AMC 10 · 2011 · #2
Grade 6 arithmeticJosanna's test scores to date are 90,80,70,60, and 85. Her goal is to raise here test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
Pick an answer.
AMC 10 2011 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: Josanna has five test scores: $90, 80, 70, 60,$ and $85$. She wants her average, after taking one more test, to be at least $3$ points higher than her current average. Find the smallest score on that next test that makes this happen.
Givens: Current test scores: $90, 80, 70, 60, 85$ (five tests); She will take one more test, making six tests in all; The new six-test average must be at least $3$ points above the current five-test average; Answer choices: (A) $80$, (B) $82$, (C) $85$, (D) $90$, (E) $95$
Unknowns: The minimum score on the sixth test that raises the average by at least $3$ points
Understand
Restated: Josanna has five test scores: $90, 80, 70, 60,$ and $85$. She wants her average, after taking one more test, to be at least $3$ points higher than her current average. Find the smallest score on that next test that makes this happen.
Givens: Current test scores: $90, 80, 70, 60, 85$ (five tests); She will take one more test, making six tests in all; The new six-test average must be at least $3$ points above the current five-test average; Answer choices: (A) $80$, (B) $82$, (C) $85$, (D) $90$, (E) $95$
Plan
Primary tool: #11 Work Backwards
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
The goal is stated in terms of the final average, and we must recover a single missing score. Tool #11 (Work Backwards) runs the average formula in reverse: from the target average, find the total the six tests must reach, then subtract what she already has. Tool #7 (Identify Subproblems) splits the job into current-average, target-total, and needed-score. Tool #3 (Eliminate Possibilities) uses the answer choices to confirm that anything below $95$ falls short.
Execute — Answer: E
6.SP.B.5 Step 1 Find the current average
- Add the five scores, then divide by $5$.
- $90+80+70+60+85 = 385$, and $385\div5 = 77$.
- So her current average is $77$.
💡 An average is just the total shared equally across all the tests.
6.EE.B.5 Step 2 Set the target average
- Raising the average by at least $3$ points means the new average must be at least $77+3=80$.
- To make the next score as small as possible, aim for exactly $80$.
💡 The lowest passing score comes from just barely reaching the target, not overshooting it.
6.SP.B.5 Step 3 Work backwards to the six-test total
- After the next test she has $6$ tests.
- For their average to be $80$, the six scores must add to $80\times6=480$.
- This is the total she needs to reach.
💡 If the average must be $80$ over six tests, the points must pile up to $80$ six times over.
4.OA.A.3 Step 4 Subtract to find the needed score
- She already has $385$ points from five tests.
- The sixth score must cover the gap up to $480$: $480-385=95$.
- Because "at least $3$ points" allows going higher, any score of $95$ or more works, so the minimum is $95$ — choice (E).
- Every smaller choice, $80, 82, 85, 90$, leaves the six-test total under $480$ and misses the goal.
💡 The missing score is exactly the leftover distance between what she has and the total she needs.
6.SP.B.5 Add the five scores, then divide by $5$. $90+80+70+60+85 = 385$, and $385\div5 = 6.EE.B.5 Raising the average by at least $3$ points means the new average must be at leas 6.SP.B.5 After the next test she has $6$ tests. For their average to be $80$, the six sco 4.OA.A.3 She already has $385$ points from five tests. The sixth score must cover the gap Review
Reasonableness: Check the winning score directly: with a $95$, the six scores total $385+95=480$, and $480\div6=80$, which is exactly $77+3$ — the average rose by $3$. Try the next choice down, $90$: the total is $475$ and the average is $475\div6\approx79.2$, only about $2.2$ points up, short of the goal. So $95$ is genuinely the smallest score that works, confirming (E).
Alternative: Introduce a variable: let $x$ be the next score and require $\dfrac{385+x}{6}\ge 80$. Multiply both sides by $6$ to get $385+x\ge 480$, so $x\ge 95$. The smallest integer meeting this is $95$, the same answer (E).
CCSS standards used (min grade 6)
6.SP.B.5Summarize numerical data sets in relation to their context, including measures of center such as the mean (Computing the current average of the five scores and turning the target average into the six-test total.)6.EE.B.5Understand solving an inequality as finding which values make it true (Reading "raise the average at least 3 points" as a minimum condition and aiming for the exact threshold.)4.OA.A.3Solve multistep word problems using the four operations (Subtracting the current total from the needed total to find the missing sixth score.)
⭐ To find one missing score for a target average, figure out the total all the tests must reach, then subtract the points you already have.
⭐ To find one missing score for a target average, figure out the total all the tests must reach, then subtract the points you already have.
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