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Chapter 4: Problem 681

In the following exercises, simplify. $$ -\frac{5}{6} \div \frac{5}{12} $$

Short Answer

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Step by step solution

01

Rewrite the division as multiplication

To simplify the division of fractions, rewrite the division as multiplication by the reciprocal of the second fraction. \[ -\frac{5}{6} \times \frac{12}{5} \text{(reciprocal of } \frac{5}{12} \text{ is } \frac{12}{5}) \]
02

Multiply the numerators

Multiply the numerators of the fractions:\[ -5 \times 12 = -60 \]
03

Multiply the denominators

Multiply the denominators of the fractions:\[6 \times 5 = 30 \]
04

Simplify the resulting fraction

Simplify the resulting fraction by dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD of 60 and 30 is 30:\[ \frac{-60}{30} = -2 \]

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Key Concepts

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

fraction division
Understanding how to divide fractions is essential for solving fraction problems. When we talk about fraction division, we actually mean multiplying by the reciprocal. The reciprocal of a fraction is obtained by swapping its numerator (top number) and denominator (bottom number). For example, the reciprocal of \(\frac{5}{12}\) is \(\frac{12}{5}\). Let's consider the exercise where we had to simplify \( -\frac{5}{6} \div \frac{5}{12} \).
To solve this, we converted the division into a multiplication problem: \( -\frac{5}{6} \times \frac{12}{5} \). By doing this, we've restructured our problem in a way that's easier to solve.
multiplication of fractions
Multiplying fractions is straightforward once you understand the process. You simply multiply the numerators together and the denominators together. With our example, \(\-\frac{5}{6} \times \frac{12}{5}\), we first multiply the top numbers: -5 and 12, giving us -60.
Next, we multiply the bottom numbers: 6 and 5, resulting in 30. So, we are left with the fraction \(\frac{-60}{30}\).
Always remember to keep track of your negative signs during multiplication to ensure accuracy.
greatest common divisor
Finally, to completely simplify your fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that evenly divides both numbers. In our fraction \(\frac{-60}{30}\), the GCD of 60 and 30 is 30. By dividing both the numerator and the denominator by 30, we simplify the fraction: \(\frac{-60 \div 30}{30 \div 30} = -2\).
Simplifying using the GCD ensures that your fraction is in its simplest form, making it easier to understand and work with.

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Most popular questions from this chapter

In the following exercises, simplify. $$ -5 \frac{7}{12} \cdot 4 \frac{4}{11} $$

In the following exercises, determine whether the each number is a solution of the given equation. $$ x-\frac{1}{2}=\frac{1}{6} $$ $$ (a)x=1 $$ $$ (b)x=\frac{2}{3} $$ $$ (c)x=-\frac{1}{3} $$

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{1}{12}$$

In the following exercises, simplify. $$ -\frac{3}{10}+\left(-\frac{5}{8}\right) $$

In the following exercises, simplify. $$12 \left(\frac{9}{20}-\frac{4}{15}\right)$$
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