Understanding the Problem

The problem involves dividing a total sum of Rs. 15525 among three people: Sunil, Anil, and Jamil. We are given two conditions related to their shares:

1. If certain amounts are subtracted from their original shares, the remaining sums are in a specific ratio (7:10:13).
2. We need to find the ratio of their shares if different amounts were added to their original shares.

Let's denote the original shares of Sunil, Anil, and Jamil as S, A, and J respectively. The total sum is Rs. 15525.

So, we know that \(S + A + J = 15525\).

Calculating the Original Shares

According to the first condition, if Rs. 22 is diminished from Sunil's share, Rs. 35 from Anil's, and Rs. 48 from Jamil's, their remaining sums are in the ratio 7 : 10 : 13.

This can be written as:

\((S - 22) : (A - 35) : (J - 48) = 7 : 10 : 13\)

We can represent these remaining amounts using a common multiple, say \(k\):

• \(S - 22 = 7k\)
• \(A - 35 = 10k\)
• \(J - 48 = 13k\)

From these equations, we can express the original shares S, A, and J in terms of \(k\):

• \(S = 7k + 22\)
• \(A = 10k + 35\)
• \(J = 13k + 48\)

Now, we use the total sum \(S + A + J = 15525\) to find the value of \(k\).

\((7k + 22) + (10k + 35) + (13k + 48) = 15525\)

Combine the terms with \(k\) and the constant terms:

\((7k + 10k + 13k) + (22 + 35 + 48) = 15525\)

\(30k + 105 = 15525\)

Subtract 105 from both sides:

\(30k = 15525 - 105\)

\(30k = 15420\)

Divide by 30 to find \(k\):

\(k = \frac{15420}{30}\)

\(k = 514\)

Now that we have the value of \(k\), we can calculate the original shares:

• Sunil's share (S) = \(7k + 22 = 7 \times 514 + 22 = 3598 + 22 = 3620\)
• Anil's share (A) = \(10k + 35 = 10 \times 514 + 35 = 5140 + 35 = 5175\)
• Jamil's share (J) = \(13k + 48 = 13 \times 514 + 48 = 6682 + 48 = 6730\)

Let's quickly verify the total sum: \(3620 + 5175 + 6730 = 15525\). This matches the given total sum.

Finding the New Shares and Ratio

The second part of the question asks for the ratio of their sums if Rs. 16, Rs. 77, and Rs. 37 were added to their original shares respectively.

• Sunil's new share = Original share + 16 = \(3620 + 16 = 3636\)
• Anil's new share = Original share + 77 = \(5175 + 77 = 5252\)
• Jamil's new share = Original share + 37 = \(6730 + 37 = 6767\)

The ratio of their new sums is \(3636 : 5252 : 6767\).

To simplify this ratio, we need to find the greatest common divisor (GCD) of these three numbers. Let's try dividing by small numbers or looking at the options to see if there's a common factor related to them.

Let's check if 101 is a common factor (often used in such problems):

• \(3636 \div 101 = 36\)
• \(5252 \div 101 = 52\)
• \(6767 \div 101 = 67\)

Since 101 divides all three numbers perfectly, the simplified ratio is \(36 : 52 : 67\).

Final Ratio

The ratio of their sums if Rs. 16, Rs. 77, and Rs. 37 were added to their original shares is 36 : 52 : 67.

Revision Table: Key Concepts in Ratio and Proportion

Additional Information: Algebraic Approach to Ratio Problems

Ratio problems involving sums of money or quantities can often be effectively solved using algebraic methods. By representing the unknown quantities (like shares) using variables and incorporating a common factor (like \(k\)), we can translate the word problem into equations.

Steps typically involved:

• Identify the variables (e.g., original shares).
• Write down the given conditions as equations involving the variables.
• If ratios are involved, use a common multiple (\(k\)) to express the quantities in terms of \(k\).
• Use total sums or other constraints to form an equation and solve for \(k\).
• Substitute the value of \(k\) back to find the specific quantities.
• Use the calculated quantities to find the required ratio or value for the second part of the problem.
• Always simplify ratios to their simplest form by dividing all terms by their greatest common divisor (GCD).

This systematic approach helps break down complex ratio and proportion problems into manageable steps.