Problem 73 - Prealgebra

Question

A pair of jeans is \(32 \frac{1}{2}\) inches long. How long are the jeans after they have been washed if they shrink \(1 \frac{1}{3}\) inches?

Step-by-Step Solution

Answer

The jeans are 31 \(\frac{1}{6}\) inches long after being washed.

Step 1: Understand the Problem

We are given the original length of a pair of jeans as \(32 \frac{1}{2}\) inches, and we know they shrink by \(1 \frac{1}{3}\) inches. The task is to find the new length of the jeans after the shrinkage.

Step 2: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers into improper fractions to make calculations easier. \(32 \frac{1}{2}\) becomes \(\frac{65}{2}\) (since \(32 \times 2 + 1 = 65\)). Similarly, \(1 \frac{1}{3}\) becomes \(\frac{4}{3}\) (since \(1 \times 3 + 1 = 4\)).

Step 3: Subtract the Fractions

To find the new length, subtract the shrinkage from the original length: \(\frac{65}{2} - \frac{4}{3}\). To do this, find a common denominator, which is 6. Thus, convert \(\frac{65}{2}\) to \(\frac{195}{6}\) and \(\frac{4}{3}\) to \(\frac{8}{6}\). Now perform the subtraction: \(\frac{195}{6} - \frac{8}{6} = \frac{187}{6}\).

Step 4: Convert the Resulting Fraction Back to a Mixed Number

Convert \(\frac{187}{6}\) back to a mixed number. Divide 187 by 6 which gives a quotient of 31 and a remainder of 1. So, \(\frac{187}{6} = 31 \frac{1}{6}\).

Step 5: State the Final Result

After performing the subtraction and conversion, the new length of the jeans after they have been washed is \(31 \frac{1}{6}\) inches.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often seen in everyday contexts, such as measuring ingredients or lengths. For example, the mixed number \(32 \frac{1}{2}\) represents 32 whole units plus half of a unit. Converting mixed numbers into improper fractions can simplify operations like addition and subtraction.

To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fraction part and then add the numerator. This sum becomes the new numerator while the denominator remains unchanged.

• Example: Convert \(3 \frac{2}{5}\) to an improper fraction.
• Multiply: \(3 \times 5 = 15\)
• Add: \(15 + 2 = 17\)
• The improper fraction is \(\frac{17}{5}\).

Improper Fractions

Improper fractions have numerators that are equal to or greater than their denominators. This makes them particularly useful for performing arithmetic operations, as they eliminate the complexity of mixed numbers.

For example, take the improper fraction \(\frac{65}{2}\). The numerator, 65, is greater than the denominator, 2.
To perform arithmetic like subtraction with fractions, they often need to have a common denominator.

When dealing with subtraction, like \(\frac{65}{2} - \frac{4}{3}\), both fractions must be expressed with the same denominator. This process involves:

• Finding the least common multiple of the denominators (here, 2 and 3).
• Converting each fraction to an equivalent fraction with this common denominator.

Unit Conversion

Unit conversion is the process of changing a measure to another unit without altering its value.

In fraction operations, converting mixed numbers to improper fractions is a form of unit conversion that helps simplify arithmetic operations. Ensuring all measurements are in a comparable form is key, similar to how you might change meters to centimeters to work with common units in a calculation.

In our example, converting \(32 \frac{1}{2}\) inches to \(\frac{65}{2}\) inches is necessary for straightforward subtraction when finding the new jean length.

Unit conversion not only simplifies calculations but also helps keep track of units in real-world problems.
This approach allows you to ensure all your measures relate correctly for comparison or computation.

Length Measurement

Length measurement is a way of quantifying how long or tall something is. It's frequently expressed in units like inches, feet, and meters.

When measuring or considering changes in length, precision is key—especially in practical situations like clothing measurements.

In situations involving shrinkage, as illustrated with the jeans, precision becomes critical for accurate results. Every fractional inch matters.

For example, knowing that a pair of jeans shrinks by \(1 \frac{1}{3}\) inches is crucial for deciding the right fit. Quite often, we deal with length measurements using fractions and mixed numbers because real-world measurements are rarely neat whole numbers.

This highlights the importance of understanding and correctly manipulating fractions and mixed numbers in calculations involving changes in length.