AMC 8 · 1999 · #7
Easy mode Grade 4Problem
Picture a long highway with numbered exits.
On this highway, Exit 3 is at milepost 40. Exit 10 is at milepost 160.
A service center sits on the highway between these two exits. It is three-fourths of the way from Exit 3 to Exit 10.
At which milepost is the service center?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: 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?
Givens: Exit $3$ is at milepost $40$; Exit $10$ is at milepost $160$; Service center is $\tfrac{3}{4}$ of the way from exit $3$ toward exit $10$; Answer choices: (A) $90$, (B) $100$, (C) $110$, (D) $120$, (E) $130$
Unknowns: The milepost number of the service center
Understand
Restated: 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?
Givens: Exit $3$ is at milepost $40$; Exit $10$ is at milepost $160$; Service center is $\tfrac{3}{4}$ of the way from exit $3$ toward exit $10$; Answer choices: (A) $90$, (B) $100$, (C) $110$, (D) $120$, (E) $130$
Plan
Primary tool: #7 Break into Subproblems
Secondary: #1 Draw a Diagram
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 $\tfrac{3}{4}$ 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 $\tfrac{3}{4}$ of $160$ instead of $\tfrac{3}{4}$ of the gap.
Execute — Answer: E
4.OA.A.3 Step 1 Subproblem 1: find the distance between the two exits by subtracting their mileposts.
💡 Distance on a number line is the difference of the endpoints — a Grade 4 multi-step setup move.
4.NF.B.4 Step 2 - Subproblem 2: take three-fourths of that $120$-mile gap.
- Multiplying a fraction by a whole number is the same as $3 \times \tfrac{120}{4}$.
💡 Grade 4 fraction-of-a-whole: divide by the denominator, multiply by the numerator.
4.OA.A.3 Step 3 - Subproblem 3: start at exit $3$'s milepost and travel $90$ miles further along the highway.
- Add to get the service center's milepost.
💡 Mileposts increase in the direction of travel, so adding distance to the starting milepost gives the new milepost.
4.OA.A.3 Subproblem 1: find the distance between the two exits by subtracting their milep 4.NF.B.4 Subproblem 2: take three-fourths of that $120$-mile gap. Multiplying a fraction 4.OA.A.3 Subproblem 3: start at exit $3$'s milepost and travel $90$ miles further along t Review
Reasonableness: 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 $\tfrac{40+160}{2} = 100$, 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) $120 = 40 + \tfrac{1}{4}(120 \cdot \text{wrong})$ also fails the $3 : 1$ check.
Alternative: Tool #1 (Draw a Diagram): sketch a number line from $40$ to $160$ and divide the $120$-mile gap into four equal pieces of $30$ miles each. The tick marks land at $40, 70, 100, 130, 160$. "Three-fourths of the way" is the third tick after $40$, which is $130$ — answer (E) without doing any fraction multiplication, just counting equal jumps.
CCSS standards used (min grade 4)
4.OA.A.3Solve multistep word problems with whole numbers using the four operations (Chaining subtraction (find the gap), multiplication (take a fraction of it), and addition (shift from the starting milepost) across three subproblems.)4.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number (Computing $\tfrac{3}{4} \times 120 = 90$ as "divide by $4$, multiply by $3$" to get three-fourths of the $120$-mile gap.)
⭐ "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.
⭐ "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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