2003 AMC 10B Problem 2

P1-10~1 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $\$1$ more than a pink pill, and Al's pills cost a total of $\$546$ for the two weeks. How much does one green pill cost?

Show solution without answering

Key insight

Divide by the 14 days first: one green plus one pink costs 39 dollars. Two prices that add to 39 and differ by 1 are 20 and 19.

Solution

1. Two weeks is $14$ days, and every day Al buys the same thing: one green and one pink pill. So first find one day's cost:
$546 \div 14 = 39 \text{ dollars}.$
2. So a green pill and a pink pill together cost $39$ dollars, and the green one costs $1$ dollar more.
3. If the two prices were equal, each would be $19.50$ . The green is $50$ cents above that and the pink is $50$ cents below, so green $= 20$ and pink $= 19$ . (With algebra: pink $= p$ , green $= p+1$ , so $2p + 1 = 39$ and $p = 19$ .)
4. Check: $20 + 19 = 39$ , and $14 \times 39 = 546$ .

The answer is $\boxed{\textbf{(D)}\ \$20}$ .

Why this works

The total is the same daily purchase repeated $14$ times, so dividing by $14$ shrinks the problem to one day. Then it is a "sum and difference" puzzle: the bigger number is (sum plus difference) divided by $2$ , here $(39+1)/2 = 20$ . When a total comes from repeating the same purchase, find the cost of one repeat first.

Concise solution

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

Two weeks is $14$ days, and each day Al buys exactly one pill of each color. The daily cost of the pair is therefore
$\frac{546}{14} = 39 \text{ dollars}.$

Let the pink pill cost $p$ dollars; the green pill costs $p + 1$ . Then $p + (p+1) = 39$ , so $2p = 38$ and $p = 19$ . The green pill costs $19 + 1 = 20$ dollars.

Check: $14 \cdot (20 + 19) = 14 \cdot 39 = 546$ .

The answer is $\boxed{\textbf{(D)}\ \$20}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)