Understanding the Ratio Distribution Problem

This problem involves finding the share of one individual (D) when a sum of money is divided among four people (A, B, C, D) based on given ratios between pairs of individuals. We are also given the difference between the shares of two individuals (A and C).

Ratio Analysis for Share Distribution

We need to determine the proportion of the total sum that each person receives. The initial ratios provided are:

• Ratio of A's share to B's share (A : B) = 1 : 3
• Ratio of B's share to C's share (B : C) = 2 : 5
• Ratio of C's share to D's share (C : D) = 2 : 3

To solve this, we first need to find a common ratio that represents the shares of A, B, C, and D together (A : B : C : D).

Combining the Given Ratios

To combine the ratios, we need to ensure the common terms in the pairs are equal. Let's start by combining A:B and B:C.

• A : B = 1 : 3
• B : C = 2 : 5

The common term is B. The values for B are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.

• To make B's share 6 in the first ratio, multiply both parts by 2: A : B = (1 $\times$ 2) : (3 $\times$ 2) = 2 : 6
• To make B's share 6 in the second ratio, multiply both parts by 3: B : C = (2 $\times$ 3) : (5 $\times$ 3) = 6 : 15

Now we can combine these to get A : B : C = 2 : 6 : 15.

Next, we combine this with the ratio C : D.

• A : B : C = 2 : 6 : 15
• C : D = 2 : 3

The common term is C. The values for C are 15 and 2. The LCM of 15 and 2 is 30.

• To make C's share 30 in the A : B : C ratio, multiply all parts by 2: A : B : C = (2 $\times$ 2) : (6 $\times$ 2) : (15 $\times$ 2) = 4 : 12 : 30
• To make C's share 30 in the C : D ratio, multiply both parts by 15: C : D = (2 $\times$ 15) : (3 $\times$ 15) = 30 : 45

Combining these gives the final ratio for all four individuals:

A : B : C : D = 4 : 12 : 30 : 45

Calculating the Value of One Ratio Part

The problem states that the difference between the shares of A and C is Rs. 3,510.

From the combined ratio, the difference in parts between C's share and A's share is:

Difference in parts = C's parts - A's parts = 30 - 4 = 26 parts

We are given that this difference corresponds to Rs. 3,510.

Therefore,

26 parts = Rs. 3,510

Now, we can find the value of 1 part:

1 part = $\\frac{\\text{Rs. } 3,510}{26}$

1 part = Rs. 135

Determining the Share of D

The combined ratio shows that D's share is 45 parts.

To find the actual amount of D's share, we multiply the value of 1 part by the number of parts D has:

Share of D = D's parts $\\times$ Value of 1 part

Share of D = 45 $\\times$ Rs. 135

Share of D = Rs. 6,075

Thus, the share of D is Rs. 6,075.