Problem 48 - Prealgebra

Question

Building Shelves You are building a small bookcase. You need three shelves, each with a length of \(4 \frac{7}{8}\) ft. You bought a piece of wood that is \(15 \mathrm{ft}\) long. Will this board be long enough?

Step-by-Step Solution

Answer

Yes, the 15 ft board is long enough to make the three shelves.

Step 1: Convert Mixed Number to Improper Fraction

The given length of each shelf is \(4 \frac{7}{8}\) ft. First, convert this mixed number to an improper fraction. Multiply the whole number by the denominator and add the numerator: \(4 \times 8 + 7 = 39\). Thus, the length of each shelf in fraction form is \(\frac{39}{8}\) ft.

Step 2: Calculate Total Length of Wood Needed

You need three shelves of \(\frac{39}{8}\) ft each. Multiply the length of one shelf by three to find the total length of wood needed: \(3 \times \frac{39}{8} = \frac{117}{8}\) ft.

Step 3: Convert Improper Fraction to Mixed Number

Convert the fraction \(\frac{117}{8}\) to a mixed number to better understand the length: Divide 117 by 8. The quotient is 14 with a remainder of 5. Therefore, \(\frac{117}{8}\) is equivalent to \(14 \frac{5}{8}\) ft.

Step 4: Compare Total Length with Available Wood

The board available is 15 ft long. Compare this to the total length needed, \(14 \frac{5}{8}\) ft. Since \(14 \frac{5}{8} < 15\), the board is long enough.

Key Concepts

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It's a way to express a quantity greater than one that isn't a whole number itself. For example,

• The mixed number \(4 \frac{7}{8}\) represents 4 whole units plus an additional fraction of \(\frac{7}{8}\) of one unit.

This format is useful for visualizing quantities in everyday scenarios, like measuring lengths. In this exercise, each shelf is \(4 \frac{7}{8}\) ft long. You'll need to understand this number in both mixed and improper fraction forms to do calculations easily.

Improper Fractions Explained

An improper fraction has a numerator larger than the denominator. It's essentially the same quantity as a mixed number but expressed differently. To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Place the total over the original denominator.

For example, \(4 \frac{7}{8}\) becomes \(\frac{39}{8}\) because \[4 \times 8 + 7 = 39. \]Improper fractions are preferred for multiplication because handling fractions without whole numbers simplifies the math.

How Measurement Conversion Works

Measurement conversion is crucial when dealing with different units or combining quantities. In this scenario, it’s about maintaining consistent metric systems while performing operations.
When we multiply the fraction \(\frac{39}{8}\) by 3 (for three shelves), we stick with fractions to keep accuracy:

• Multiply numerators: \(39 \times 3 = 117\).
• Use the same denominator: 8.

Thus, the total length is \(\frac{117}{8}\) ft. Changing this improper fraction back to a mixed number can help better compare the wood lengths.

The Art of Length Comparison

Length comparison involves estimating which lengths are greater, equal, or lesser. After converting \(\frac{117}{8}\) to the mixed number \(14 \frac{5}{8}\) ft, you set it against the length of the available wood, which is 15 ft.
Since:

• \(14 \frac{5}{8}\) ft is visibly less than 15 ft,

This means you have enough wood. This process helps in practical applications like ensuring enough material is on hand for certain tasks.