Problem 216 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{10 y}{13} \cdot \frac{8}{15 y} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{16}{39} \)

Step 1 - Write Down the Original Expression

Write the given expression clearly: \[ \frac{10y}{13} \times \frac{8}{15y} \]

Step 2 - Multiply the Numerators

Multiply the numerators of the fractions: \[ 10y \times 8 = 80y \]

Step 3 - Multiply the Denominators

Multiply the denominators of the fractions: \[ 13 \times 15y = 195y \]

Step 4 - Form the New Fraction

Combine the results from Step 2 and Step 3 to form a new fraction: \[ \frac{80y}{195y} \]

Step 5 - Simplify the Fraction

Since \( y \) appears in both the numerator and denominator, cancel out the \( y \): \[ \frac{80}{195} \]

Step 6 - Simplify Further (if needed)

Check if 80 and 195 have a common factor. The greatest common divisor (GCD) of 80 and 195 is 5. Divide both the numerator and the denominator by 5 to get: \[ \frac{80 \/ 5}{195 \/ 5} = \frac{16}{39} \]

Key Concepts

Multiplying Fractions

Multiplying fractions is simple once you understand the process. To multiply fractions, follow these steps:

• Multiply the numerators (the top numbers) of the fractions together.
• Multiply the denominators (the bottom numbers) of the fractions together.

For example, in the expression \( \frac{10y}{13} \times \frac{8}{15y} \) step-by-step, it looks like this:

• Multiplying the numerators: \( 10y \times 8 = 80y \)
• Multiplying the denominators: \( 13 \times 15y = 195y \)

Now, the new fraction formed is \( \frac{80y}{195y} \).

Give this method a try with different fractions to get the hang of it!

Canceling Common Factors

Canceling common factors means reducing the fraction by removing the same numbers from the numerator and the denominator. In algebraic fractions, this often involves variables too.
In our example, \( \frac{80y}{195y} \), you can see that 'y' appears in both the numerator and the denominator.

• Since \( y \) is common, we cancel it: \( \frac{80y}{195y} \rightarrow \frac{80}{195} \)

This process makes the fraction simpler.
Double-check each fraction you work with for common factors before canceling.

Practicing with different fractions will make this easy.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.
For the fraction \( \frac{80}{195} \), we need to find the GCD of 80 and 195:

• List the factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
• List the factors of 195: 1, 3, 5, 13, 15, 39, 65, 195.
• The largest common factor is 5.

We divide both the numerator and the denominator by their GCD (5):
\( \frac{80}{195} \rightarrow \frac{80 \/ 5}{195 \/ 5} = \frac{16}{39} \).
By this, we get our simplified fraction \( \frac{16}{39} \).

This method saves time and helps avoid errors in simplifying fractions. Keep practicing to get better at spotting the GCD quickly.