2007 AMC 10A Problem 6

P1-10~2 minPrint

Ratios, Percents & Averages Statistics & Data Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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)