Problem 26 - Beginning and Intermediate Algebra

Question

Find each product. \(\frac{3}{7}\left(-1 \frac{5}{9}\right)\)

Step-by-Step Solution

Answer

-\frac{2}{3}

Step 1 - Convert Mixed Number to Improper Fraction

First, convert the mixed number \(-1 \frac{5}{9}\) to an improper fraction. Multiply the integer part (-1) by the denominator (9) and add the numerator (5). Remember to keep the negative sign: \(-1 \frac{5}{9} = -\frac{9 \cdot 1 + 5}{9} = -\frac{9 + 5}{9} = -\frac{14}{9}\).

Step 2 - Multiply the Fractions

Multiply the fractions \frac{3}{7}\ and \frac{-14}{9}\: \(\frac{3}{7} \cdot \frac{-14}{9}\). To do this, multiply the numerators together and the denominators together: \(\frac{3 \cdot -14}{7 \cdot 9} = \frac{-42}{63}\).

Step 3 - Simplify the Fraction

Simplify the resulting fraction \(\frac{-42}{63}\). Find the greatest common divisor (GCD) of 42 and 63, which is 21. Divide both the numerator and the denominator by 21: \(\frac{-42 \div 21}{63 \div 21} = \frac{-2}{3}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

A mixed number has both an integer and a fractional part. To multiply a mixed number by another fraction, it's easier to first convert it to an improper fraction.
To do this, multiply the integer part by the denominator, add the numerator, and keep the denominator the same. For example, for \(-1 \frac{5}{9}\), you multiply -1 by 9 and then add 5, keeping the negative sign in mind.
This gives you: \(-\frac{14}{9}\). This improper fraction means the same thing as the original mixed number.

Multiplying Fractions

After converting, the task becomes simpler. Now you have \(\frac{3}{7} \times \-\frac{14}{9}\).
Multiply the numerators together: 3 and -14, giving you -42.
Then, multiply the denominators: 7 and 9, giving you 63.
This results in \(-\frac{42}{63}\).

Simplifying Fractions

The last step is to simplify the fraction. You need to find the greatest common divisor (GCD) of the numerator and the denominator.
For \-42 \ and \63\, the GCD is 21.
Divide both the numerator and the denominator by 21: \(-\frac{42}{63}\) becomes \(-\frac{42 \div 21}{63 \div 21} = \- \frac{2}{3}\).
This simplified fraction is much easier to work with and interpret.