Questions: A recipe calls for 1/2 cup of ingredient A for every 1 2/5 cups of ingredient B. You use 3 cups of ingredient A. How many cups of ingredient B do you need? You need cups of ingredient B. (Simplify your answer. Type an integer, proper fraction, or mixed number.)

Solution Steps

To solve this problem, we need to determine the ratio of ingredient A to ingredient B and then use that ratio to find out how many cups of ingredient B are needed when 3 cups of ingredient A are used.

1. Convert the mixed number for ingredient B to an improper fraction.
2. Determine the ratio of ingredient A to ingredient B.
3. Use the ratio to calculate the required amount of ingredient B when 3 cups of ingredient A are used.

The recipe calls for \( \frac{1}{2} \) cup of ingredient A for every \( 1 \frac{2}{5} \) cups of ingredient B. First, we convert the mixed number \( 1 \frac{2}{5} \) to an improper fraction: \[ 1 \frac{2}{5} = \frac{5 \cdot 1 + 2}{5} = \frac{7}{5} \]

Next, we find the ratio of ingredient A to ingredient B: \[ \text{Ratio} = \frac{\frac{1}{2}}{\frac{7}{5}} = \frac{1}{2} \cdot \frac{5}{7} = \frac{5}{14} \]

Now, we need to calculate how many cups of ingredient B are needed when using 3 cups of ingredient A: \[ \text{Ingredient B needed} = \frac{\text{Ingredient A used}}{\text{Ratio}} = \frac{3}{\frac{5}{14}} = 3 \cdot \frac{14}{5} = \frac{42}{5} \] To express \( \frac{42}{5} \) as a mixed number: \[ \frac{42}{5} = 8 \frac{2}{5} \]

Final Answer