AMC 8 · 2003 · #5

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

percentageratio-proportionfraction-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: fraction-decimal-conversionlinear-equations-one-var

📏 Short solution 💡 2 insights

Problem

If 20% of some number equals 12, what is 30% of the same number?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Set Up an Equation

The problem is a one-sentence percent statement with one unknown. Tool #4 (Introduce a Variable) names the mystery number N, and Tool #3 (Set Up an Equation) turns the sentence "20% of N is 12" directly into the equation 0.20 N = 12. Once N is known, finding 30% of it is one more multiplication. No guess-and-check or pattern hunt is needed — the translation is immediate.

1STEP 1

Let N be the unknown number that the percents are taken from.

N = the unknown number

2STEP 2

Since 20% = $\frac{1}{5}$ , "20% of N is 12" becomes $\frac{N}{5}$ = 12.

\frac{1}{5}

N = 12

3STEP 3

Multiply both sides by 5 to get N = 60.

N = 5 × 12 = 60

4STEP 4

Take 30% of 60: three-tenths of 60 is 18.

\frac{3}{10}

× 60 = 3 × 6 = 18 → (B)

Answer

Check the size. Since 30% is more than 20%, the answer should be larger than 12. It is: 18 ＞ 12. Also, 30% is one and a half times 20%, so 30% of the number should be one and a half times 12, which is 1.5 × 12 = 18. The trap answers map to common mistakes: (A) 15 adds

\frac{1}{4}

of 12 instead of

\frac{1}{2}

, (C) 20 confuses the answer with the number itself, (D) 24 doubles 12 (treating 30% as twice 20%), and (E) 30 uses 30 as if it were a count, not a percent.

💡Key takeaway

When a percent of an unknown number is given, find the whole number first, then take the new percent of it. Or notice the ratio between the two percents and multiply 12 by it directly — both paths give 18.

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