Problem 79 - Elementary Algebra

Question

Solve the word problems. One of Arlene's recipes calls for \(3 \frac{1}{2}\) cups of milk. If she wants to make one-half of the recipe, how much milk should she use?

Step-by-Step Solution

Answer

Arlene should use \(1 \frac{3}{4}\) cups of milk.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(3 \frac{1}{2}\) to an improper fraction. Multiply the whole number part (3) by the denominator of the fraction (2) and add the numerator (1): \(3 \times 2 + 1 = 7\). So, the improper fraction is \(\frac{7}{2}\).

Step 2: Calculate Half of the Fraction

Next, find one-half of the fraction \(\frac{7}{2}\). To do this, multiply \(\frac{7}{2}\) by \(\frac{1}{2}\): \(\frac{7}{2} \times \frac{1}{2} = \frac{7}{4}\). This fraction represents the amount of milk needed.

Step 3: Convert Improper Fraction to Mixed Number

Now, convert the improper fraction \(\frac{7}{4}\) back to a mixed number. Divide 7 by 4, which gives 1 with a remainder of 3. So, \(\frac{7}{4}\) is equal to \(1 \frac{3}{4}\).

Key Concepts

Mixed Numbers to Improper Fractions

Understanding how to convert mixed numbers into improper fractions is essential when dealing with fraction word problems, like those you may encounter in recipes. A mixed number consists of a whole number and a fraction. For Arlene's recipe, she needs to use \(3 \frac{1}{2}\) cups of milk. Let's break down the conversion process to ensure clarity.

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction part. In Arlene's case, multiply \(3\) (the whole number) by \(2\) (the denominator) to get \(6\).
• Add this result to the numerator of the fraction. Here, add \(6\) to \(1\), resulting in \(7\).
• The improper fraction is formed by placing this sum over the original denominator, thus, \(\frac{7}{2}\).

This method helps in simplifying calculations since improper fractions are often easier to use in arithmetic operations, like multiplication.

Improper Fractions to Mixed Numbers

Converting improper fractions back to mixed numbers can make it easier to understand quantities. Once you have performed a calculation, like the one Arlene did to find out how much milk she needs for half her recipe, it might result in an improper fraction, such as \(\frac{7}{4}\). Here’s how you convert it back:

• Divide the numerator by the denominator. For \(\frac{7}{4}\), divide \(7\) by \(4\).
• This division results in a quotient that becomes the whole number (\(1\)) and a remainder that forms the numerator of the fraction part (\(3\)).
• Write the mixed number as \(1 \frac{3}{4}\), using the quotient as the whole number and the remainder over the original denominator.

Understanding both forms helps to visualize amounts better, especially in practical problems like cooking.

Fraction Multiplication

Fraction multiplication is a straightforward process that is a key skill when adjusting quantities in recipes. If you need to halve a recipe, as Arlene did, multiplying fractions can help. Take her fractional amount \(\frac{7}{2}\) and multiply it by \(\frac{1}{2}\) to find half the quantity.

To multiply fractions:

• Multiply the numerators together. For \(\frac{7}{2} \times \frac{1}{2}\), multiply \(7\) by \(1\) to get \(7\).
• Multiply the denominators together. Multiply \(2\) by \(2\) to get \(4\).
• Combine these results to form a new fraction: \(\frac{7}{4}\).

Multiply across numerators and denominators separately. It's efficient and ensures that operations with fractions remain manageable. This skill is especially handy when scaling recipes up or down.