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2021 AMC Fall 10A Problem 4

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2021 AMC Fall 10A Problem 4AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Mr. Lopez has a choice of two routes to get to work. Route A is 66 miles long, and his average speed along this route is 3030 miles per hour. Route B is 55 miles long, and his average speed along this route is 4040 miles per hour, except for a 12\frac{1}{2} -mile stretch in a school zone where his average speed is 2020 miles per hour. By how many minutes is Route B quicker than Route A?

  1. A)

    2342 \frac{3}{4}

  2. B)

    3343 \frac{3}{4}

  3. C)

    4124 \frac{1}{2}

  4. D)

    5125 \frac{1}{2}

  5. E)

    6346 \frac{3}{4}

Answer

B

Key insight

Convert each leg to minutes: Route A takes 12 minutes, Route B takes 6.75 + 1.5 = 8.25 minutes, so the difference is 3.75.

Solution

Work in minutes, using time == distance ÷\div speed and 11 hour =60= 60 minutes.

Route A: 630\dfrac{6}{30} hour =15= \dfrac{1}{5} hour =12= 12 minutes.

Route B has two legs. The school zone is 12\tfrac12 mile at 2020 mph: 1/220=140\dfrac{1/2}{20} = \dfrac{1}{40} hour =1.5= 1.5 minutes. The remaining 4.54.5 miles at 4040 mph: 4.540=980\dfrac{4.5}{40} = \dfrac{9}{80} hour =6.75= 6.75 minutes. Total: 1.5+6.75=8.251.5 + 6.75 = 8.25 minutes.

Route B is quicker by 12−8.25=3.75=33412 - 8.25 = 3.75 = 3\tfrac{3}{4} minutes.

The answer is (B) 334\boxed{\textbf{(B)}\ 3\frac{3}{4}}.

Why this works

Different speeds on different parts of a trip mean separate time computations per leg; there is no shortcut through an "average" speed. Picking minutes as the unit from the start keeps the numbers clean and matches the form of the answer choices.

The trap

Using the full 5 miles at 40 mph and adding the school-zone time on top, instead of splitting Route B into a 4.5-mile stretch and a 0.5-mile stretch.

Common mistakes

  • Using the full 5 miles at 40 mph and adding the school-zone time on top, instead of splitting Route B into a 4.5-mile stretch and a 0.5-mile stretch.
  • Leaving times in hours (0.0625) and misconverting to minutes, or answering the Route B time itself.

Techniques

Set up the equation/formula and compute; no special trick needed