Problem 163 - Intermediate Algebra

Question

In the following exercises, perform the indicated operation. $$ \frac{8 u}{15} \div \frac{12 y}{25} $$

Step-by-Step Solution

Answer

\frac{10u}{9y}

Step 1 - Understand the operation

The operation to be performed here is the division of two fractions: \( \frac{8u}{15} \div \frac{12y}{25} \).

Step 2 - Invert and multiply

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{12y}{25} \) is \( \frac{25}{12y} \). Thus, \(\frac{8u}{15} \div \frac{12y}{25} = \frac{8u}{15} \times \frac{25}{12y} \).

Step 3 - Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{8u}{15} \times \frac{25}{12y} = \frac{8u \times 25}{15 \times 12y} = \frac{200u}{180y} \).

Step 4 - Simplify the fraction

Both the numerator and the denominator can be divided by their greatest common divisor (GCD), which is 20: \(\frac{200u}{180y} = \frac{200u \div 20}{180y \div 20} = \frac{10u}{9y} \).

Key Concepts

Reciprocal

Let's start by understanding what a reciprocal is. The reciprocal of a fraction is simply flipping the numerator (top number) and the denominator (bottom number). For example, the reciprocal of \(\frac{12y}{25}\) is \(\frac{25}{12y}\).

• Numerator becomes the denominator.
• Denominator becomes the numerator.

This concept is essential when dividing fractions because dividing by a fraction is the same as multiplying by its reciprocal. Always double-check your reciprocal to avoid errors.

Multiplying Fractions

Once you have the reciprocal, the next step is to multiply the fractions. To do this:

• Multiply the numerators together.
• Multiply the denominators together.

So, \(\frac{8u}{15} \times \frac{25}{12y}\) involves multiplying 8u by 25 and 15 by 12y. This gives us: \(\frac{8u \times 25}{15 \times 12y} = \frac{200u}{180y}\). Keep track of each multiplication step to ensure accuracy.

Simplifying Fractions

After multiplying, you may end up with a complex fraction like \(\frac{200u}{180y}\). The next step is to simplify this fraction. Simplifying means making the fraction as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 200 and 180 is 20. So we divide both by 20: \(\frac{200u \div 20}{180y \div 20} = \frac{10u}{9y}\). Keep practicing to get a good grasp of simplifying fractions.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator. Finding the GCD helps in simplifying fractions. Here's an easy way to find the GCD:

• List out the factors of both numbers.
• Identify the largest common factor.

Let’s take 200 and 180 as an example: The factors of 200 are 1, 2, 4, 5, 10, 20, 25, 40, 50, 100, 200. The factors of 180 are 1, 2, 3, 6, 9, 10, 12, 15, 18, 30, 45, 60, 90, 180. Hence, the GCD is 20. By dividing both the numerator and the denominator by the GCD, we simplify the fraction to \(\frac{10u}{9y}\).