Problem 41 - Prealgebra

Question

A set of measuring cups has measures of 1 cup, \(\frac{3}{4}\) cup, \(\frac{1}{2} \operatorname{cup}, \frac{1}{3} \operatorname{cup},\) and \(\frac{1}{4}\) cup. How could you get \(\frac{1}{6}\) cup of milk by using these measures?

Step-by-Step Solution

Answer

Fill the \( \frac{1}{3} \) cup halfway to get \( \frac{1}{6} \) cup.

Step 1: Identify the Goal

We need to measure out exactly \( \frac{1}{6} \) cup of milk using the available measuring cups: 1 cup, \( \frac{3}{4} \) cup, \( \frac{1}{2} \) cup, \( \frac{1}{3} \) cup, and \( \frac{1}{4} \) cup.

Step 2: Find Compatible Fractions

To find \( \frac{1}{6} \) cup, we need to consider fractions that can combine to make \( \frac{1}{6} \). We know \( \frac{1}{3} \) can be split in half to get \( \frac{1}{6} \) because \( \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \).

Step 3: Use Available Measures

Since the \( \frac{1}{3} \) cup measure is available, we'll use the \( \frac{1}{3} \) cup and somehow measure half of it to get \( \frac{1}{6} \).

Step 4: Calculate Fractional Measurement

If we fill the \( \frac{1}{3} \) cup halfway, we effectively have \( \frac{1}{6} \) cup, because the operation is \( \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \), where the \( \frac{1}{2} \) represents filling the \( \frac{1}{3} \) cup halfway.

Key Concepts

Measuring Cups

Measuring cups are essential tools in both cooking and baking, allowing us to precisely measure ingredients. When working with recipes or tasks that require accuracy, using measuring cups can make a huge difference in the final result. Each measuring cup is marked with specific fractional amounts, such as 1 cup, \( \frac{3}{4} \) cup, \( \frac{1}{2} \) cup, and so on. This assists in ensuring the proper proportions are used.
Measuring small quantities, such as \( \frac{1}{6} \) cup in the given exercise, can be challenging without the exact measure. By understanding the fractions, we can manipulate other measuring cup sizes to achieve the desired amount.

• Knowing how to combine different cup measures is key.
• Identifying the need to split a measure for smaller, specific amounts aids in precision cooking.
• Accuracy means if you know halves and thirds, you can create sixths or other fractions.

Fraction Addition

Understanding fraction addition is crucial when combining different measuring cups to achieve the desired ingredient amount. Fractions represent parts of a whole and can be combined or decomposed to create new fractions when needed.When adding fractions, ensure they have a common denominator. This allows for seamless addition. For example, to add \( \frac{1}{3} \) and \( \frac{1}{2} \), the common denominator is 6. Therefore:

• Convert \( \frac{1}{3} \) to \( \frac{2}{6} \).
• Convert \( \frac{1}{2} \) to \( \frac{3}{6} \).
• Add: \[ \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \]

This approach can be applied to measuring ingredients, ensuring precise amounts. Even without a specific measuring cup for \( \frac{1}{6} \), understanding how fractions work allows you to achieve the accurate measure through addition or subtraction of available measures.

Fraction Multiplication

Fraction multiplication is not only useful in mathematical problems but also practical when using measuring cups for cooking. In the exercise, to achieve \( \frac{1}{6} \) cup using a \( \frac{1}{3} \) cup, you can multiply the \( \frac{1}{3} \) cup by \( \frac{1}{2} \), meaning fill it halfway.When multiplying fractions, multiply the numerators together and the denominators together:

• Numerator: \( 1 \times 1 = 1 \).
• Denominator: \( 3 \times 2 = 6 \).
• Result: \[ \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \]

This skill helps in recognizing how to use a part of a measure, for instance, filling half of a \( \frac{1}{3} \) cup to get the \( \frac{1}{6} \) needed. Mastering fraction multiplication can transform a basic understanding of cooking measurements, allowing for flexibility and creativity in the kitchen.