Problem 47 - Prealgebra

Question

Find the quotients. (Divide.) $$\frac{4}{5} \div \frac{7}{8}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{32}{35} \).

Step 1: Understand the Division of Fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction. This means that \( \frac{a}{b} \div \frac{c}{d} \) becomes \( \frac{a}{b} \times \frac{d}{c} \).

Step 2: Identify the Reciprocals

The reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \). In this problem, \( \frac{4}{5} \div \frac{7}{8} \) becomes \( \frac{4}{5} \times \frac{8}{7} \).

Step 3: Multiply the Fractions

To multiply \( \frac{4}{5} \times \frac{8}{7} \), multiply the numerators together and the denominators together. This gives: \( \frac{4 \times 8}{5 \times 7} = \frac{32}{35} \).

Step 4: Simplify If Needed

Check if the fraction \( \frac{32}{35} \) can be simplified. The greatest common divisor of 32 and 35 is 1, so \( \frac{32}{35} \) is already in its simplest form.

Key Concepts

Reciprocal

Understanding the concept of the **reciprocal** is fundamental when working with fraction division. The reciprocal of a number is what you get when you swap the numerator (the top number) and the denominator (the bottom number) of a fraction. So, if you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept is particularly useful in fraction division because you convert the division into multiplication by taking the reciprocal of the divisor. When you face a problem like \( \frac{4}{5} \div \frac{7}{8} \), you do not really "divide" in the traditional sense. Instead, you change the division sign to multiplication and flip the second fraction (the divisor). Thus, it becomes \( \frac{4}{5} \times \frac{8}{7} \).

• Always remember, flipping makes the reciprocal.
• Reciprocal turns division into multiplication.

Multiply Fractions

Once you have transformed the division into a multiplication problem using the reciprocal, the next step is to **multiply fractions**. Multiplying fractions is straightforward—multiply the numerators together and the denominators together.Using the example \( \frac{4}{5} \times \frac{8}{7} \), you would:

• Multiply the numerators: \( 4 \times 8 = 32 \).
• Multiply the denominators: \( 5 \times 7 = 35 \).

Putting these results together gives you \( \frac{32}{35} \). This method ensures that you are consistent, and it's a universal approach regardless of the fractions you're dealing with. Just remember:

• Top numbers together (numerators).
• Bottom numbers together (denominators).

Simplify Fractions

After multiplying fractions, the final step is often to **simplify fractions**, although it might not always be necessary. Simplifying means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.To simplify, you find the greatest common divisor (GCD) of both the numerator and the denominator. In the case of our example \( \frac{32}{35} \), you check:

• The factors of 32 are 1, 2, 4, 8, 16, 32.
• The factors of 35 are 1, 5, 7, 35.

The only common factor is 1. Therefore, \( \frac{32}{35} \) is already in its simplest form. Here are a few tips:

• Use the GCD to determine if simplification is needed.
• Some fractions are already simple and need no further reduction.