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2007 AMC 10B Problem 16

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2007 AMC 10B Problem 16AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P16-20

A teacher gave a test to a class in which 10%10\% of the students are juniors and 90%90\% are seniors. The average score on the test was 84.84. The juniors all received the same score, and the average score of the seniors was 83.83. What score did each of the juniors receive on the test?

  1. A)

    8585

  2. B)

    8888

  3. C)

    9393

  4. D)

    9494

  5. E)

    9898

Answer

C

Key insight

The class average is a weighted average: 0.1 j + 0.9(83) = 84, so the juniors' common score is j = 93.

Solution

The class size does not matter, so suppose there are 1010 students: 11 junior and 99 seniors.

The class total is 10⋅84=84010 \cdot 84 = 840 points. The seniors contribute 9⋅83=7479 \cdot 83 = 747 points. The junior's score is the rest:
840−747=93.840 - 747 = 93.

Since every junior scored the same, each junior received 9393.

The answer is (C) 93\boxed{\textbf{(C)}\ 93}.

Why this works

An overall average is a weighted average of group averages, with weights equal to the group sizes. Because only proportions are given, choosing a convenient class size (here 1010) makes the arithmetic concrete. Equivalently, the seniors are 11 point below the mean and outnumber the juniors 99 to 11, so the juniors must sit 99 points above it.

Alternative approach

Balance the deviations from 8484: the 90%90\% of seniors each contribute −1-1, total −0.9-0.9 per student overall, so the 10%10\% of juniors must contribute +0.9+0.9: 0.1(j−84)=0.90.1(j - 84) = 0.9, giving j=93j = 93.

The trap

Averaging the two group averages evenly, or reasoning 'seniors are 1 below, so juniors are 1 above' and answering 85.

Common mistakes

  • Averaging the two group averages evenly, or reasoning 'seniors are 1 below, so juniors are 1 above' and answering 85.
  • Weighting backwards (juniors at 90%90\%), which gives j≈84.1j \approx 84.1 and no matching choice.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer