Let the shares of A, B, and C be denoted by \(A\), \(B\), and \(C\), respectively.

The total sum of money to be divided is ₹ 1250.

So, the sum of their shares is:

\(A + B + C = 1250 \quad \text{(Equation 1)}\)

We are given the following relationships between their shares:

• A gets \(\frac{2}{9}\) of B's share.
• C gets \(\frac{3}{4}\) of A's share.

We can write these relationships as equations:

\(A = \frac{2}{9}B \quad \text{(Equation 2)}\)

\(C = \frac{3}{4}A \quad \text{(Equation 3)}\)

Our goal is to find the values of \(A\), \(B\), and \(C\). We can express \(B\) and \(C\) in terms of \(A\) and substitute them into Equation 1.

From Equation 2, we can express \(B\) in terms of \(A\):

\(9A = 2B\)

\(B = \frac{9}{2}A \quad \text{(Equation 4)}\)

From Equation 3, \(C\) is already expressed in terms of \(A\):

\(C = \frac{3}{4}A \quad \text{(Equation 3)}\)

Now, substitute Equation 4 and Equation 3 into Equation 1:

\(A + B + C = 1250\)

\(A + \frac{9}{2}A + \frac{3}{4}A = 1250\)

To solve for \(A\), find a common denominator for the fractions, which is 4.

\(\frac{4}{4}A + \frac{18}{4}A + \frac{3}{4}A = 1250\)

Combine the terms on the left side:

\(\frac{4A + 18A + 3A}{4} = 1250\)

\(\frac{25A}{4} = 1250\)

Now, solve for \(A\):

\(25A = 1250 \times 4\)

\(25A = 5000\)

\(A = \frac{5000}{25}\)

\(A = 200\)

So, A's share is ₹ 200.

Now, use the value of \(A\) to find \(B\)'s share using Equation 4:

\(B = \frac{9}{2}A\)

\(B = \frac{9}{2} \times 200\)

\(B = 9 \times 100\)

\(B = 900\)

So, B's share is ₹ 900.

Next, use the value of \(A\) to find \(C\)'s share using Equation 3:

\(C = \frac{3}{4}A\)

\(C = \frac{3}{4} \times 200\)

\(C = 3 \times 50\)

\(C = 150\)

So, C's share is ₹ 150.

Let's verify if the sum of the shares is ₹ 1250:

\(A + B + C = 200 + 900 + 150 = 1100 + 150 = 1250\)

The sum is correct.

The shares of A, B, and C are ₹ 200, ₹ 900, and ₹ 150 respectively.

Therefore, the shares are 200, 900, and 150.

Revision Table: Dividing Money by Ratio

Additional Information: Ratio and Proportion Basics

What is a Ratio?

A ratio is a comparison of two or more quantities of the same kind by division. It shows how much of one quantity is present compared to another. Ratios are often written using a colon (:), e.g., A:B, or as a fraction, e.g., \(\frac{A}{B}\).

What is Proportion?

A proportion is an equality between two ratios. If two ratios \(\frac{a}{b}\) and \(\frac{c}{d}\) are equal, we say they are in proportion, written as \(a:b :: c:d\) or \(\frac{a}{b} = \frac{c}{d}\).

Solving Problems with Ratios and Shares:

Many problems involve dividing a total amount based on given ratios or relationships between parts. A common approach is to:

• Represent the unknown parts using variables (like \(Ax\), \(Bx\), \(Cx\), or by expressing all parts in terms of one variable as done in this problem).
• Form an equation based on the total amount.
• Solve the equation to find the value of the variable or the common multiplier.
• Calculate the individual parts using the value found.

In this specific problem, the relationships were given as fractions ("A gets 2/9 of B"), which implies a ratio A:B = 2:9. Similarly, C gets 3/4 of A, implying C:A = 3:4. By linking these ratios through the common variable A, we were able to express all shares in terms of A.