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2007 AMC 10A Problem 6

2007 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: October 28, 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

A percent increase compares the gain to the starting number. From 60 to 66 is exactly 10%, and every other year's gain is less than 10% of its start.

Solution

Percent increase = (gain) $\div$ (number in the earlier year). A big gain on a big starting number can still be a small percent.

List the gains and starting numbers.

Years	Start	Gain	Gain $\div$ Start
2002 to 2003	$60$	$6$	$\frac{6}{60}$
2003 to 2004	$66$	$4$	$\frac{4}{66}$
2004 to 2005	$70$	$6$	$\frac{6}{70}$
2005 to 2006	$76$	$2$	$\frac{2}{76}$
2006 to 2007	$78$	$7$	$\frac{7}{78}$

Use $10\%$ as a yardstick. The first one is $\frac{6}{60} = \frac{1}{10}$ , exactly $10\%$ .
Compare each of the others with $10\%$ . Ten percent of the starting number is $6.6$ , $7$ , $7.6$ and $7.8$ . The gains $4$ , $6$ , $2$ and $7$ are each smaller than those. So every other year is below $10\%$ .
The largest percent increase is from $2002$ to $2003$ .

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

Why this works

Percent change depends on the gain and on what you started with. The biggest gain in students ( $7$ ) came on the biggest start ( $78$ ), so it is not the biggest percent. Comparing every fraction with an easy benchmark like $10\%$ saves you from working out five decimals.

The trap

Picking the year with the biggest gain in students (7, from 2006 to 2007) instead of the biggest gain compared with the starting number.

Concise 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 $\boxed{\textbf{(A)}\ 2002\ \text{and}\ 2003}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Prealgebra, Chapter 13: Data and Statistics
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)

Techniques

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