A sum of Rs. 3,170 is divided among X, Y and Z. If Rs. 13, Rs. 12 and Rs. 18 are subtracted from the shares of X, Y and Z respectively, then their resulting shares are in the ratio 20 : 18 : 21. What is the initial share of Z in rupees (that is, before the deduction)?

Introduction / Context:
This is another ratio and distribution problem involving adjusted shares. The total amount is fixed, but the ratio provided applies to the amounts after specific deductions from each person's share. We need to use the given ratio of the adjusted amounts to reconstruct the original shares and then find Z's initial share before the deduction of Rs. 18.

Given Data / Assumptions:

• Total amount divided among X, Y and Z is Rs. 3,170.
• When Rs. 13, Rs. 12 and Rs. 18 are subtracted from X, Y and Z respectively, their resulting shares are in the ratio 20 : 18 : 21.
• We are asked to find Z's original share before the deduction.

Concept / Approach:
We let the adjusted shares of X, Y and Z be 20k, 18k and 21k respectively. Then we add back the deducted amounts to express the original shares in terms of k. The sum of these original shares must equal the total amount given, 3,170. Solving for k gives us the adjusted and original shares, including Z's initial share.

Step-by-Step Solution:
Step 1: Let X's reduced share be 20k, Y's reduced share be 18k and Z's reduced share be 21k.Step 2: Then X's original share is 20k + 13.Step 3: Y's original share is 18k + 12.Step 4: Z's original share is 21k + 18.Step 5: The sum of original shares equals 3,170, so (20k + 13) + (18k + 12) + (21k + 18) = 3,170.Step 6: Combine like terms: 20k + 18k + 21k = 59k, and 13 + 12 + 18 = 43, so 59k + 43 = 3,170.Step 7: Subtract 43 from both sides to get 59k = 3,170 - 43 = 3,127.Step 8: Solve for k: k = 3,127 / 59 = 53.Step 9: Z's original share = 21k + 18 = 21 * 53 + 18 = 1,113 + 18 = Rs. 1,131.

Verification / Alternative check:
With k = 53, we can compute the original shares. X gets 20 * 53 + 13 = 1,060 + 13 = 1,073. Y gets 18 * 53 + 12 = 954 + 12 = 966. Z gets 1,131 as calculated. Summing these, 1,073 + 966 + 1,131 = Rs. 3,170, which matches the given total. The reduced shares are 1,073 - 13 = 1,060, 966 - 12 = 954 and 1,131 - 18 = 1,113, which are in ratio 20 : 18 : 21 when divided by 53. This confirms that our solution is consistent.

Why Other Options Are Wrong:
Values such as 1,530, 910 and 1,350 do not satisfy the equation when substituted back into the distribution. They lead to incorrect totals or ratios that do not match 20 : 18 : 21 after adjusting for the deductions. 'None of these' is incorrect because 1,131 fits all given conditions precisely.

Common Pitfalls:
One frequent mistake is to apply the ratio 20 : 18 : 21 directly to 3,170 without considering the subtractions, which leads to wrong answers. Another error is to forget to include all three deduction amounts when forming the total equation. Always define the post deduction amounts in terms of a common variable and then add back the deducted values before equating to the total.

Final Answer:
The initial share of Z is Rs. 1,131.