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2007 AMC 10A Problem 6

2007 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ and 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase?

A)
$2002 \text{and} 2003$
B)
$2003 \text{and} 2004$
C)
$2004 \text{and} 2005$
D)
$2005 \text{and} 2006$
E)
$2006 \text{and} 2007$

Answer

Key insight

Percent increase is the change divided by the starting value; 6/60 = 10% beats every other year's ratio, which are all under 10%.

Solution

Percent increase is the gain divided by the earlier year's count. The yearly gains are $6, 4, 6, 2, 7$ , so the five ratios are
$\frac{6}{60},\quad \frac{4}{66},\quad \frac{6}{70},\quad \frac{2}{76},\quad \frac{7}{78}.$
The first equals exactly $10\%$ . Each of the others is less than $10\%$ : $\frac{4}{66} < \frac{4}{40}$ , $\frac{6}{70} < \frac{6}{60}$ , $\frac{2}{76}$ is tiny, and $\frac{7}{78} < \frac{7}{70} = 10\%$ .

So the biggest percentage jump is from $2002$ to $2003$ .

The answer is $\boxed{\textbf{(A)}\ 2002\ \text{and}\ 2003}$ .

Why this works

Percent change depends on both the size of the change and the base it is measured against. Here the largest raw gain ( $7$ ) sits on the largest base ( $78$ ), while a slightly smaller gain ( $6$ ) sits on the smallest base ( $60$ ) and wins. Comparing each fraction to a clean benchmark like $10\%$ avoids computing five decimals.

The trap

Comparing raw increases (7 from 2006 to 2007 is the largest) instead of increases relative to the starting number.

Common mistakes

Comparing raw increases ( $7$ from 2006 to 2007 is the largest) instead of increases relative to the starting number.
Dividing by the later year's count rather than the earlier one, which changes the percentages and can reorder close cases.

Techniques

Bound the quantity above/below or estimate to pin it down · Set up the equation/formula and compute; no special trick needed