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Chapter 4: Problem 156

Divide, and write the answer in simplified form. \(\frac{7 p}{12} \div \frac{21 p}{8}\)

Short Answer

Expert verified
\( \frac{2}{9} \)

Step by step solution

01

Understand Division of Fractions

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

Find the Reciprocal of the Second Fraction

The reciprocal of \(\frac{21p}{8}\) is \(\frac{8}{21p}\).
03

Convert Division to Multiplication

Rewrite the division problem as a multiplication problem: \(\frac{7p}{12} \times \frac{8}{21p}\).
04

Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{7p \times 8}{12 \times 21p} = \frac{56p}{252p}\).
05

Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor (GCD): The GCD of 56 and 252 is 28, and p cancels out. \(\frac{56p \div 28p}{252p \div 28p} = \frac{2}{9}\).

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Key Concepts

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

Reciprocals
Understanding reciprocals is key when dividing fractions. A reciprocal of a fraction is created by swapping its numerator and denominator. For instance, the reciprocal of \( \frac{21p}{8} \) is \( \frac{8}{21p} \). This flip is crucial because dividing by a fraction is equivalent to multiplying by its reciprocal. Remember, anytime you divide by a fraction, you'll use its reciprocal!
Simplification
Simplification makes fractions easier to work with. After converting division into multiplication by the reciprocal, you'll often need to simplify the resulting fraction. In our exercise, we multiplied \( \frac{7p}{12} \) by \( \frac{8}{21p} \) to get \( \frac{56p}{252p} \). The next step is to simplify this fraction by dividing both the numerator and denominator by their common factors. Simplification reduces fractions to their smallest, most manageable form.
Greatest Common Divisor
When simplifying fractions, the greatest common divisor (GCD) is very important. The GCD is the highest number that divides both the numerator and the denominator without leaving a remainder. For \( \frac{56p}{252p} \), the GCD of 56 and 252 is 28. By dividing both the numerator and denominator by 28, we simplify the fraction to \( \frac{2}{9} \). Simplifying fractions by using the GCD makes them easier to understand and compare.

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Most popular questions from this chapter

In the following exercises, solve the equation. $$ \frac{m}{-2}=-16 $$

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