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2009 AMC 10A Problem 8

2009 AMC 10A Problem 8AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$ % discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $\$6.00$ , is paying for everyone. How many dollars must he pay?

A)
$34$
B)
$36$
C)
$42$
D)
$46$
E)
$48$

Answer

Key insight

A senior ticket is 75% of full price, so full price is 6 / 0.75 = 8; then the six tickets cost 2(8) + 2(6) + 2(4).

Solution

A senior ticket is the full price with $25\%$ removed, i.e. $75\%$ of full price. Since that equals $\$6$ , the full price is
$\frac{6}{0.75} = 8 \text{ dollars}.$

Now price each generation:
- Middle generation (no discount): $2 \times 8 = 16$ .
- Oldest generation (seniors): $2 \times 6 = 12$ .
- Youngest generation (half price): $2 \times 4 = 8$ .

Total: $16 + 12 + 8 = 36$ dollars.

The answer is $\boxed{\textbf{(B)}\ 36}$ .

Why this works

All three ticket types are fixed fractions of one unknown, the full price, so recover that first. Reversing a discount means dividing by the retained fraction ( $0.75$ ), not multiplying by $1.25$ . Once the base price is known, the total is a short organized sum.

The trap

Recovering the full price as 6 x 1.25 = 7.50 instead of 6 / 0.75 = 8, because a 25% discount is not undone by adding 25%.

Common mistakes

Recovering the full price as 6 x 1.25 = 7.50 instead of 6 / 0.75 = 8, because a 25% discount is not undone by adding 25%.
Forgetting that there are two people in each generation, or treating $\$6$ as the full price.

Techniques

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