Problem 83 - Prealgebra

Question

Write your answers as whole numbers, proper fractions, or mixed numbers. Find each product. (Multiply) $$1 \frac{1}{2} \cdot 2 \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is \(3 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(1 \frac{1}{2}\), multiply the whole number 1 by the denominator 2, then add the numerator 1. This gives \(1 \times 2 + 1 = 3\), so \(1 \frac{1}{2}\) becomes \(\frac{3}{2}\). For \(2 \frac{1}{3}\), multiply the whole number 2 by the denominator 3, then add the numerator 1. This gives \(2 \times 3 + 1 = 7\), so \(2 \frac{1}{3}\) becomes \(\frac{7}{3}\).

Step 2: Multiply the Improper Fractions

Now multiply the two improper fractions together. \(\frac{3}{2} \cdot \frac{7}{3} = \frac{3 \times 7}{2 \times 3} = \frac{21}{6}\).

Step 3: Simplify the Fraction

Simplify \(\frac{21}{6}\) by finding the greatest common divisor of 21 and 6, which is 3. Divide both the numerator and the denominator by 3. \(\frac{21 \div 3}{6 \div 3} = \frac{7}{2}\).

Step 4: Convert to Mixed Number

Convert the improper fraction \(\frac{7}{2}\) into a mixed number by dividing the numerator by the denominator. \(7 \div 2 = 3\) with a remainder of 1. Therefore, \(\frac{7}{2} = 3 \frac{1}{2}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value greater than or equal to one whole. Understanding improper fractions is crucial, especially when dealing with mixed numbers or when multiplying fractions.
When working with mixed numbers, the first step is to convert them into improper fractions. This conversion is necessary because it simplifies the multiplication process later on. For instance, if you have a mixed number, like \(1 \frac{1}{2}\), you convert it by multiplying the whole number by the fraction's denominator, then adding the numerator.

• Multiply the whole number by the denominator: \(1 \times 2 = 2\)
• Add the numerator: \(2 + 1 = 3\)

Thus, \(1 \frac{1}{2}\) becomes \(\frac{3}{2}\). This same method applies to any mixed number and is essential for operations like multiplication.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator are the smallest possible integers that maintain the same value. This process makes fractions easier to understand and work with, whether you'd like to perform operations or simply express the number more cleanly.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without a remainder.
• Divide both the numerator and denominator by the GCD to get the simplified fraction.

For example, if you have \(\frac{21}{6}\), the GCD of 21 and 6 is 3. Dividing both by 3 gives you \(\frac{7}{2}\), which is the simplified form. This step is fundamental in ensuring the fraction is as straightforward and reduced as possible.

Converting Mixed Numbers

Converting between mixed numbers and improper fractions is a handy skill, especially when performing operations like multiplication. A mixed number combines a whole number and a fraction, such as \(3 \frac{1}{2}\).
To convert an improper fraction back into a mixed number, you divide the numerator by the denominator:

• The quotient becomes the whole number.
• The remainder is the new numerator, and the denominator remains the same.

Consider the improper fraction \(\frac{7}{2}\). Dividing 7 by 2, you get a quotient of 3 and a remainder of 1. Thus, \(\frac{7}{2} = 3 \frac{1}{2}\), a mixed number.
Understanding this conversion is particularly helpful, as sometimes, you need results in mixed number form to communicate amounts more precisely, especially in real-life contexts like cooking or construction.