AMC 10 · 2012 · #7

Grade 4 rate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
ratio-proportionfraction-arithmetic easier-related-problem ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
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Problem
In a bag of marbles, 3/5 of the marbles are blue and the rest are red. Then the number of red marbles is doubled while the number of blue marbles stays the same. What fraction of the marbles are red afterward?

Pick an answer.

(A)
$\frac{2}{5}$
(B)
$\frac{3}{7}$
(C)
$\frac{4}{7}$
(D)
$\frac{3}{5}$
(E)
$\frac{4}{5}$

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

The fractions are abstract and no real count is given, so Tool #9 (Solve an Easier Related Problem) replaces them with a concrete total of 5 marbles, turning fractions into objects you can count. Tool #16 (Count the Complement) gets the starting red share as everything that is not blue. Tool #3 (Eliminate Possibilities) guards the traps (A) 2/5 and (D) 3/5, which are just the original red and blue shares.

1STEP 1

Find the starting red fraction

Blue and red are the only colors, so the reds are whatever is left: 1 - 3/5 = 2/5 of the marbles start red.

1 - 3/5 = 5/5 - 3/5 = 2/5
2STEP 2

Pick a friendly total of 5 marbles

Counts are not given and ratios survive doubling, so let the total be 5 marbles: 3 blue, 2 red.

5 marbles → 3 blue, 2 red
3STEP 3

Double the red marbles

Doubling gives 2 × 2 = 4 red; the 3 blue are untouched, so the bag now holds 7 marbles.

2 × 2 = 4 red, 3 blue, 4 + 3 = 7 total
4STEP 4

Write the new red fraction

Red over the new total gives 4/7, choice (C); 2/5 (A) is the old red share and 3/5 (D) the blue one.

4/(4+3) = 4/7 → (C)
Answer
4/7
Check the direction: doubling the reds should push the red share above its old 2/5 = 0.4, but the blues do not vanish, so red must land between 0.4 and 1. 4/7 ≈ 0.57 sits there, with blue 3/7 ≈ 0.43, and 4/7 + 3/7 = 1 as it must. That rules out (A) 2/5 (no change) and (E) 4/5 (too big), confirming (C).
💡Key takeaway

When a problem gives only fractions, pick a small total that makes them whole — here 5 marbles — count what happens, then read off the new fraction: 4/7.

  • Find the starting red fraction
  • Pick a friendly total of 5 marbles
  • Double the red marbles
  • Write the new red fraction

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