Dividing Money Using Ratios: Finding C's Share

This problem involves dividing a total amount of money (Rs. 2420) among three individuals, A, B, and C, based on given ratios between pairs of individuals (A:B and B:C). To solve this, we first need to find a combined ratio A:B:C and then use this combined ratio to determine each person's share of the total amount.

Step-by-Step Solution to Divide Money by Ratios

1. Combine the Given Ratios

We are given two separate ratios:

• Ratio 1: A : B = 5 : 4
• Ratio 2: B : C = 9 : 10

To combine these ratios into a single A:B:C ratio, we need to make the 'B' part common in both ratios. The values for B are 4 and 9. The least common multiple (LCM) of 4 and 9 is 36.

• Adjust Ratio 1 (A:B = 5:4) so that B becomes 36. Multiply both parts of the ratio by \( \frac{36}{4} = 9 \). \( A : B = (5 \times 9) : (4 \times 9) = 45 : 36 \)
• Adjust Ratio 2 (B:C = 9:10) so that B becomes 36. Multiply both parts of the ratio by \( \frac{36}{9} = 4 \). \( B : C = (9 \times 4) : (10 \times 4) = 36 : 40 \)

Now that the 'B' part is 36 in both adjusted ratios, we can combine them to get the comprehensive ratio A:B:C.

\( A : B : C = 45 : 36 : 40 \)

2. Calculate the Total Number of Ratio Parts

The combined ratio A:B:C = 45:36:40 means that the total amount is divided into \( 45 + 36 + 40 \) equal parts.

Total parts = \( 45 + 36 + 40 = 121 \)

3. Determine the Value of One Ratio Part

The total amount of money to be divided is Rs. 2420. This total amount corresponds to the total number of ratio parts (121 parts).

Value of one part = \( \frac{\text{Total Amount}}{\text{Total Parts}} = \frac{2420}{121} \)

\( \frac{2420}{121} = \frac{242 \times 10}{121} = \frac{(2 \times 121) \times 10}{121} = 2 \times 10 = 20 \)

So, the value of one ratio part is Rs. 20.

4. Calculate C's Share

From the combined ratio A:B:C = 45:36:40, C's share corresponds to 40 parts.

C's share = C's ratio parts \( \times \) Value of one part

C's share = \( 40 \times 20 = 800 \)

Therefore, C gets Rs. 800.

Let's verify the shares for A and B as well:

• A's share = \( 45 \times 20 = 900 \)
• B's share = \( 36 \times 20 = 720 \)

Total amount = A's share + B's share + C's share = \( 900 + 720 + 800 = 2420 \). This matches the total amount given, confirming our calculations are correct.

Summary of Shares

The question asks for the amount C gets, which is Rs. 800.

Revision Table: Dividing Amounts by Ratio

Additional Information on Ratio and Proportion Concepts

Ratio and Proportion are fundamental concepts in quantitative aptitude. A ratio is a comparison of two quantities by division. It shows how much of one quantity there is compared to another. For example, A:B = 5:4 means for every 5 units A has, B has 4 units.

A proportion is a statement that two ratios are equal. For instance, if A:B = 5:4 and B:C = 9:10, we used the concept of proportion to find equivalent ratios (like 45:36 and 36:40) to combine them.

When dividing a quantity in a given ratio, say P:Q:R, the total quantity is divided into \(P+Q+R\) equal parts. The share of P is \( \frac{P}{P+Q+R} \times \text{Total Quantity} \), and similarly for Q and R. In our case, the combined ratio was 45:36:40, and the total parts were 121. C's share is \( \frac{40}{121} \) of the total amount, which is \( \frac{40}{121} \times 2420 \).

\( \frac{40}{121} \times 2420 = 40 \times \frac{2420}{121} = 40 \times 20 = 800 \)

Understanding how to combine ratios and calculate shares is essential for solving many problems related to division of assets, profit sharing, mixtures, etc.