A family spends on wheat, meat, and vegetables in the ratio 12 : 17 : 3. If their prices increase by 20%, 30%, and 50% respectively (quantities unchanged), by what percentage does the total expense increase?

Introduction / Context:
This is a weighted-percentage increase problem. Different items contribute differently to the total expense, so their price hikes must be combined using the original spending weights.

Given Data / Assumptions:

• Spending ratio (wheat : meat : vegetables) = 12 : 17 : 3.
• Price increases: 20%, 30%, 50% respectively.
• Quantities consumed remain unchanged, so spending on each item scales by its price factor.

Concept / Approach:
Let original spends be 12x, 17x, and 3x (total 32x). After the increases, new spends are 12x*1.20, 17x*1.30, and 3x*1.50. Sum these to get the new total and compare with 32x.

Step-by-Step Solution:

Verification / Alternative check:
Compute weighted average of increases using weights 12, 17, and 3 over 32 total. Weighted increase = (12*20% + 17*30% + 3*50%) / 32 = (240 + 510 + 150) / 32% = 900/32% = 28.125%.

Why Other Options Are Wrong:
23 1/3%, 27 1/8%, and 25 1/7% result from incorrect weights or arithmetic; 30% is the largest single-item increase, not the weighted total.

Common Pitfalls:
Taking a simple average of 20%, 30%, 50% or using the largest increase directly instead of a weighted calculation.

Final Answer:
28 1/8%