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Chapter 7: Problem 46

Divide and, if possible, simplify. $$\frac{4}{9} \div \frac{5}{7}$$

Short Answer

Expert verified
\(\frac{4}{9} \div \frac{5}{7} = \frac{28}{45}\).

Step by step solution

01

Understand the Operation

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
02

Write Down the Reciprocal

The reciprocal of \(\frac{5}{7}\) is \(\frac{7}{5}\).
03

Set Up the Multiplication

Rewrite the division problem \(\frac{4}{9} \div \frac{5}{7}\) as a multiplication problem: \(\frac{4}{9} \times \frac{7}{5}\).
04

Multiply the Numerators

Multiply the numerators of the two fractions: \(4 \times 7 = 28\).
05

Multiply the Denominators

Multiply the denominators of the two fractions: \(9 \times 5 = 45\).
06

Write the Resulting Fraction

Combine the results of the multiplications from steps 4 and 5 to get: \(\frac{28}{45}\).
07

Simplify the Fraction

Check if the fraction \(\frac{28}{45}\) can be simplified. Since 28 and 45 have no common factors other than 1, the fraction is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal of a Fraction
To get the reciprocal of a fraction, you simply swap its numerator and denominator. For example, the reciprocal of \( \frac{5}{7} \) is \( \frac{7}{5} \). It's important to understand that the product of a fraction and its reciprocal is always 1. This is because \[ \frac{a}{b} \times \frac{b}{a} = \frac{ab}{ba} = 1. \] So, whenever you need to divide fractions, you'll multiply by the reciprocal of the second fraction.
Multiplying Fractions
Multiplying fractions is straightforward. You multiply the numerators together and the denominators together. For example, consider multiplying \( \frac{4}{9} \) and \( \frac{7}{5} \). First, multiply the numerators: \[ 4 \times 7 = 28. \] Next, multiply the denominators: \[ 9 \times 5 = 45. \] The result is \( \frac{28}{45} \).
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number. However, if the GCD is 1, the fraction is already in its simplest form. For instance, in \( \frac{28}{45} \), 28 and 45 have no common factors other than 1. Thus, \( \frac{28}{45} \) cannot be simplified further.

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