AMC 8 · 1999 · #7

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

fraction-multiplicationinterval-arithmeticequal-spacing identify-subproblems ↑ Prerequisites: fraction-multiplicationmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Exit 3 on a highway is at milepost 40 and exit 10 is at milepost 160. A service center sits three-fourths of the way from exit 3 to exit 10. What is its milepost?

Pick an answer.

(A)

(B)

100

(C)

110

(D)

120

(E)

130

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break into Subproblems

The sentence asks one question but hides three short jobs, which is the signal for Tool #7 (Break into Subproblems): (i) find the total distance between the two exits, (ii) take $\frac{3}{4}$ of that distance, (iii) add the result to the starting milepost. Tool #1 (Draw a Diagram) backs this up — a simple number line from 40 to 160 makes "three-fourths of the way" visible as a tick mark, not just words, and prevents the common slip of computing $\frac{3}{4}$ of 160 instead of $\frac{3}{4}$ of the gap.

1STEP 1

Subtract the two mileposts to find the gap between the exits: 120 miles.

160 - 40 = 120 miles

2STEP 2

Take three-fourths of that gap — divide by 4, multiply by 3 — which is 90 miles.

\frac{3}{4}

× 120 = 3 ×

\frac{120}{4}

= 3 × 30 = 90 miles

3STEP 3

Start at exit 3's milepost 40 and add the 90 miles to land at milepost 130 — choice (E).

40 + 90 = 130 → (E)

Answer

130

Sanity check the position: three-fourths of the way from 40 to 160 should be much closer to 160 than to 40. Our answer 130 is 90 past exit 3 and only 30 short of exit 10, and 90 : 30 = 3 : 1 — exactly the "three parts done, one part to go" split that "three-fourths of the way" describes. The midpoint would be

\frac{40+160}{2}

= 100, so the service center must lie between 100 and 160; that immediately rules out (A) 90, (B) 100, (C) 110, and the quarter-of-the-way trap (D) 120 = 40 +

\frac{1}{4}

(120 · wrong) also fails the 3 : 1 check.

💡Key takeaway

"Fraction of the way from A to B" always means fraction of the gap B - A, then add back to A. Subtract, take the fraction, add — three Grade 4 moves and this AMC 8 problem is done.

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