Problem 55 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{12}{5} \cdot \frac{10}{9} $$

Step-by-Step Solution

Answer

\( \frac{12}{5} \cdot \frac{10}{9} = \frac{8}{3} \).

Step 1 - Multiply the Numerators

First, multiply the numerators of the given fractions. The numerators are 12 and 10. So, we have: \( 12 \times 10 = 120 \).

Step 2 - Multiply the Denominators

Next, multiply the denominators of the given fractions. The denominators are 5 and 9. So, we have: \( 5 \times 9 = 45 \).

Step 3 - Form the New Fraction

Now, combine the results from the previous steps to form a new fraction: \( \frac{120}{45} \).

Step 4 - Simplify the Fraction

Next, simplify the fraction \( \frac{120}{45} \). Find the greatest common divisor (GCD) of 120 and 45. Both 120 and 45 are divisible by 15. \( \frac{120 \div 15}{45 \div 15} \) simplifies to \( \frac{8}{3} \).

Step 5 - Verify Using a Calculator

Finally, check the result using a calculator. Multiply \( \frac{12}{5} \cdot \frac{10}{9} \rightarrow 2.6667 \) and divide \( \frac{8}{3} \rightarrow 2.6667 \). Both values are equal, confirming the result.

Key Concepts

Numerator

In a fraction, the numerator is the number above the fraction line. It indicates how many parts of the whole are being considered. For example, in the fraction \(\frac{12}{5}\), the numerator is 12 because it represents the number of parts we have. When multiplying fractions, you multiply the numerators together. In our problem, we multiplied 12 by 10 to get a new numerator of 120.

Denominator

The denominator is the number below the fraction line. It shows the total number of equal parts the whole is divided into. For instance, in the fraction \(\frac{12}{5}\), the denominator is 5, denoting that the whole is divided into five equal parts. To multiply fractions, you multiply the denominators together. In our example, we multiplied 5 by 9, resulting in a new denominator of 45.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This helps make calculations easier and results clearer. To simplify \(\frac{120}{45}\), we need to find the greatest common divisor (GCD) of the numerator and denominator. Once we determine that both 120 and 45 are divisible by 15, we divide both by 15. This gives us \(\frac{120 \div 15}{45 \div 15} = \frac{8}{3}\), which is the simplest form of our fraction.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator. It is crucial when simplifying fractions. In our problem, the GCD of 120 and 45 is 15 because 15 is the biggest number that evenly divides them. By dividing both the numerator and the denominator by this GCD, we simplify our fraction to its lowest terms, resulting in \(\frac{8}{3}\).