AMC 10 · 2017 · #5

Grade 7 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State
linear-equations-one-varratio-proportion convert-to-algebraguess-and-check ↑ Prerequisites: linear-equations-one-var
📏 Short solution 💡 1 insight
Problem
Camilla starts with twice as many blueberry jelly beans as cherry ones. She eats 10 blueberry and 10 cherry. After that, she has three times as many blueberry as cherry. The question asks how many blueberry jelly beans she started with.

Pick an answer.

(A)
10
(B)
20
(C)
30
(D)
40
(E)
50

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Two amounts move together under one rule, so Tool #4 (Introduce a Variable) leads: call the cherry count c, and the blueberry count becomes 2c for free. Tool #13 (Convert to Algebra) turns the after-eating sentence into a single equation 2c-10=3(c-10). Solving that one equation gives c, and then 2c answers the question. Tool #3 (Eliminate Possibilities) offers a fast multiple-choice backup: the original blueberry count minus 10 must be a positive multiple of 3, which only one choice satisfies. Tool #6 (Guess and Check) lets us plug the result back to confirm the wording holds.

1STEP 1

Name the two counts with one variable

Let c be the cherry count. Since blueberry is twice cherry, the blueberry count is 2c.

cherry=c, blueberry=2c
2STEP 2

Translate the after-eating sentence

Eating 10 of each drops the counts to 2c-10 and c-10; blueberry being three times cherry gives 2c-10=3(c-10).

2c-10=3(c-10)
3STEP 3

Solve for the cherry count

Expand and collect the c terms: 2c-10=3c-30, so c=20 cherry jelly beans.

2c-10=3c-30 → 20=c
4STEP 4

Find the original blueberry count

Double the cherry count: blueberry = 2c = 2×20 = 40, choice (D).

blueberry=2c=2×20=40 → (D)
Answer
40
Plug back: start with 40 blueberry and 20 cherry, and indeed 40=2×20. Eat 10 of each, leaving 30 blueberry and 10 cherry, and 30=3×10, exactly three times as many. Both sentences hold, so (D) 40 is correct.
💡Key takeaway

Call cherry c and blueberry 2c, write what happens after eating 10 of each as one equation, solve it, and double the answer — 40, choice (D).

  • Name the two counts with one variable
  • Translate the after-eating sentence
  • Solve for the cherry count
  • Find the original blueberry count

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