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Chapter 6: Problem 8

Simplify each complex fraction. Use either method. $$ \frac{-\frac{5}{6}}{\frac{5}{4}} $$

Short Answer

Expert verified
-\frac{2}{3}

Step by step solution

01

- Understand the Problem

We need to simplify the complex fraction \(\frac{-\frac{5}{6}}{\frac{5}{4}}\). This involves dividing two fractions.
02

- Invert the Divisor

To divide by a fraction, we multiply by its reciprocal. Thus, the complex fraction becomes: \(\frac{-5/6}{5/4} = -\frac{5}{6} \times \frac{4}{5}\).
03

- Multiply the Fractions

Multiply the numerators together and the denominators together: \(-\frac{5 \times 4}{6 \times 5} = -\frac{20}{30}\).
04

- Simplify the Result

Reduce the fraction \(-\frac{20}{30}\) to its simplest form. Both the numerator and the denominator can be divided by their greatest common divisor, which is 10. Therefore, \(-\frac{20}{30} = -\frac{2}{3}\).

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Key Concepts

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

reciprocal
The reciprocal of a number is very useful in simplifying fractions. To find the reciprocal, simply flip the fraction. For example, the reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\). The same rule applies to any fraction. You change the numerator and the denominator with each other.

This method helps a lot, especially when dividing fractions. Why? Because dividing by a fraction is the same as multiplying by its reciprocal. Imagine this, instead of dividing \(\frac{-\frac{5}{6}}{\frac{5}{4}}\), we turn it into a multiplication problem with the reciprocal and proceed with our solution.
multiplying fractions
Multiplying fractions is straightforward. Follow these steps:
  • First, multiply the numerators (top numbers).
  • Second, multiply the denominators (bottom numbers).
For instance, in our problem, once we turned the division into multiplication, we multiplied \(-\frac{5}{6} \times \frac{4}{5}\). That means we did:

\(-5 \times 4 = -20\ \ ,\ \ and\ \ ,\ 6 \times 5 = 30\). So, \(-\frac{20}{30}\).

Simple as that! Remember always to multiply across, not cross-multiply.
greatest common divisor
Reducing fractions helps make them simpler and cleaner. To do this, use the greatest common divisor (GCD). The GCD of two numbers is the largest number that divides both without a remainder.

In our problem, we needed to reduce \(-\frac{20}{30}\). First, find the GCD of 20 and 30. The GCD is 10 because 10 is the largest number that can evenly divide both 20 and 30.

Once you know the GCD, divide both the numerator and the denominator by it: \(\frac{20 \/ 10}{30 \/ 10} = -\frac{2}{3}\). So, our simplified fraction is \(-\frac{2}{3}\).

Reducing fractions with the GCD makes final answers easier to understand.

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Most popular questions from this chapter

Simplify each complex fraction. Use either method. $$ \frac{\frac{12}{x+2}+2}{\frac{18}{x+2}-2} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{14 x^{2}+14 y^{2}}{21}}{\frac{x^{4}-y^{4}}{27}} $$

Simplify each fraction. $$ \frac{1+x^{-1}-12 x^{-2}}{1-x^{-1}-20 x^{-2}} $$

Simplify each complex fraction. Use either method. $$ \frac{2+\frac{1}{x}-\frac{28}{x^{2}}}{3+\frac{13}{x}+\frac{4}{x^{2}}} $$

Simplify each fraction. $$ \frac{1+t^{-1}-56 t^{-2}}{1-t^{-1}-72 t^{-2}} $$
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