AMC 10 · 2006 · #9

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

ratio-proportionrateunit-conversion dimensional-analysis ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

129

(B)

137

(C)

174

(D)

233

(E)

411

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Francesca mixes 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water. The lemon juice holds 25 calories, the sugar holds 386 calories, and the water holds none. Find how many calories are in a 200-gram serving of the mixed lemonade.

Givens: 100 grams of lemon juice with 25 calories; 100 grams of sugar with 386 calories; 400 grams of water with 0 calories; The serving to measure is 200 grams of the finished lemonade

Unknowns: The number of calories in 200 grams of the lemonade

Understand

Givens: 100 grams of lemon juice with 25 calories; 100 grams of sugar with 386 calories; 400 grams of water with 0 calories; The serving to measure is 200 grams of the finished lemonade

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

Calories are spread evenly through the lemonade, so what matters is the calories-per-gram rate. Find the total calories and the total grams of the whole batch, then scale that rate down to the 200-gram serving. Splitting the work into 'total calories' and 'total grams' keeps each number easy.

Execute — Answer: B

#7 Identify Subproblems 4.NBT.B.4 Step 1

Add up the calories

Only two ingredients carry calories.
The lemon juice has 25 and the sugar has 386, while the water has none.
Add them to get the calories in the whole batch.

$$25 + 386 + 0 = 411 \text{ calories}$$

💡 The whole batch holds the calories of its ingredients combined.

#7 Identify Subproblems 4.NBT.B.4 Step 2

Add up the grams

The batch is 100 grams of lemon juice plus 100 grams of sugar plus 400 grams of water.
Add the masses to find how much lemonade she made in total.

$$100 + 100 + 400 = 600 \text{ grams}$$

💡 Every gram poured in stays in, so the total mass is just the sum.

#8 Analyze the Units 6.RP.A.3 Step 3

Scale down to 200 grams

The whole 600-gram batch holds 411 calories, evenly spread.
A 200-gram serving is 200 out of 600 grams, which is one-third of the batch.
So it holds one-third of the calories: 411 divided by 3 is 137.
The answer is (B).

$$\frac{200}{600} = \frac{1}{3}, \quad 411 \times \frac{1}{3} = \frac{411}{3} = 137$$

💡 One-third of the lemonade carries one-third of the calories.

[1] #7 4.NBT.B.4 Only two ingredients carry calories. The lemon juice has 25 and the sugar has 38

[2] #7 4.NBT.B.4 The batch is 100 grams of lemon juice plus 100 grams of sugar plus 400 grams of

[3] #8 6.RP.A.3 The whole 600-gram batch holds 411 calories, evenly spread. A 200-gram serving i

Review

Reasonableness: The serving of 200 grams is one-third of the 600-gram batch, and 137 is very close to one-third of 411, so the size of the answer fits. Choice (E) 411 is the whole batch and (D) 233 is more than half, both too big for a one-third serving; (A) 129 and (C) 174 do not divide 411 evenly by 3. Only 137 gives exactly 411 back when tripled: 137 × 3 = 411.

Alternative: Find the rate first. Dividing 411 calories by 600 grams gives 0.685 calories per gram. Multiply by the 200-gram serving: 0.685 × 200 = 137 calories, the same result.

CCSS standards used (min grade 6)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Summing the ingredient calories to 411 and the ingredient masses to 600 grams)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Recognizing that 200 grams is one-third of the batch and taking one-third of the total calories)

⭐ When something is mixed evenly, a serving carries the same fraction of the calories as it is of the whole batch.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Add up the calories

Add up the grams

Scale down to 200 grams

Review

CCSS standards used (min grade 6)

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