Problem 41 - Prealgebra

Question

CARPENTRY A 3-foot long shelf is to be installed between two walls that are \(32 \frac{5}{8}\) inches apart. How much of the shelf must be cut off so that it fits between the walls?

Step-by-Step Solution

Answer

Cut off 3 \(\frac{3}{8}\) inches from the shelf.

Step 1: Convert Feet to Inches

First, convert the length of the shelf from feet to inches. Since there are 12 inches in a foot, multiply 3 feet by 12 to get the length of the shelf in inches. \[3 \text{ feet} \times 12 \text{ inches per foot} = 36 \text{ inches}\]

Step 2: Determine the Shelf Length to Be Removed

Subtract the wall-to-wall distance from the shelf length (both in inches) to find out how much of the shelf needs to be cut off:\[36 \text{ inches} - 32 \frac{5}{8} \text{ inches}\]

Step 3: Perform the Subtraction

First, convert the mixed number to an improper fraction to make subtraction easier:\[32 \frac{5}{8} = \frac{256}{8} + \frac{5}{8} = \frac{261}{8}\]Now subtract this from the shelf length:\[36 = \frac{288}{8}\]Then,\[\frac{288}{8} - \frac{261}{8} = \frac{27}{8} \text{ inches}\]

Step 4: Simplify the Resulting Fraction

Convert the fraction \(\frac{27}{8}\) into a mixed number to express the length to cut off:\[\frac{27}{8} = 3 \frac{3}{8} \text{ inches}\]This means you need to cut off 3 and \(\frac{3}{8}\) inches from the shelf.

Key Concepts

Unit Conversion

When performing carpentry calculations like cutting a shelf to fit between two walls, understanding how to convert units of measurement is essential. In this exercise, we need to convert feet into inches. This is because the space between walls is given in inches, so the shelf’s length must also be in inches for accurate subtraction.Here’s the basic principle:- 1 foot is equal to 12 inches.- To convert feet to inches, multiply the number of feet by 12.
So, for a 3-foot long shelf, the calculation would be:\[3 \text{ feet} \times 12 \text{ inches per foot} = 36 \text{ inches}\]This tells us that the shelf is 36 inches long.
Remember, unit conversion allows us to perform calculations in the same units, which is crucial for precise measurement.

Mixed Numbers

Mixed numbers are used frequently in everyday mathematics, especially when dealing with measurements that include fractions. A mixed number combines a whole number part with a fraction part, like in the wall measurement of the exercise: \(32 \frac{5}{8}\).Here’s what to do with mixed numbers:- Convert them to improper fractions for easier arithmetic operations.- An improper fraction is where the numerator is greater than or equal to the denominator.
For example, to convert \(32 \frac{5}{8}\) into an improper fraction, you multiply the whole number by the denominator and add the numerator:\[32 \frac{5}{8} = \frac{32 \times 8 + 5}{8} = \frac{261}{8}\]Knowing how to convert mixed numbers is a practical skill for simplifying calculations, especially when dealing with real-life scenarios like cutting materials to fit specific dimensions.

Subtraction of Fractions

The subtraction of fractions might seem complex, but by following a few key steps, it becomes manageable. In this exercise, we are subtracting one length from another to find out how much to cut from a shelf.To perform subtraction with fractions, here’s what you should do:- Make sure the fractions have a common denominator.- Convert whole numbers to fractions. Here, 36 inches is converted to \(\frac{288}{8}\).
Given our improper fraction \(\frac{261}{8}\), the subtraction becomes:\[\frac{288}{8} - \frac{261}{8} = \frac{27}{8}\]To finish, you can simplify the fraction result, \(\frac{27}{8}\), into a mixed number for clarity:\[\frac{27}{8} = 3 \frac{3}{8}\]This shows that 3 and \(\frac{3}{8}\) inches need to be removed from the shelf. A systematic approach to fraction subtraction helps with precise and reliable results in carpentry and many other situations involving measurements.