Problem 58 - Prealgebra

Question

A piece of wood is \(1 \frac{9}{16}\) inches thick. A layer of padding \(\frac{15}{16}\) inch thick is placed on top. What is the total thickness of the wood and the padding? F. \(1 \frac{3}{8} \text { in }\) G. \(1 \frac{1}{2} \text { in. }\) H. \(1 \frac{24}{16} \mathrm{in.}\) J. \(2 \frac{1}{2} \mathrm{in.}\)

Step-by-Step Solution

Answer

The total thickness is \(2 \frac{1}{2} \mathrm{in.}\), option J.

Step 1: Convert Mixed Number to Improper Fractions

The piece of wood is given as a mixed number: \( 1 \frac{9}{16} \). First, convert this to an improper fraction by multiplying the whole number by the denominator and adding the numerator: \(1 \times 16 + 9 = 25\). Hence, \(1 \frac{9}{16} = \frac{25}{16}\).

Step 2: Represent the Padding as a Fraction

The padding thickness is already given as a simple fraction: \(\frac{15}{16}\).

Step 3: Add the Fractions

Now add the two fractions: \(\frac{25}{16}\) (wood) + \(\frac{15}{16}\) (padding). Since the denominators are the same, you can directly add the numerators: \(25 + 15 = 40\). This gives you a new fraction: \(\frac{40}{16}\).

Step 4: Simplify the Fraction

Simplify \(\frac{40}{16}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 8: \(\frac{40}{16} = \frac{40 \div 8}{16 \div 8} = \frac{5}{2}\).

Step 5: Convert the Improper Fraction to a Mixed Number

Convert the improper fraction \(\frac{5}{2}\) into a mixed number by dividing the numerator by the denominator: \(5 \div 2 = 2\) with a remainder of 1. This results in \(2 \frac{1}{2}\).

Step 6: Confirm the Answer with Given Options

Check the calculated mixed number \(2 \frac{1}{2}\) against the provided answer choices. It matches option J, which is \(2 \frac{1}{2} \mathrm{in.}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that include both a whole number and a fraction part. They are very useful in everyday situations where simplicity and clarity are needed. For example, a mixed number like \(1 \frac{9}{16}\) represents more than just one or two; it shows exactly how much more. Breaking it down, when you see a mixed number, you should understand:

• The whole number part, which tells you how many complete units there are.
• The fraction part, which explains how much extra there is beyond the whole number.

To convert a mixed number to an improper fraction, simply multiply the whole number by the denominator and add the numerator. This gives the new numerator, while the denominator remains unchanged. Understanding mixed numbers helps in measuring objects or ingredients that are not exactly at a whole number, making these situations easier to interpret.

Improper Fractions

An improper fraction is a fraction where the numerator is larger than the denominator. In simple terms, it tells you how many parts you have in total, where each part is smaller than a whole. The fraction \(\frac{25}{16}\) is an example, where you have 25 parts, each being 1/16th of a whole.
Improper fractions might look a bit strange because we're used to seeing numerators smaller than denominators. However, they are very useful, particularly for arithmetic operations like adding and subtracting fractions.
To convert an improper fraction into a mixed number, divide the numerator by the denominator. The quotient will give you the whole number part, while the remainder becomes the numerator of the fraction part. This allows you to express the number more clearly and understandably, especially for measurements and calculations.

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible. This involves reducing both the numerator and the denominator to the smallest possible values, while still maintaining the fraction's value. The goal is to make calculations easier and results more understandable.

• To simplify, find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both numbers by their GCD to get the simplified fraction.

In our example problem, the fraction \(\frac{40}{16}\) can be simplified by dividing both 40 and 16 by 8, resulting in \(\frac{5}{2}\). By simplifying fractions, especially after performing an operation like addition or subtraction, you ensure the result is presented in the simplest, and often most meaningful form. Simplifying fractions is an essential step in making mathematical results clearer and easier to use.