2016 AMC 10A Problem 3

P1-10~2 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

David spends 75 cents for each dollar Ben spends, so the 12.50 dollar gap is one quarter of Ben's total. Ben spent 50 dollars and David 37.50.

Solution

Start by turning the sentence into a simple rule.

For each dollar Ben spends, David spends $25$ cents less, which is $75$ cents. So David always spends $\tfrac34$ of what Ben spends.
Try a small example to see the gap: if Ben spent $\$4$ , David spent $\$3$ , and the gap is $\$1$ , one quarter of Ben's amount. The gap is always $\tfrac14$ of Ben's total.
The real gap is $\$12.50$ , so Ben's total is $4 \cdot 12.50 = \$50$ .
David spent $\tfrac34 \cdot 50 = \$37.50$ .
The question asks for the two of them together: $50 + 37.50 = \$87.50$ .

The answer is $\boxed{\textbf{(C)}\ \$87.50}$ .

Why this works

"For every dollar" describes a fixed ratio between two totals. Once one amount is a known fraction of the other, the difference is also a known fraction of it, and one multiplication finishes the job. Always re-read the question at the end: here it asks for the total, not either person's amount.

Alternative approach

Think in quarters of a dollar. For every $4$ quarters Ben spends, David spends $3$ . So their spending splits into "parts" with Ben $4$ , David $3$ and gap $1$ . The gap of $1$ part is $\$12.50$ , so the total of $4 + 3 = 7$ parts is $7 \cdot 12.50 = \$87.50$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

"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 $\boxed{\textbf{(C)}\ \$87.50}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)