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2007 AMC 10B Problem 14

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2007 AMC 10B Problem 14AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P11-15

Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40%40\% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30%30\% of the group are girls. How many girls were initially in the group?

  1. A)

    44

  2. B)

    66

  3. C)

    88

  4. D)

    1010

  5. E)

    1212

Answer

C

Key insight

The group size never changes, so the drop from 40% to 30% of the same total equals the two girls who left: 10% of the group is 2.

Solution

Let nn be the number of people in the group. Two girls leave and two boys arrive, so nn does not change.

Initially there are 0.4n0.4n girls; afterward there are 0.4n−20.4n - 2 girls, and this is 30%30\% of the same group:
0.4n−2=0.3n⟹0.1n=2⟹n=20.0.4n - 2 = 0.3n \quad\Longrightarrow\quad 0.1n = 2 \quad\Longrightarrow\quad n = 20.
The initial number of girls is 0.4⋅20=80.4 \cdot 20 = 8.

Check: 88 girls out of 2020 is 40%40\%; after the swap, 66 girls out of 2020 is 30%30\%.

The answer is (C) 8\boxed{\textbf{(C)}\ 8}.

Why this works

Percentages are only comparable when you know what they are percentages of. Noticing that the total is unchanged lets the two percentages be subtracted directly: a 1010 percentage-point drop equals two people, so the group has 2020 people. Look for the invariant quantity before writing any equation.

Alternative approach

Test the choices. With 88 girls at 40%40\% the group has 2020 people; after the swap 6/20=30%6/20 = 30\%. The other choices give group sizes 10,15,25,3010, 15, 25, 30, and none produce 30%30\% after removing two girls.

The trap

Changing the group total after the swap (it stays the same since two leave and two arrive) or reporting the final number of girls, 6.

Common mistakes

  • Changing the group total after the swap (it stays the same since two leave and two arrive) or reporting the final number of girls, 6.
  • Setting 0.4n−2=0.3(n−2)0.4n - 2 = 0.3(n - 2) or similar, as if the group shrank by two.

Techniques

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