A recipe for pie crust calls for 2/3 cup of water for every 3 cups of flour. If you doubled the recipe, you would need 1 and 1/3 cups of water for 6 cups of flour. Which proportions involving complex fractions could represent these ratios?
The correct answer and explanation is:
To find the proportion involving complex fractions that represents the given ratios, let’s analyze the information step-by-step.
Given Ratios:
- Original recipe: 23 cups of water for 3 cups of flour\frac{2}{3} \text{ cups of water for } 3 \text{ cups of flour}
- Doubled recipe: 43 (which is 113) cups of water for 6 cups of flour\frac{4}{3} \text{ (which is } 1 \frac{1}{3}) \text{ cups of water for } 6 \text{ cups of flour}
Step 1: Write both as ratios (fractions):
- Original: 233\frac{\frac{2}{3}}{3}
- Doubled: 436\frac{\frac{4}{3}}{6}
These are complex fractions because the numerator is a fraction and the denominator is a whole number.
Step 2: Determine whether they are equivalent:
Let’s simplify both complex fractions:
Simplify Original:
233=23÷3=23×13=29\frac{\frac{2}{3}}{3} = \frac{2}{3} \div 3 = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}
Simplify Doubled:
436=43÷6=43×16=418=29\frac{\frac{4}{3}}{6} = \frac{4}{3} \div 6 = \frac{4}{3} \times \frac{1}{6} = \frac{4}{18} = \frac{2}{9}
Since both simplify to 29\frac{2}{9}, the proportions are equivalent, which means the complex fractions: 233=436\frac{\frac{2}{3}}{3} = \frac{\frac{4}{3}}{6}
are a valid representation of the ratio of water to flour in the original and doubled recipe.
✅ Correct Answer:
233=436\frac{\frac{2}{3}}{3} = \frac{\frac{4}{3}}{6}
Explanation:
This type of proportion is called a complex fraction because a fraction is divided by a whole number. These proportions are useful when comparing unit rates or scaling recipes.
In this example, the ratio of water to flour remains constant, meaning the recipe scales correctly. So, using complex fractions like: waterflour=233\frac{\text{water}}{\text{flour}} = \frac{\frac{2}{3}}{3}
and 436\frac{\frac{4}{3}}{6}
ensures that the proportion is preserved whether you are using the original or doubled recipe.