2006 AMC 10B Problem 9

P1-10~1 minPrint

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

Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade?

Show solution without answering

Key insight

The batch weighs 600 grams and has 411 calories. A 200-gram glass is one third of the batch, so it has 411 divided by 3 = 137.

Solution

Weigh the whole batch: $100 + 100 + 400 = 600$ grams. The water has no calories, but it still counts toward the weight.
Count all the calories: $25$ from lemon juice, $386$ from sugar, and $0$ from water, for a total of $25 + 386 = 411$ calories.
The lemonade is mixed evenly, so every gram has the same share of calories. A $200$ -gram glass is $\frac{200}{600} = \frac{1}{3}$ of the batch.
So it has one third of the calories: $411 \div 3 = 137$ .

The answer is $\boxed{\textbf{(B)}\ 137}$ .

Why this works

In a well-mixed drink, any part of it has the same calories per gram as the whole batch. So find the totals for the whole batch (every ingredient's weight, even the ones with no calories), then take the right fraction.

Concise solution

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

Total mass of the lemonade: $100 + 100 + 400 = 600$ grams.

Total calories: $25$ from the lemon juice, $386$ from the sugar, $0$ from the water, so $411$ calories.

A $200$ -gram serving is $\frac{200}{600} = \frac{1}{3}$ of the batch, so it contains $\frac{411}{3} = 137$ calories.

The answer is $\boxed{\textbf{(B)}\ 137}$ .

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)