AMC 10 · 2007 · #1
Grade 6 arithmeticOne ticket to a show costs $$$20$ at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?
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: One ticket costs $\$20$ at full price. Susan buys $4$ tickets with a coupon for $25\%$ off. Pam buys $5$ tickets with a coupon for $30\%$ off. Find how many more dollars Pam pays than Susan.
Givens: Full price of one ticket is $\$20$; Susan buys $4$ tickets with a $25\%$ discount; Pam buys $5$ tickets with a $30\%$ discount; Answer choices: (A) $2$, (B) $5$, (C) $10$, (D) $15$, (E) $20$
Unknowns: How many more dollars Pam pays than Susan
Understand
Restated: One ticket costs $\$20$ at full price. Susan buys $4$ tickets with a coupon for $25\%$ off. Pam buys $5$ tickets with a coupon for $30\%$ off. Find how many more dollars Pam pays than Susan.
Givens: Full price of one ticket is $\$20$; Susan buys $4$ tickets with a $25\%$ discount; Pam buys $5$ tickets with a $30\%$ discount; Answer choices: (A) $2$, (B) $5$, (C) $10$, (D) $15$, (E) $20$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
The question compares two people, so Tool #7 (Identify Subproblems) splits it into three clean pieces: what Susan pays, what Pam pays, and the difference between them. Inside each piece the work is the same shape — full-price total, then apply that buyer's percent discount. Tool #8 (Analyze the Units) keeps every number in dollars so the discount is taken on a dollar total and the final subtraction compares dollars to dollars.
Execute — Answer: C
4.OA.A.3 Step 1 Full-price cost for each buyer
At $\$20$ a ticket, Susan's $4$ tickets cost $4\times20=\$80$ before any discount, and Pam's $5$ tickets cost $5\times20=\$100$ before any discount.
💡 Same price on every ticket, so total cost is just price times number of tickets.
6.RP.A.3 Step 2 Apply Susan's 25% discount
A $25\%$ discount takes off a quarter of the price, so Susan pays $75\%$ of her $\$80$. One quarter of $80$ is $20$, so the discount is $\$20$ and Susan pays $80-20=\$60$.
💡 Percent means per hundred, so $25\%$ off a total is a quarter of that total taken away.
6.RP.A.3 Step 3 Apply Pam's 30% discount
A $30\%$ discount means Pam pays $70\%$ of her $\$100$. Since $30\%$ of $100$ is $30$, the discount is $\$30$ and Pam pays $100-30=\$70$.
💡 A percent of $100$ is that many dollars, because a percent is already a count per hundred.
4.OA.A.3 Step 4 Compare the two totals
- Pam pays $\$70$ and Susan pays $\$60$.
- The difference is $70-60=\$10$, so Pam pays $\$10$ more than Susan, which is choice (C).
- The trap choices come from comparing the wrong numbers: (E) $20$ is the gap in full-price totals ($100-80$) before any discount, and the smaller values appear if a discount is applied to a single ticket instead of the whole order.
💡 "How much more" is a subtraction of the two final amounts.
4.OA.A.3 At $\$20$ a ticket, Susan's $4$ tickets cost $4\times20=\$80$ before any discoun 6.RP.A.3 A $25\%$ discount takes off a quarter of the price, so Susan pays $75\%$ of her 6.RP.A.3 A $30\%$ discount means Pam pays $70\%$ of her $\$100$. Since $30\%$ of $100$ is 4.OA.A.3 Pam pays $\$70$ and Susan pays $\$60$. The difference is $70-60=\$10$, so Pam pa Review
Reasonableness: The numbers are small and easy to sanity-check: Susan pays $\$60$ and Pam pays $\$70$, both under their full-price totals of $\$80$ and $\$100$, and both are whole dollars because the percents divide evenly. Pam buys more tickets and still pays only $\$10$ more even with a bigger discount, which fits — her extra ticket adds cost while the deeper discount pulls it back down. The answer $\$10$ sits in the middle of the choices, matching the modest gap you'd expect.
Alternative: Work with the paid fractions directly instead of the discounts. Susan pays $75\%$ of $\$80$, which is $0.75\times80=\$60$; Pam pays $70\%$ of $\$100$, which is $0.70\times100=\$70$. Subtracting gives $70-60=\$10$, the same result.
CCSS standards used (min grade 6)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Finding each full-price total ($4\times20$ and $5\times20$) and taking the final difference $70-60$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Applying each percent discount to a dollar total: $25\%$ off $\$80$ leaves $\$60$, and $30\%$ off $\$100$ leaves $\$70$.)
⭐ Find each person's full-price total first, take their percent off, then subtract the two final prices to compare.
⭐ Find each person's full-price total first, take their percent off, then subtract the two final prices to compare.
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