AMC 10 · 2015 · #4

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 is a multiple of their difference. Find their ratio.

Pick an answer.

(A)
$\frac{5}{4}$
(B)
$\frac{3}{2}$
(C)
$\frac{9}{5}$
(D)
2
(E)
$\frac{5}{2}$
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 L/S falls out — without ever needing the actual numbers.

1STEP 1

Name the two numbers

Two letters name the two numbers.

L = larger, S = smaller
2STEP 2

Turn the sentence into an equation

The sentence becomes one equation.

L + S = 5(L - S)
3STEP 3

Expand and collect the terms

Collecting terms leaves a simple proportion.

L + S = 5L - 5S → 6S = 4L
4STEP 4

Read off the ratio

The ratio is 3/2, choice (B).

L/S = 6/4 = 3/2 → (B)
Answer
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 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) 5/2.
💡Key takeaway

When a problem gives only how two unknowns compare, name them with letters, write the one sentence 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

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