AMC 10 · 2015 · #6

Grade 7 rate-ratio

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

ratio-proportionlinear-equations-two-varsystems-of-equations convert-to-algebra ↑ Prerequisites: linear-equations-two-var

📏 Medium solution 💡 2 insights

Problem

Two positive numbers have a sum that equals 5 times their difference. Find the ratio of the larger number to the smaller number.

Pick an answer.

(A)

$\frac{5}{4}$

(B)

$\frac{3}{2}$

(C)

$\frac{9}{5}$

(D)

(E)

$\frac{5}{2}$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The numbers themselves are never given, only how their sum and difference compare, so Tool #4 (Introduce a Variable) names the larger number L and the smaller number S to hold those two unknown quantities. Tool #13 (Convert to Algebra) then rewrites the sentence 'the sum is 5 times the difference' as one equation, L + S = 5(L - S), which can be simplified until the ratio $\frac{L}{S}$ falls out — without ever needing the actual numbers.

1STEP 1

Name the two numbers

Call the numbers L (larger) and S (smaller): sum = L + S, difference = L - S.

L = larger, S = smaller

2STEP 2

Turn the sentence into an equation

Translate the sentence directly: L + S = 5(L - S).

L + S = 5(L - S)

3STEP 3

Expand and collect the terms

Expand and collect terms to get 6S = 4L.

L + S = 5L - 5S → 6S = 4L

4STEP 4

Read off the ratio

So the ratio is $\frac{L}{S} = \frac{6}{4}$ = $\frac{3}{2}$ — choice (B).

\frac{L}{S} = \frac{6}{4} = \frac{3}{2}

→ (B)

Answer

\frac{3}{2}

Test the ratio with real numbers: if L = 3 and S = 2, the sum is 5 and the difference is 1, and 5 is indeed 5 times 1. The condition holds, so

\frac{3}{2}

is correct. It also makes sense that the ratio is only a little above 1: the difference is small compared to the sum, so the two numbers are fairly close, ruling out a large ratio like (E)

\frac{5}{2}

💡Key takeaway

When a problem gives only how two unknowns compare, name them with letters, write it as an equation, and the ratio drops out.

Name the two numbers
Turn the sentence into an equation
Expand and collect the terms
Read off the ratio

Name the two numbers

Turn the sentence into an equation

Expand and collect the terms

Read off the ratio

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