AMC 10 · 2010 · #12
Grade 6 algebraAt the beginning of the school year, 50% of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and 50% answered "No." At the end of the school year, 70% answered "Yes" and 30% answered "No." Altogether, x% of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of x?
Pick an answer.
AMC 10 2010 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: At the start of the year, half of a class said "Yes" to loving math and half said "No." By the end, $70\%$ said "Yes" and $30\%$ said "No." Some students changed their answer between the two times; call that share $x\%$. Find the difference between the largest and smallest possible values of $x$.
Givens: At the start, $50\%$ said "Yes" and $50\%$ said "No"; At the end, $70\%$ said "Yes" and $30\%$ said "No"; $x\%$ of students gave a different answer at the end than at the start; Answer choices: (A) $0$, (B) $20$, (C) $40$, (D) $60$, (E) $80$
Unknowns: The difference between the maximum and minimum possible values of $x$
Understand
Restated: At the start of the year, half of a class said "Yes" to loving math and half said "No." By the end, $70\%$ said "Yes" and $30\%$ said "No." Some students changed their answer between the two times; call that share $x\%$. Find the difference between the largest and smallest possible values of $x$.
Givens: At the start, $50\%$ said "Yes" and $50\%$ said "No"; At the end, $70\%$ said "Yes" and $30\%$ said "No"; $x\%$ of students gave a different answer at the end than at the start; Answer choices: (A) $0$, (B) $20$, (C) $40$, (D) $60$, (E) $80$
Plan
Primary tool: #14 Extreme Principle
Secondary: #4 Introduce a Variable, #15 Organize Information in More Ways
The question asks for the biggest and smallest a quantity can be, which is the signature of Tool #14 (Extreme Principle): solve by finding the boundary cases. Tool #15 (Organize Information in More Ways) sorts every student into a four-way start-to-end table so nobody is double counted. Tool #4 (Introduce a Variable) names the one free choice — how many students flip from "Yes" to "No" — turning the switch count into a single expression whose two ends give the answer.
Execute — Answer: D
6.RP.A.3 Step 1 Sort students into four groups
- Track what each student says at the start and at the end.
- Everyone lands in exactly one of four groups: stayed Yes (Yes$\to$Yes), flipped Yes to No (Yes$\to$No), flipped No to Yes (No$\to$Yes), or stayed No (No$\to$No).
- The two flip groups are the students who changed, so $x\%$ is those two groups added together.
💡 Splitting the class into start-then-end groups turns "changed their answer" into an exact, countable set instead of a vague idea.
6.EE.B.6 Step 2 Name the flips and link them
- Let $s$ be the percent who flip Yes$\to$No.
- The Yes side must grow from $50\%$ to $70\%$, a net gain of $20$.
- Each No$\to$Yes flip adds one to Yes and each Yes$\to$No flip removes one, so (No$\to$Yes) minus (Yes$\to$No) must equal $20$.
- That forces No$\to$Yes $= s + 20$.
💡 The Yes total only shifts by the difference between students walking in and students walking out, so that difference is locked at $20$.
6.EE.B.6 Step 3 Write the switch count as one expression
Add the two flip groups to get the total percent who changed, then substitute No$\to$Yes $= s + 20$.
💡 With one variable controlling both flip groups, the whole answer rides on the single number $s$.
6.EE.B.8 Step 4 Push $s$ to its extremes
- Now apply the Extreme Principle to $s$.
- The smallest $s$ can be is $0$ — nobody is forced to flip Yes$\to$No.
- The largest is capped by the end: only $30\%$ say No at the end, and those No-sayers are the Yes$\to$No flippers plus the No$\to$No stayers.
- Setting the stayers to $0$ lets all $30\%$ be flippers, so $s \le 30$.
💡 Squeezing a group down to $0$ — either no forced flips, or no leftover stayers — is exactly what pins each boundary case.
6.EE.B.8 Step 5 Compare the extremes
- Plug the two boundary values of $s$ into $x = 2s + 20$.
- The minimum is at $s = 0$ and the maximum at $s = 30$.
- Subtract to get the difference the problem asks for.
💡 The gap between the biggest and smallest switch counts is the whole question — it comes out to $60$, choice (D).
6.RP.A.3 Track what each student says at the start and at the end. Everyone lands in exac 6.EE.B.6 Let $s$ be the percent who flip Yes$\to$No. The Yes side must grow from $50\%$ t 6.EE.B.6 Add the two flip groups to get the total percent who changed, then substitute No 6.EE.B.8 Now apply the Extreme Principle to $s$. The smallest $s$ can be is $0$ — nobody 6.EE.B.8 Plug the two boundary values of $s$ into $x = 2s + 20$. The minimum is at $s = 0 Review
Reasonableness: Test both extreme cases against every total. Minimum ($s = 0$): Yes$\to$No $= 0$, No$\to$Yes $= 20$, Yes$\to$Yes $= 50$, No$\to$No $= 30$. Start Yes $= 50+0 = 50$, end Yes $= 50+20 = 70$, end No $= 0+30 = 30$, and changers $= 0+20 = 20$. Maximum ($s = 30$): Yes$\to$No $= 30$, No$\to$Yes $= 50$, Yes$\to$Yes $= 20$, No$\to$No $= 0$. Start Yes $= 20+30 = 50$, end Yes $= 20+50 = 70$, end No $= 30+0 = 30$, and changers $= 30+50 = 80$. Both cases satisfy all four totals, and $80 - 20 = 60$, confirming (D).
Alternative: Reason without algebra. Minimum: the Yes share must climb $20$ points, so at least $20\%$ have to switch; leave everyone else unchanged and exactly $20\%$ change. Maximum: flip as many as possible — all $50\%$ who started No switch to Yes, and of the original $50\%$ Yes, $30\%$ switch to No while $20\%$ stay so the end shows $70\%$ Yes and $30\%$ No. That is $50 + 30 = 80\%$ changed. The difference is $80 - 20 = 60$, matching (D) by direct construction.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Reading the $50\%$/$70\%$/$30\%$ figures as parts of the whole class and sorting them into the four start-to-end groups.)6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Letting $s$ be the Yes$\to$No flippers, writing No$\to$Yes as $s + 20$, and building the switch count $x = 2s + 20$.)6.EE.B.8Write an inequality of the form x > c or x < c and graph on a number line (Bounding the free flip count with $0 \le s \le 30$ to locate the minimum and maximum values of $x$.)
⭐ Sort everyone into who-switched groups, notice the Yes side only moves by the net in-minus-out, then push the free flip count to its lowest and highest to bracket the answer.
⭐ Sort everyone into who-switched groups, notice the Yes side only moves by the net in-minus-out, then push the free flip count to its lowest and highest to bracket the answer.
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