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2016 AMC 10A Problem 3

2016 AMC 10A Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 2 min · position P1-10

For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together?

A)
$37.50
B)
$50.00
C)
$87.50
D)
$90.00
E)
$92.50

Answer

Key insight

David spends 3/4 of what Ben spends, so the $12.50 gap is 1/4 of Ben's total; Ben spent $50, David $37.50.

Solution

"For every dollar Ben spent, David spent $25$ cents less" means David spent $\$0.75$ per Ben-dollar, i.e. David's total is $\tfrac{3}{4}$ of Ben's.

Let Ben's total be $B$ . Then David's total is $\tfrac34 B$ , and the difference is
$B - \tfrac34 B = \tfrac14 B = 12.50,$
so $B = 50$ . David therefore spent $\tfrac34 \cdot 50 = 37.50$ .

Together they spent $50 + 37.50 = 87.50$ . The answer is $\boxed{\textbf{(C)}\ \$87.50}$ .

Why this works

A "per dollar" comparison is a fixed ratio between two totals. Once one quantity is a known fraction of the other, the difference is also a known fraction of it, and one division finishes. Translate word ratios into a single variable rather than two.

Alternative approach

Think in units: each unit is $\$0.25$ of Ben's spending. Ben has $4$ units per dollar-block and David $3$ , so the gap of $1$ unit equals $\$12.50$ , and the total of $7$ units is $7 \cdot 12.50 = 87.50$ .

The trap

Reporting Ben's $50 or David's $37.50 alone instead of the combined total.

Common mistakes

Reporting Ben's $50 or David's$ 37.50 alone instead of the combined total.
Interpreting "25 cents less" as David spending $\$12.50 - 0.25$ or as a flat $\$0.25$ less overall rather than per dollar.

Techniques

Set up the equation/formula and compute; no special trick needed