AMC 10 · 2007 · #6
Grade 7 arithmeticAt 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?
Pick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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. Decide which pair of consecutive years had the largest percentage increase in that number.
Givens: The counts by year: $2002{:}\,60$, $2003{:}\,66$, $2004{:}\,70$, $2005{:}\,76$, $2006{:}\,78$, $2007{:}\,85$; The five consecutive-year intervals are the answer choices: (A) 2002-2003, (B) 2003-2004, (C) 2004-2005, (D) 2005-2006, (E) 2006-2007
Unknowns: Which single one-year interval has the greatest percentage increase
Understand
Restated: 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. Decide which pair of consecutive years had the largest percentage increase in that number.
Givens: The counts by year: $2002{:}\,60$, $2003{:}\,66$, $2004{:}\,70$, $2005{:}\,76$, $2006{:}\,78$, $2007{:}\,85$; The five consecutive-year intervals are the answer choices: (A) 2002-2003, (B) 2003-2004, (C) 2004-2005, (D) 2005-2006, (E) 2006-2007
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #7 Identify Subproblems, #8 Analyze the Units
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.
Execute — Answer: A
7.RP.A.3 Step 1 Fix what 'percentage increase' means
- Percentage increase from one year to the next is the increase in students divided by the starting count, written as a percent: $\frac{\text{new}-\text{old}}{\text{old}}\times 100\%$.
- The key point is the denominator: the same jump of $+6$ students counts for MORE when the starting number is small than when it is large.
- So the biggest jump in raw students need not be the biggest percentage increase.
💡 A gain feels bigger when you started with less, so we always divide by the earlier year's count.
6.RP.A.3 Step 2 Compute each interval's percentage increase
- Do the five intervals one at a time.
- (A) 2002-2003: $\frac{66-60}{60}=\frac{6}{60}=\frac{1}{10}=10\%$.
- (B) 2003-2004: $\frac{70-66}{66}=\frac{4}{66}\approx 6.1\%$.
- (C) 2004-2005: $\frac{76-70}{70}=\frac{6}{70}\approx 8.6\%$.
- (D) 2005-2006: $\frac{78-76}{76}=\frac{2}{76}\approx 2.6\%$.
- (E) 2006-2007: $\frac{85-78}{78}=\frac{7}{78}\approx 9.0\%$.
💡 Turning each interval into one small fraction lets you read its growth off a single number.
6.NS.B.3 Step 3 Compare the five percentages and choose the largest
- Line up the results: $10\%,\ 6.1\%,\ 8.6\%,\ 2.6\%,\ 9.0\%$.
- The largest is $10\%$, from 2002 to 2003.
- Notice the trap: the biggest jump in actual students is the $+7$ from 2006 to 2007, but its starting count of $78$ is large, so it is only about $9.0\%$ — just short of the $\frac{6}{60}=10\%$ from the smaller base of $60$.
- The largest percentage increase is between 2002 and 2003, which is choice (A).
💡 Once every interval is a percent, the winner is just the biggest number in the list.
7.RP.A.3 Percentage increase from one year to the next is the increase in students divide 6.RP.A.3 Do the five intervals one at a time. (A) 2002-2003: $\frac{66-60}{60}=\frac{6}{6 6.NS.B.3 Line up the results: $10\%,\ 6.1\%,\ 8.6\%,\ 2.6\%,\ 9.0\%$. The largest is $10\ Review
Reasonableness: The two real contenders are 2002-2003 and 2006-2007, since those have the largest jumps ($+6$ and $+7$). Compare their fractions directly instead of trusting the rounding: $\frac{6}{60}=\frac{1}{10}$ and $\frac{7}{78}$. Cross-multiplying, $6\times 78=468$ while $60\times 7=420$, and $468>420$, so $\frac{6}{60}>\frac{7}{78}$. 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 $10\%$.
Alternative: Because a $10\%$ mark is easy to test, screen the intervals against it directly (Tool #3): does the increase reach one tenth of the starting count? 2002-2003 needs $\ge 6$ and gets exactly $6$ — yes, $10\%$. For every other interval, one tenth of the start is $6.6$, $7.0$, $7.6$, or $7.8$, but the actual increases are only $4$, $6$, $2$, and $7$ — all below their thresholds. So only 2002-2003 reaches $10\%$, and it is the largest without computing every decimal.
CCSS standards used (min grade 7)
7.RP.A.3Use proportional relationships to solve multistep ratio and percent problems, including percent increase (Defining percentage increase as the gain divided by the starting count, the core comparison the problem asks for.)6.RP.A.3Use ratio and rate reasoning, including finding a percent as a rate per 100 (Turning each interval's increase-over-base into a percent, one small fraction per interval.)6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals (Evaluating and comparing the five ratios (including the cross-multiplication check $6\times 78$ vs $60\times 7$) to pick the largest.)
⭐ Percentage growth means the increase divided by where you started, so a smaller starting count can beat a bigger raw jump.
⭐ Percentage growth means the increase divided by where you started, so a smaller starting count can beat a bigger raw jump.
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