Finding Shares in a Ratio

The problem asks us to divide a total amount of Rs. 750 among three people, A, B, and C, based on given ratios between pairs of them. We are given the ratio of A to B as 5:2 and the ratio of B to C as 7:13.

Combining Ratios A:B and B:C

To find the individual shares of A, B, and C from the total amount, we first need to combine the given ratios A:B and B:C into a single combined ratio A:B:C. The link between the two ratios is the share of B.

• The first ratio is A : B = 5 : 2.
• The second ratio is B : C = 7 : 13.

To combine these, we need to make the value corresponding to B the same in both ratios. The least common multiple (LCM) of 2 and 7 (the values for B) is 14.

• Multiply the first ratio (A : B = 5 : 2) by 7: \(A : B = (5 \times 7) : (2 \times 7) = 35 : 14\)
• Multiply the second ratio (B : C = 7 : 13) by 2: \(B : C = (7 \times 2) : (13 \times 2) = 14 : 26\)

Now that the value for B is the same (14) in both resulting ratios, we can combine them to get the ratio A : B : C.

\(A : B : C = 35 : 14 : 26\)

Calculating the Total Ratio Parts

The combined ratio A : B : C is 35 : 14 : 26. The total number of ratio parts is the sum of these individual parts.

Total ratio parts = \(35 + 14 + 26 = 75\)

Calculating A's Share

The total amount to be divided is Rs. 750. A's share is the fraction of the total amount corresponding to A's ratio part out of the total ratio parts.

• A's ratio part is 35.
• Total ratio parts are 75.
• Total amount is Rs. 750.

A's share = \(\left(\frac{\text{A's ratio part}}{\text{Total ratio parts}}\right) \times \text{Total amount}\)

A's share = \(\left(\frac{35}{75}\right) \times 750\)

Now, we can calculate the value:

A's share = \(35 \times \left(\frac{750}{75}\right)\)

Since 750 divided by 75 is 10:

A's share = \(35 \times 10\)

A's share = Rs. 350

Summary of Shares

Based on the combined ratio A:B:C = 35:14:26 and a total of Rs. 750:

• A's share = Rs. 350
• B's share = \(\left(\frac{14}{75}\right) \times 750 = 14 \times 10 = \text{Rs. } 140\)
• C's share = \(\left(\frac{26}{75}\right) \times 750 = 26 \times 10 = \text{Rs. } 260\)

Total shares = \(350 + 140 + 260 = 750\), which matches the total amount.

Therefore, A's share is Rs. 350.