Question

helena needs 3.5 cups of flour per loaf of bread and 2.5 cups of flour per batch of muffins. she also needs 0.75 cup of sugar per loaf of bread and 0.75 cup of sugar per batch of muffins. helena has 17 cups of flour and 4.5 cups of sugar available for baking.
which combination of loaves of bread and batches of muffins could helena bake?
2 loaves of bread and 4 batches of muffins
3 loaves of bread and 3 batches of muffins
4 loaves of bread and 2 batches of muffins
5 loaves of bread and 1 batch of muffins

Explanation:

Step1: Define variables

Let \( b \) be the number of loaves of bread and \( m \) be the number of batches of muffins. The flour constraint is \( 3.5b + 2.5m \leq 17 \) and the sugar constraint is \( 0.75b + 0.75m \leq 4.5 \).

Step2: Simplify sugar constraint

Divide the sugar inequality by \( 0.75 \): \( b + m \leq 6 \).

Step3: Check each option

• Option 1: 2 loaves, 4 batches

Flour: \( 3.5(2) + 2.5(4) = 7 + 10 = 17 \) (≤17).
Sugar: \( 2 + 4 = 6 \) (≤6). Valid? Wait, but let's check others.

• Option 2: 3 loaves, 3 batches

Flour: \( 3.5(3) + 2.5(3) = 10.5 + 7.5 = 18 \) (>17). Invalid.

• Option 3: 4 loaves, 2 batches

Flour: \( 3.5(4) + 2.5(2) = 14 + 5 = 19 \) (>17). Invalid.

• Option 4: 5 loaves, 1 batch

Flour: \( 3.5(5) + 2.5(1) = 17.5 + 2.5 = 20 \) (>17). Invalid.

Wait, recheck Option 1: Flour is exactly 17, sugar is 6 (exactly 4.5? Wait no—wait, sugar per loaf/batch is 0.75. So 2 loaves: \( 0.75(2) = 1.5 \), 4 batches: \( 0.75(4) = 3 \). Total sugar: \( 1.5 + 3 = 4.5 \) (which matches). Flour: \( 3.5(2) = 7 \), \( 2.5(4) = 10 \), total 17. So it works. Wait, but earlier I thought Option 2's flour was 18, which is over 17. So Option 1 is valid. Wait, but let me re-express the sugar calculation correctly (per item, not sum of counts).

Correct sugar check for Option 1: \( 0.75(2) + 0.75(4) = 0.75(6) = 4.5 \) (good). Flour: \( 3.5(2) + 2.5(4) = 7 + 10 = 17 \) (good).

Option 2: Sugar: \( 0.75(3) + 0.75(3) = 4.5 \) (good). Flour: \( 10.5 + 7.5 = 18 \) (over 17). Invalid.

So only Option 1 satisfies both constraints. Wait, but the initial check for Option 1: flour is 17 (okay), sugar is 4.5 (okay). So it's valid.

Answer:

A. 2 loaves of bread and 4 batches of muffins