2014 AMC 10A Problem 3

P1-10~2 minPrint

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

Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $\$2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$0.75$ for her to make. In dollars, what is her profit for the day?

Show solution without answering

Key insight

Follow the loaves in three groups, 24, 16 and 8, to get 88 dollars of sales. Then subtract what all 48 loaves cost to make, 36 dollars.

Solution

Profit means money taken in minus money spent. Work out each part separately.

Money taken in. Follow the loaves through the day.

Morning: half of $48$ is $24$ loaves. At $\$2.50$ each, that is $24 \cdot 2.50 = \$60$ .
Afternoon: $48 - 24 = 24$ loaves are left. Two thirds of $24$ is $16$ loaves. Half price is $\$1.25$ , so she takes in $16 \cdot 1.25 = \$20$ .
Late afternoon: $24 - 16 = 8$ loaves are left, at $\$1$ each, so $\$8$ .
Total taken in: $60 + 20 + 8 = \$88$ .

Money spent. She made all $48$ loaves at $\$0.75$ each: $48 \cdot 0.75 = \$36$ .

Profit. $88 - 36 = 52$ dollars.

The answer is $\boxed{\textbf{(E)}\ 52}$ .

Why this works

A problem where things are sold in stages is just careful bookkeeping: at each stage, write down how many are left and what price they sell for. The one idea to remember is that profit subtracts the cost of everything that was made, not only the items sold at full price.

Concise solution

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

Follow the loaves through the day.

Morning: half of $48$ is $24$ loaves at $\$2.50$ , earning $24 \cdot 2.50 = \$60$ .
Afternoon: $24$ loaves remain; two thirds of them is $16$ loaves at half price, $\$1.25$ , earning $16 \cdot 1.25 = \$20$ .
Late afternoon: the last $8$ loaves at $\$1$ each earn $\$8$ .

Total revenue: $60 + 20 + 8 = \$88$ .

Every loaf cost $\$0.75$ to make, so the total cost is $48 \cdot 0.75 = \$36$ .

Profit $= 88 - 36 = 52$ . The answer is $\boxed{\textbf{(E)}\ 52}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)