2009 AMC 10A Problem 8

P1-10~2 minPrint

Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

A senior ticket is 75% of the full price, so the full price is 6 divided by 0.75, which is 8 dollars; a child ticket is half of that.

Solution

All three kinds of ticket are based on the same full price, so find the full price first.

1. A senior gets $25\%$ off, so a senior pays $75\%$ of the full price. That is $\$6$ :
$0.75 \times (\text{full price}) = 6, \qquad \text{full price} = \frac{6}{0.75} = 8 \text{ dollars}.$
2. A child gets $50\%$ off, so a child ticket costs half of $8$ , which is $4$ dollars.
3. Add up two tickets of each kind:

Generation	Price each	Two tickets
Middle (full price)	$8$	$16$
Oldest (senior)	$6$	$12$
Youngest (child)	$4$	$8$

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

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

Why this works

When every price is a fixed part of one unknown price, find that unknown first. To undo a discount, divide by the part that is left ( $0.75$ ); adding the $25\%$ back on to $6$ gives $7.50$ , because $25\%$ of $6$ is smaller than $25\%$ of $8$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(B)}\ 36}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)