Problem 88 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 3 \frac{1}{2} \cdot \frac{1}{5} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{7}{10}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number into an improper fraction. The mixed number given is \(3 \frac{1}{2}\). To convert it, multiply the whole number 3 by the denominator 2, and then add the numerator 1. This gives: \(3 \times 2 + 1 = 7\). So, \(3 \frac{1}{2}\) becomes \(\frac{7}{2}\).

Step 2: Perform the Multiplication

Now, multiply the improper fraction \(\frac{7}{2}\) by the fraction \(\frac{1}{5}\). To multiply fractions, multiply the numerators and the denominators: \(\frac{7}{2} \times \frac{1}{5} = \frac{7 \times 1}{2 \times 5} = \frac{7}{10}\).

Step 3: Simplify the Result

The result of the multiplication is \(\frac{7}{10}\). Check if this fraction can be simplified further. Since 7 and 10 have no common factors other than 1, \(\frac{7}{10}\) is already in its simplest form.

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. You will typically see them in this form: 3 \(\frac{1}{2}\), where 3 is the whole number, and \(\frac{1}{2}\) is the fraction part.
They can be useful for representing amounts larger than a whole. For example, if you ate 3 whole apples and half of another, you ate 3 \(\frac{1}{2}\) apples.

To work with mixed numbers in equations, especially in multiplication or division, you often need to convert them into improper fractions. This makes calculations easier and more straightforward.

Improper Fractions Explained

Improper fractions are fractions where the numerator is larger than the denominator. For example, \(\frac{7}{2}\) is an improper fraction because 7 (numerator) is greater than 2 (denominator).
Converting a mixed number to an improper fraction involves a simple process. Multiply the whole number by the fraction's denominator, add the fraction's numerator, and place the result over the original denominator.
- For example, to convert \(3 \frac{1}{2}\) into an improper fraction: - Multiply 3 (whole number) by 2 (denominator): 3 \(\times\) 2 = 6. - Add 1 (numerator): 6 + 1 = 7. - The improper fraction is \(\frac{7}{2}\).

Improper fractions are often more versatile than mixed numbers in mathematical problems, especially when doing operations like multiplication or division.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and denominator by their GCD.

In the example of \(\frac{7}{10}\), the numbers 7 and 10 have no common factors other than 1, meaning it is already in its simplest form.
Simplification is crucial as it makes numbers easier to work with and understand. Always check if fractions in your results can be simplified, as it often makes further calculations cleaner and more intuitive.