Francesca uses grams of lemon juice, grams of sugar, and grams of water to make lemonade. There are calories in grams of lemon juice and calories in grams of sugar. Water contains no calories. How many calories are in grams of her lemonade?
- A)
- B)
- C)
- D)
- E)
Answer
B
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: grams. The water has no calories, but it still counts toward the weight.
- Count all the calories: from lemon juice, from sugar, and from water, for a total of calories.
- The lemonade is mixed evenly, so every gram has the same share of calories. A -gram glass is of the batch.
- So it has one third of the calories: .
The answer is .
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.
The trap
Forgetting that the 400 grams of water still adds weight. Without it the batch seems to be 200 grams, and you would answer 411.
Concise solution
Total mass of the lemonade: grams.
Total calories: from the lemon juice, from the sugar, from the water, so calories.
A -gram serving is of the batch, so it contains calories.
The answer is .
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)
Techniques
Set up the equation/formula and compute; no special trick needed