AMC 8 · 2008 · #7

Grade 4 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

ratio-proportionfraction-arithmeticlinear-equations-one-var identify-subproblems ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Short solution 💡 2 insights

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Problem

The fractions

\frac{3}{5}

\frac{M}{45}

, and

\frac{60}{N}

are all equal. Find M + N.

Pick an answer.

(A)

(B)

(C)

(D)

105

(E)

127

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Write an Equation

The chain $\frac{3}{5}$ = $\frac{M}{45}$ = $\frac{60}{N}$ is really two independent statements glued together. Tool #7 (Identify Subproblems) splits it into one equation for M and one equation for N. Each piece is a Grade 4 equivalent-fraction question: scale the top and bottom of $\frac{3}{5}$ by the same factor. Tool #3 (Write an Equation) lets us name that factor in each case and solve quickly.

1STEP 1

Split the chain into two separate equations, one for M and one for N.

\frac{3}{5}

\frac{M}{45}

and

\frac{3}{5}

\frac{60}{N}

2STEP 2

The denominator 5 → 45 is ×9, so scale the numerator by the same 9: M = 27.

5 × 9 = 45 → M = 3 × 9 = 27

3STEP 3

The numerator 3 → 60 is ×20, so scale the denominator by the same 20: N = 100.

3 × 20 = 60 → N = 5 × 20 = 100

4STEP 4

Add the two results: M + N = 27 + 100 = 127.

M + N = 27 + 100 = 127 → (E)

Answer

127

Check each piece is really equal to

\frac{3}{5}

\frac{27}{45}

: divide top and bottom by 9 to get

\frac{3}{5}

\frac{60}{100}

: divide top and bottom by 20 to get

\frac{3}{5}

. Both match, so M = 27 and N = 100 are correct, and M + N = 127. Answer choices (A) 27 and (D) 105 are traps for students who stop after finding M or who add only one of the two unknowns to something else.

💡Key takeaway

When two fractions are equal, the top and bottom always scale by the same number — find that scale factor and the unknown falls out.

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