A sum of Rs 15525 is divided among Sunil, Anil and Jamil so that if Rs 22, Rs 35 and Rs 48 are subtracted from their respective shares, the remaining amounts are in the ratio 7 : 10 : 13. What would be the ratio of their shares if instead Rs 16, Rs 77 and Rs 37 were added to their original shares?

Introduction / Context:
This problem requires handling a sum divided among three persons with conditions involving both subtraction and addition of different amounts. Initially, subtracting certain sums yields a given ratio. We must first use this information to determine the original individual shares. Then we apply a different set of additions to those original shares and find the new ratio. This tests linear equation setup and careful manipulation of constants in distribution problems.

Given Data / Assumptions:
Total sum divided among Sunil, Anil and Jamil is Rs 15525. When Rs 22, Rs 35 and Rs 48 are subtracted from their shares respectively, the remaining amounts are in the ratio 7 : 10 : 13. We are asked for the ratio of their shares if Rs 16, Rs 77 and Rs 37 are added to their original shares. There are no other conditions on time or profit; this is a pure division problem.

Concept / Approach:
We denote the original shares by variables and use the given ratio after subtraction to express them in terms of a common factor. This relation, combined with the total sum condition, allows us to solve for that factor and thus obtain concrete values of each share. Then we add the new constants Rs 16, Rs 77 and Rs 37 to these shares and form a new ratio. Finally, we simplify this ratio to its lowest integer form to match the answer options.

Step-by-Step Solution:
Step 1: Let original shares of Sunil, Anil and Jamil be S, A and J respectively. Step 2: After subtracting Rs 22, Rs 35 and Rs 48, we are told (S - 22) : (A - 35) : (J - 48) = 7 : 10 : 13. Step 3: Let common factor be k, so S - 22 = 7k, A - 35 = 10k and J - 48 = 13k. Step 4: Therefore S = 7k + 22, A = 10k + 35 and J = 13k + 48. Step 5: Total sum S + A + J = 15525, so (7k + 22) + (10k + 35) + (13k + 48) = 15525. Step 6: Combine like terms: (7k + 10k + 13k) + (22 + 35 + 48) = 30k + 105. Step 7: So 30k + 105 = 15525. Step 8: Subtract 105 from both sides: 30k = 15525 - 105 = 15420. Step 9: k = 15420 / 30 = 514. Step 10: Now compute original shares. S = 7 * 514 + 22 = 3598 + 22 = 3620. Step 11: A = 10 * 514 + 35 = 5140 + 35 = 5175. Step 12: J = 13 * 514 + 48 = 6682 + 48 = 6730. Step 13: Check total: 3620 + 5175 + 6730 = 15525, which is correct. Step 14: Now add new amounts to original shares: Sunil gets S + 16 = 3620 + 16 = 3636. Step 15: Anil gets A + 77 = 5175 + 77 = 5252. Step 16: Jamil gets J + 37 = 6730 + 37 = 6767. Step 17: New ratio is 3636 : 5252 : 6767. Step 18: To simplify, find common factor. All three numbers are divisible by 101. Step 19: 3636 / 101 = 36, 5252 / 101 = 52 and 6767 / 101 = 67. Step 20: So the new ratio is 36 : 52 : 67.

Verification / Alternative check:
We can verify the initial condition. After subtracting the earlier amounts, we have 3620 - 22 = 3598, 5175 - 35 = 5140 and 6730 - 48 = 6682. Dividing each by 7, 10 and 13 respectively, we find 3598 / 7 = 514, 5140 / 10 = 514 and 6682 / 13 = 514, confirming that a common factor k = 514 works and the ratio is correct. This consistency shows that the original shares are accurate and that the added amounts lead to the rightly simplified ratio 36 : 52 : 67.

Why Other Options Are Wrong:
Ratios 9 : 13 : 17 and 18 : 26 : 35 do not match the calculated values 3636, 5252 and 6767 when simplified, and cannot come from a simple common factor division. Option None of these is invalid because we found a valid ratio that matches option 36 : 52 : 67 exactly. Therefore 36 : 52 : 67 is the only ratio consistent with both the subtraction and addition conditions.

Common Pitfalls:
Some learners may mistakenly add or subtract the constants directly to the ratio numbers rather than to the actual shares. Others may skip checking that the original shares satisfy the total sum condition before moving to the second part of the question. Handling the algebra carefully, writing all equations explicitly and verifying both initial and modified conditions helps avoid these mistakes.

Final Answer:
After adding the new amounts to their original shares, the ratio of the sums for Sunil, Anil and Jamil is 36 : 52 : 67.