AMC 10 · 2007 · #6

Grade 7 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

percentageratio-proportionsystematic-enumeration systematic-enumeration ↑ Prerequisites: percentage

📏 Medium solution 💡 2 insights

Problem

The number of students taking the AMC 10 at one school was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and 85 in 2007. Looking only at consecutive years, decide which pair had the largest percentage increase in that number.

Pick an answer.

(A)

$2002\ \text{and}\ 2003$

(B)

$2003\ \text{and}\ 2004$

(C)

$2004\ \text{and}\ 2005$

(D)

$2005\ \text{and}\ 2006$

(E)

$2006\ \text{and}\ 2007$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Each answer choice is one of the five year-to-year intervals, so this is a finite compare-them-all problem — Tool #3 (Eliminate Possibilities) fits: evaluate every interval and keep the largest. Computing five separate percentage increases is Tool #7 (Identify Subproblems), one small ratio per interval. The subtle part is what 'percentage' measures — Tool #8 (Analyze the Units) keeps the focus on the increase as a fraction of the STARTING count, not the raw number of extra students, which is exactly the trap this problem sets.

1STEP 1

Fix what 'percentage increase' means

Percentage increase divides the rise by the starting count, so the same +6 is worth more from a smaller base.

percent increase=(new-old)/old × 100%

2STEP 2

Compute each interval's percentage increase

One ratio per interval: $\frac{6}{60}=10\%$ , $\frac{4}{66}\approx 6.1\%$ , $\frac{6}{70}\approx 8.6\%$ , $\frac{2}{76}\approx 2.6\%$ , $\frac{7}{78}\approx 9.0\%$ .

6/60=10%, 4/66≈ 6.1%, 6/70≈ 8.6%, 2/76≈ 2.6%, 7/78≈ 9.0%

3STEP 3

Compare the five percentages and choose the largest

Lined up: $10\%$ , $6.1\%$ , $8.6\%$ , $2.6\%$ , $9.0\%$ — the largest is 10%, from 2002 to 2003, so (A).

max{10%, 6.1%, 8.6%, 2.6%, 9.0%}=10% → (A)

Answer

2002 and 2003

The two real contenders are 2002-2003 and 2006-2007, since those have the largest jumps (+6 and +7). Compare their fractions directly instead of trusting the rounding: 6/60=1/10 and 7/78. Cross-multiplying, 6 × 78=468 while 60 × 7=420, and 468 ＞ 420, so 6/60 ＞ 7/78. The 2002-2003 interval genuinely wins, confirming (A). Every other interval has a smaller or equal numerator over a larger denominator, so none can beat 10%.

💡Key takeaway

Percentage growth means the increase divided by where you started, so a smaller starting count can beat a bigger raw jump.

Fix what 'percentage increase' means
Compute each interval's percentage increase
Compare the five percentages and choose the largest

Fix what 'percentage increase' means

Compute each interval's percentage increase

Compare the five percentages and choose the largest

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