Problem 51 - Basic College Mathematics

Question

A recipe for piecrust calls for \(\frac{2}{3}\) cup of flour. A baker is making \(\frac{1}{2}\) of the recipe. How much flour should the baker use?

Step-by-Step Solution

Answer

The baker should use \( \frac{1}{3} \) cup of flour.

Step 1 - Understand the problem

The recipe requires \( \frac{2}{3} \) cup of flour for a full recipe. The baker is making only \( \frac{1}{2} \) of the recipe. The task is to find out how much flour is needed for this half recipe.

Step 2 - Set up the multiplication

To find how much flour is needed, multiply \(\frac{2}{3} \) by \(\frac{1}{2} \). This means: \(\frac{2}{3} \times \frac{1}{2} \).

Step 3 - Perform the multiplication

Multiply the numerators and the denominators separately: \(\frac{2}{3} \times \frac{1}{2} = \frac{2 \times 1}{3 \times 2} = \frac{2}{6} \).

Step 4 - Simplify the fraction

Simplify \(\frac{2}{6} \) by dividing both the numerator and the denominator by 2: \(\frac{2}{6} = \frac{1}{3} \).

Step 5 - State the result

So, the baker should use \( \frac{1}{3} \) cup of flour.

Key Concepts

Simplifying Fractions

When dealing with fractions, it's important to know how to simplify them.
This makes the fractions easier to work with and understand. Simplifying a fraction means reducing it to its smallest form.

To simplify a fraction, you divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
Let's look at the exercise example: After multiplying \( \frac{2}{3} \) by \( \frac{1}{2} \), we got the fraction \( \frac{2}{6} \).

To simplify \( \frac{2}{6} \), we find the GCD of 2 and 6, which is 2. Then, divide both the numerator and the denominator by 2:
\[ \frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3} \] Now, our fraction is in its simplest form: \( \frac{1}{3} \).

This fraction tells us that the baker needs \( \frac{1}{3} \) cup of flour, which is easier to measure and understand.

Recipe Math

When following or modifying recipes, math often becomes necessary. Understanding fractions and how to multiply them is crucial.
For instance, if you want to make a smaller or larger portion than what the recipe suggests, you'll need to adjust the ingredient amounts accordingly.

In our exercise, the original recipe called for \( \frac{2}{3} \) cup of flour, but the baker only wanted to make half of the recipe.
This means multiplying the flour amount by \( \frac{1}{2} \) to get the correct amount: \( \frac{2}{3} \) \( \times \) \( \frac{1}{2} \).

By doing this, we ensure that the final product has the right proportions. This kind of adjustment is common in baking and cooking, and it helps achieve the desired results whether you're scaling up or down.

Proportional Reasoning

Proportional reasoning is about understanding the relationship between different quantities. In the context of recipes, it means knowing how to adjust ingredient amounts based on changes in the recipe size.
This concept is essential for accurate cooking and baking.

In our example, the baker is working with half the original recipe. To keep the proportions correct, all ingredient amounts must be halved.
Therefore, if a full recipe needs \( \frac{2}{3} \) cup of flour, half the recipe needs that amount divided by 2.
This is done through fraction multiplication: \[ \frac{2}{3} \ \times \ \frac{1}{2} \] Multiplied together, this fraction reduces to \( \frac{1}{3} \).

Understanding proportional reasoning helps us make these adjustments smoothly, ensuring consistency in the final dish's taste and texture.