ratio of the length to the breadth is $3:2$.

The ratio of the breadth to the perimeter is $1:5$.

Given:

Cost of $8$ notebooks = $\textsf{₹}160$

To Find:

Cost of $12$ notebooks using the concept of proportion.

Solution:

Let the cost of $12$ notebooks be $\textsf{₹}x$.

The number of notebooks and their cost are in direct proportion, meaning as the number of notebooks increases, the cost also increases proportionally.

We can set up a proportion based on the ratio of the number of notebooks and the ratio of their costs:

Ratio of notebooks: $8 : 12$

Ratio of costs: $160 : x$

Since these ratios are equal, they form a proportion:

$8 : 12 :: 160 : x$

This can be written in fractional form:

Using the property of proportion that the product of the extremes is equal to the product of the means:

Now, we solve for $x$ by dividing both sides of the equation by $8$:

Let's perform the division:

The value of $x$ is $240$, which represents the cost in Rupees.

Therefore, the cost of $12$ notebooks is $\textsf{₹}240$.

Given:

The ratio of two numbers is $4:7$.

The sum of the two numbers is $132$.

To Find:

The two numbers.

Solution:

Let the two numbers be represented by the ratio parts. Since the ratio is $4:7$, we can say the numbers are $4k$ and $7k$ for some common factor $k$.}

The sum of the two numbers is given as $132$.

So, we can write the equation:

Combine the terms on the left side:

Now, solve for $k$ by dividing both sides by $11$:

Perform the division:

Now that we have the value of $k$, we can find the two numbers.

The first number is $4k$:

The second number is $7k$:

The two numbers are $48$ and $84$.

Check: The ratio of $48$ to $84$ is $\frac{48}{84} = \frac{\cancel{48}^{4}}{\cancel{84}_{7}} = \frac{4}{7}$, which matches the given ratio. The sum of the numbers is $48 + 84 = 132$, which matches the given sum.

Given:

Earning in $7$ days = $\textsf{₹}2100$

Number of days for which earning is required = $15$ days

The worker works for the same number of hours each day (implies constant daily earning rate).

To Find:

Total earning in $15$ days.

Solution (Using Unitary Method):

In the unitary method, we first find the value for a single unit. Here, the unit is one day. We need to find the earning for $1$ day.

Earning in $7$ days = $\textsf{₹}2100$

Earning in $1$ day = $\frac{\text{Earning in 7 days}}{\text{Number of days}}$

Now that we know the earning for $1$ day ($\textsf{₹}300$), we can find the earning for $15$ days by multiplying the daily earning by the number of days.

Earning in $15$ days = Earning in $1$ day $\times 15$

Therefore, the worker will earn $\textsf{₹}4500$ in $15$ days.

Given:

Total amount to be divided = $\textsf{₹}750$

Ratio of shares among A, B, and C = $2:3:5$

To Find:

The share of A, the share of B, and the share of C.

Solution:

The ratio $2:3:5$ means that the total amount $\textsf{₹}750$ is divided into parts, where A gets $2$ parts, B gets $3$ parts, and C gets $5$ parts.

The total number of parts in the ratio is the sum of the individual ratio terms:

Now, we find the value of one ratio part by dividing the total amount by the total number of parts.

Now, we calculate the share for each person by multiplying their respective ratio part by the value of one part.

Share of A = A's ratio part $\times$ Value of one part

Share of B = B's ratio part $\times$ Value of one part

Share of C = C's ratio part $\times$ Value of one part

The shares of A, B, and C are $\textsf{₹}150$, $\textsf{₹}225$, and $\textsf{₹}375$ respectively.

Check: Sum of shares = $\textsf{₹}150 + \textsf{₹}225 + \textsf{₹}375 = \textsf{₹}750$, which matches the total amount.

Given:

Original recipe ratio of flour to sugar = $2$ cups of flour : $1$ cup of sugar

New amount of flour to be used = $6$ cups

To Find:

The amount of sugar needed for the larger cake.

Explanation and Solution (Using Ratio and Proportion):

The recipe gives us a fixed ratio between the amount of flour and the amount of sugar needed to maintain the correct taste and texture of the cake. This ratio is $2:1$ (flour : sugar).

When we want to make a larger cake using $6$ cups of flour, we must keep the same relationship (ratio) between flour and sugar as in the original recipe. This situation can be represented as a proportion, which states that two ratios are equal.

Let the amount of sugar needed for the larger cake be $x$ cups.

The new ratio of flour to sugar is $6:x$.

Since the ratios must be equal, we can set up the proportion:

Original Ratio = New Ratio

Flour : Sugar :: Flour : Sugar

$2 : 1 :: 6 : x$

We can write this proportion as an equation of fractions:

To solve for $x$, we use the property of proportion: the product of the extremes equals the product of the means.

Extremes: $2$ and $x$

Means: $1$ and $6$

Now, divide both sides by $2$ to find $x$:

The value of $x$ is $3$, which represents the amount of sugar in cups.

Alternatively, using reasoning about the ratio:

The original ratio is $2$ parts flour to $1$ part sugar.

We are using $6$ cups of flour, which is $3$ times the original amount of flour ($6 \div 2 = 3$).

To maintain the same ratio (proportion), the amount of sugar must also be $3$ times the original amount of sugar.

Original sugar = $1$ cup

New sugar = $1 \text{ cup} \times 3 = 3$ cups

Therefore, if you use $6$ cups of flour, you will need $3$ cups of sugar to maintain the same ratio as in the original recipe.