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Chapter 10: Problem 28

Divide. $$ -\frac{5}{9} \div\left(-\frac{5}{6}\right) $$

Short Answer

Expert verified
The result is \( \frac{2}{3} \).

Step by step solution

01

Understand the Problem

You need to divide \( -\frac{5}{9} \) by \( -\frac{5}{6} \). Division of fractions involves multiplying by the reciprocal of the second fraction.
02

Find the Reciprocal

The reciprocal of \( -\frac{5}{6} \) is \( -\frac{6}{5} \).
03

Multiply the Fractions

Now multiply \( -\frac{5}{9} \) by \( -\frac{6}{5} \). When multiplying fractions, multiply the numerators together and the denominators together:\[-\frac{5}{9} \times -\frac{6}{5} = \frac{(-5) \times (-6)}{9 \times 5} = \frac{30}{45}\]
04

Simplify the Fraction

To simplify \( \frac{30}{45} \), find the greatest common divisor (GCD) of 30 and 45, which is 15. Then, divide both the numerator and the denominator by their GCD:\[\frac{30 \div 15}{45 \div 15} = \frac{2}{3}\]

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Key Concepts

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

Reciprocal of a Fraction
Understanding the reciprocal of a fraction is a key step in fraction division. The reciprocal, or multiplicative inverse, of a fraction is obtained by flipping the numerator and denominator. For example, the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\). This means that if you multiply these two fractions together, you get 1:
  • Reciprocal of \[ \frac{a}{b} \] is \[ \frac{b}{a} \]
  • The sign remains the same, even for negative fractions
Finding the reciprocal is essential because dividing by a fraction is equivalent to multiplying by its reciprocal.
In our problem, we first found the reciprocal of \-\frac{5}{6}\, which is \-\frac{6}{5}\.
This allows us to turn the division problem into a multiplication problem, which is much easier to solve.
Multiplying Fractions
Once you have the reciprocal, the next step is to multiply the fractions. Multiplying fractions involves multiplying their numerators and denominators together. In our problem:
  • \[-\frac{5}{9} \times -\frac{6}{5}\rightarrow\frac{(-5) \times (-6)}{9 \times 5}\]
  • Always multiply the numerators together and the denominators together:

When multiplying \-\frac{5}{9}\ and \-\frac{6}{5}\, we get:
\[ \frac{30}{45} \]
Here, \(- 5 \times - 6 = 30\) and \( 9 \times 5 = 45\).
Remember, multiplying negative numbers gives a positive result, hence the result is \( \frac{30}{45} \)
Simplifying Fractions
Simplifying a fraction means reducing it to its simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).

In our problem, the fraction we need to simplify is \(\frac{30}{45}\):
  • First, find the GCD of 30 and 45, which is 15
  • Next, divide both the numerator and denominator by 15:
  • \[ \frac{30 \div 15}{45 \div 15} = \frac{2}{3} \]
Thus, \(\frac{30}{45}\) simplifies to \(\frac{2}{3} \). Simplifying fractions helps in obtaining the most reduced and manageable form, making it easier to understand and work with.

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