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2007 AMC 10A Problem 1

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2007 AMC 10A Problem 1AMC 10 Step by Step
Name:                             Date: October 28, 2026
Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

One ticket to a show costs $20\$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?

  1. A)22
  2. B)55
  3. C)1010
  4. D)1515
  5. E)2020

Answer

C

Key insight

Turn each discount into a price per ticket first: 25% off 20 dollars is 15 dollars, and 30% off is 14 dollars. Then multiply by the number of tickets.

Solution

Work out what one ticket costs for each person, then multiply by how many tickets they buy.

  1. Susan's ticket price. A 25%25\% discount on $20\$20 takes off 14\frac{1}{4} of 2020, which is 55. So each ticket costs her 20−5=1520 - 5 = 15 dollars.
  2. Susan's total. She buys 44 tickets: 4×15=604 \times 15 = 60 dollars.
  3. Pam's ticket price. A 30%30\% discount on $20\$20 takes off 0.3×20=60.3 \times 20 = 6. So each ticket costs her 20−6=1420 - 6 = 14 dollars.
  4. Pam's total. She buys 55 tickets: 5×14=705 \times 14 = 70 dollars.
  5. Difference. Pam pays 70−60=1070 - 60 = 10 dollars more.

The answer is (C) 10\boxed{\textbf{(C)}\ 10}.

Why this works

A discount changes the price of each ticket, so find the new price of one ticket before doing anything else. After that it is ordinary multiplication and subtraction. Pam gets the bigger discount, but she buys one more ticket, and the extra ticket costs more than her savings.

The trap

Comparing the discounts or the ticket counts instead of the total dollars each person actually pays.

Concise solution

A 25%25\% discount means Susan pays 75%75\% of full price on each ticket:
4⋅20⋅0.75=4⋅15=60 dollars.4 \cdot 20 \cdot 0.75 = 4 \cdot 15 = 60 \text{ dollars}.
A 30%30\% discount means Pam pays 70%70\% of full price on each ticket:
5⋅20⋅0.70=5⋅14=70 dollars.5 \cdot 20 \cdot 0.70 = 5 \cdot 14 = 70 \text{ dollars}.
Pam pays 70−60=1070 - 60 = 10 dollars more.

The answer is (C) 10\boxed{\textbf{(C)}\ 10}.

Review in the AoPS books

Techniques

Set up the equation/formula and compute; no special trick needed