AMC 10 · 2018 · #6

Grade 7 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain
unit-conversionrateratio-proportiondimensional-analysis dimensional-analysiseasier-related-problem ↑ Prerequisites: ratio-proportionunit-conversion
📏 Short solution 💡 2 insights
Problem
A vending machine sells S cans of soda for Q quarters. Using the fact that one dollar equals four quarters, write an expression in S, Q, and D for how many cans D dollars buys.

Pick an answer.

(A)
$\frac{4DQ}{S}$
(B)
$\frac{4DS}{Q}$
(C)
$\frac{4Q}{DS}$
(D)
$\frac{DQ}{4S}$
(E)
$\frac{DS}{4Q}$
How to solve
Strategy Analyze the Units

Tool #8 (Analyze the Units): three quantities are in play — cans, quarters, and dollars — and every wrong choice comes from mixing them up. Converting dollars to quarters and then building a cans-per-quarter rate decides which letter belongs on top and which on the bottom. Tool #9 (Solve an Easier Related Problem): with three letters it is easy to flip a fraction without noticing, so we replace S, Q, and D with small numbers and redo the purchase by hand. Tool #3 (Eliminate Possibilities): those same small numbers can be pushed through all five choices, and only one of them produces the hand-counted result.

1STEP 1

Put the money in one unit

Put all money into quarters.

D dollars = 4D quarters
2STEP 2

Build the cans-per-quarter rate

Build the cans per quarter rate.

rate = (S cans)/(Q quarters) = S/Q cans per quarter
3STEP 3

Multiply rate by quarters

Multiply the rate by the quarters.

4D · S/Q = 4DS/Q
4STEP 4

Test with small numbers

Check it with small numbers.

4DS/Q = (4 · 3 · 2)/4 = 6 cans
5STEP 5

Knock out the other four

What remains is four D S over Q.

(A) 24, (C) 8/3, (D) 3/2, (E) 3/8, (B) 6 ✓
Answer
4DS/Q
Three checks agree. Units: S counts cans while Q and D both measure money, so a valid answer must reduce to cans times (money over money); only (B) 4DS/Q and (E) DS/4Q do, and (E) is (B) divided by 16, which would mean a dollar is worth a quarter of a quarter. Special case: if you carry exactly D = Q/4 dollars, that is exactly Q quarters, so you should get exactly S cans — and (B) gives (4 · Q/4 · S)/Q = S, while (E) gives only S/16. Direction: more dollars or a bigger bundle S should mean more cans, and a higher price Q should mean fewer, which matches D and S sitting on top of the fraction and Q on the bottom.
💡Key takeaway

Get all the money into one coin, turn the deal into a rate of S/Q cans per quarter, then multiply by the 4D quarters you actually have.

  • Put the money in one unit
  • Build the cans-per-quarter rate
  • Multiply rate by quarters
  • Test with small numbers
  • Knock out the other four

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