AMC 10 · 2016 · #3

Grade 7 arithmetic

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionlinear-equations-one-var convert-to-algebra ↑ Prerequisites: ratio-proportion

📏 Short solution 💡 2 insights

Problem

For each dollar Ben spends on bagels, David spends 25 cents less. Ben ends up paying $12.50 more than David in total. Find the combined amount they spent.

Pick an answer.

(A)

$37.50

(B)

$50.00

(C)

$87.50

(D)

$90.00

(E)

$92.50

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The phrase "for every dollar Ben spent, David spent 25 cents less" is a rate, so Tool #8 (Analyze the Units) turns it into a clean fact: David's total is $0.75 for each of Ben's dollars, i.e. 75% of Ben's total. Once that link is fixed, Tool #4 (Introduce a Variable) names Ben's total and the single fact "$12.50 more" becomes one short equation to solve. Tool #3 (Eliminate Possibilities) then checks the size against the choices.

1STEP 1

The rate 25 cents less per dollar means David's total is always 0.75 times Ben's total.

David = 0.75 × Ben

2STEP 2

Let B be Ben's dollars; David spends 0.75B, so the gap 0.25B = 12.50 gives B=50.

B-0.75B=12.50 → 0.25B=12.50 → B=50

3STEP 3

Ben spent 50 and David 37.50, so together they spent 50+37.50 = 87.50, choice (C).

50.00+0.75× 50.00 = 50.00+37.50 = 87.50 → (C)

Answer

$87.50

Check the gap: $50.00-$37.50=$12.50, exactly the stated difference, and $37.50 is indeed 75% of $50.00. The total $87.50 sits between the two individual amounts doubled, which is the right size; choices like $37.50 and $50.00 are just one person's share, not the combined total, so they are traps.

💡Key takeaway

A steady gap per dollar lets you turn the total difference into one easy equation for the bigger amount.

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