Problem 42 - Prealgebra

Question

Sewing If it takes \(1 \frac{1}{2}\) yards of material to make a pillow cover, how much material will it take to make 3 pillow covers?

Step-by-Step Solution

Answer

It will take \(4 \frac{1}{2}\) yards of material to make 3 pillow covers.

Step 1: Identify the material needed for one pillow cover

First, let's determine the amount of material needed for one pillow cover. According to the problem, it takes \(1 \frac{1}{2}\) yards of material for a single pillow cover.

Step 2: Convert the mixed number to an improper fraction

The given material for one pillow cover is \(1 \frac{1}{2}\) yards. Convert this mixed number to an improper fraction: \(1 \frac{1}{2} = \frac{3}{2}\).

Step 3: Calculate the material needed for multiple pillow covers

To find out how much material is needed for 3 pillow covers, multiply the material needed for one cover by 3: \(3 \times \frac{3}{2}\).

Step 4: Perform the multiplication

Perform the multiplication: \(3 \times \frac{3}{2} = \frac{9}{2}\) yards.

Step 5: Express the result as a mixed number

Convert the improper fraction \(\frac{9}{2}\) back to a mixed number to make it clearer: \(\frac{9}{2} = 4 \frac{1}{2}\).

Key Concepts

Mixed Numbers

A mixed number consists of a whole number and a fractional part combined together. For example, in the expression \(1 \frac{1}{2}\), 1 is the whole number and \(\frac{1}{2}\) is the fractional part. When working with mixed numbers, we often need to convert them for different operations, such as addition, subtraction, or multiplication. This makes calculations easier and more straightforward.

• Keep in mind that the whole number and the fraction represent parts of a whole, which together indicate a value greater than either the whole or fractional part alone.
• Mixed numbers are commonly used when precise amounts are hard to express with either a whole number or a simple fraction alone. For instance, cooking recipes often use mixed numbers for measurements.

Understanding and converting mixed numbers is key to fully mastering fraction-related tasks.

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This indicates that the fraction represents a value equal to or greater than one whole. For example, \(\frac{3}{2}\) is an improper fraction because 3 (numerator) is greater than 2 (denominator), and it is equivalent to \(1 \frac{1}{2}\) as a mixed number.

• Improper fractions can easily be converted into mixed numbers by dividing the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the numerator of the fractional part.
• Converting improper fractions to mixed numbers or vice versa helps in simplifying and calculating complex problems involving fractions.

By practicing the conversion between improper fractions and mixed numbers, you can enhance your problem-solving skills in dealing with diverse mathematical operations involving fractions.

Multiplication of Fractions

Multiplying fractions involves combining the numerators together and the denominators together. Let's consider the multiplication process using the example from our exercise: \(3 \times \frac{3}{2}\). The process is straightforward:

• Multiply the numerators: \(3 \times 3 = 9\).
• Multiply the denominators: The whole number 3 can be rewritten as \(\frac{3}{1}\), so \(1 \times 2 = 2\).
• Resulting improper fraction: \(\frac{9}{2}\).

After obtaining the improper fraction \(\frac{9}{2}\), it's useful to convert it into a mixed number for better clarity. In our example, \(\frac{9}{2}\) becomes \(4 \frac{1}{2}\).

Also, remember:

• When multiplying fractions, reducing fractions to their simplest form, if possible, can save you time and effort.
• Multiplication is useful in solving real-world problems where repeated additions are necessary, like calculating total ingredients in recipes or fabric amounts for multiple items.

With practice, the multiplication and simplification of fractions become second nature.