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Chapter 10: Problem 32

Divide. $$ -\frac{5}{8} \div\left(-\frac{6}{5}\right) $$

Short Answer

Expert verified
The result of \-\frac{5}{8} \div\left(-\frac{6}{5}\right) is \frac{25}{48}.

Step by step solution

01

Understand the Division of Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. For any fractions \(\frac{a}{b} \) and \(\frac{c}{d} \), \(\frac{a}{b} \div\(\frac{c}{d}\) = \frac{a}{b} \times \frac{d}{c} \).
02

Find the Reciprocal of the Second Fraction

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

Multiply the First Fraction by the Reciprocal of the Second

Now, multiply \(\frac{-5}{8} \) by \(\frac{5}{-6} \).\[ -\frac{5}{8} \times \frac{5}{-6} = \frac{-5 \times 5}{8 \times -6} \]Simplify the multiplication:\[ \frac{-25}{-48} \]
04

Simplify the Fraction

Since a negative divided by a negative is a positive, the fraction simplifies to \(\frac{25}{48} \).
05

Final Answer

The final simplified form of the division is \(\frac{25}{48} \).

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Key Concepts

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

Reciprocal
Understanding the concept of a reciprocal is essential when dividing fractions. The reciprocal of a fraction is simply obtained by swapping its numerator and denominator. For instance, the reciprocal of the fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). In our problem, we dealt with \(-\frac{6}{5} \).
To find its reciprocal, we flip the fraction, resulting in \(\frac{5}{-6}\).

The reciprocal is used in fraction division because dividing by a number is the same as multiplying by its reciprocal. This principle simplifies the operation, making complex divisions more manageable and easier to compute.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. This step makes fractions easier to understand and compare.
Consider the result from our multiplication phase: \(\frac{-25}{-48}\).
Since both the numerator and the denominator are negative, we can simplify the fraction by noting that \- / - = + \.
Thus, \(\frac{-25}{-48} \) simplifies to \(\frac{25}{48} \).

Always look for common factors between the numerator and the denominator before declaring a fraction simplified. If there are no common factors, other than 1, then the fraction is in its simplest form.
Negative Numbers
Dealing with negative numbers can be tricky, but there are clear rules to follow.
When multiplying or dividing two negative numbers, the result is positive. This rule stems from the concept that the negatives cancel each other out.
In our problem, multiplying \(\frac{-5}{8}\) by \(\frac {5}{-6}\) yields \(\frac{-25}{-48} \).

Since a negative divided by a negative gives a positive, the fraction becomes \(\frac{25}{48} \).
Here are some quick tips:
  • Negative times positive is negative
  • Positive times positive is positive
  • Negative times negative is positive
By remembering these simple rules, navigating problems involving negative numbers becomes much more straightforward.

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