# 2007 AMC 10A Problems/Problem 6

## Problem

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

## Solution 1

We compute the percentage increases:

1. $\frac{66 - 60}{60} = 10\%$
2. $\frac{70 - 66}{66} \approx 6\%$
3. $\frac{76-70}{70} \approx 8.6\%$
4. $\frac{78-76}{76} \approx 2.6\%$
5. $\frac{85-78}{78} \approx 9\%$

The answer is $\mathrm{(A)}$.

In fact, the answer follows directly from examining the differences between each year. The largest differences are $6$ and $7$. Due to the decreased starting number of students between $2002$ and $2003$, that interval will be our answer.

~edited by mobius247

## Solution 2

We make the numerator 1 and compare the denominators.

1. $\frac{66 - 60}{60} = \frac{1}{10}$
1. $\frac{70 - 66}{66} = \frac{1}{16.5}$
1. $\frac{76-70}{70} \approx \frac{1}{11.7}$
1. $\frac{78-76}{76} = \frac{1}{38}$
1. $\frac{85-78}{78} \approx \frac{1}{11.1}$

The answer is $\mathrm{(A)}$.

~thatmathsguy

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 