Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \div \frac{4}{3}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{15}{16} \).

Step 1: Identify the Division Operation

Identify that the exercise wants to divide two fractions: \( \frac{5}{4} \div \frac{4}{3} \).

Step 2: Change Division to Multiplication

Change the division operation to multiplication by taking the reciprocal of the second fraction. This will result in: \( \frac{5}{4} \times \frac{3}{4} \).

Step 3: Multiply Fractions

Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. This will result in: \( \frac{5 \times 3}{4 \times 4} = \frac{15}{16} \).

Step 4: Check if Further Simplification is Needed

Check to see if the result can be further reduced to lowest terms. In this case, \( \frac{15}{16} \) is already in its simplest form. Therefore, no further simplification is needed.

Key Concepts

Reciprocal

When you are dividing fractions, one important concept to understand is the reciprocal. The reciprocal of a fraction is just like flipping the fraction upside down. For any given fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). Switching the numerator with the denominator creates the reciprocal. Understanding reciprocals is crucial as it helps convert division problems into multiplication ones.
When you encounter a division problem with fractions, such as \( \frac{5}{4} \div \frac{4}{3} \), you switch the operation to multiplication and use the reciprocal of the second fraction. This means it changes from division \( \div \) into multiplication \( \times \), and the second fraction becomes \( \frac{3}{4} \) instead of \( \frac{4}{3} \). This is written as \( \frac{5}{4} \times \frac{3}{4} \).
So, always remember: converting division into multiplication with reciprocals simplifies the process when dealing with fractions.

Multiplying Fractions

Once you have changed the division of fractions into a multiplication problem using reciprocals, the next step is to multiply the fractions.
Multiplying fractions is actually quite simple. You multiply the numerators (the top numbers) with each other and the denominators (the bottom numbers) with each other.
Let's apply this to the example \( \frac{5}{4} \times \frac{3}{4} \):

• Multiply the numerators: \( 5 \times 3 = 15 \).
• Multiply the denominators: \( 4 \times 4 = 16 \).

This results in the fraction \( \frac{15}{16} \). Practicing multiplication of fractions helps recognize patterns and easily manage fraction operations.

Simplifying Fractions

After multiplying, it's important to check if the fraction can be simplified to its lowest terms.
Simplifying, also known as reducing a fraction, involves making the fraction as simple as possible by dividing the top and bottom numbers by any common factors they might have. This results in an equivalent fraction that cannot be simplified any further.
Consider \( \frac{15}{16} \):

• Check for common factors between the numerator and the denominator.
• In this case, 15 and 16 do not share any common factors besides 1.

Therefore, \( \frac{15}{16} \) is already in its simplest form, and no further simplification is needed. Ensuring fractions are simplified allows for the clearest and most accurate results.