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Chapter 4: Problem 55

Simplify the complex fractions. $$\frac{-\frac{21}{10}}{\frac{6}{5}}$$

Short Answer

Expert verified
\( \frac{-7}{4} \)

Step by step solution

01

Understand the Problem

The given expression is a complex fraction: dividing fractions: \( \frac{-\frac{21}{10}}{\frac{6}{5}} \). The goal is to simplify this expression.
02

Simplify the Expression

Rewrite the division of fractions as the multiplication of the first fraction by the reciprocal of the second fraction: division by a fraction is the same as multiplication by its reciprocal, so \( \frac{-\frac{21}{10}}{\frac{6}{5}} = -\frac{21}{10} \times \frac{5}{6} \).
03

Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{-21 \times 5}{10 \times 6} = \frac{-105}{60}\).
04

Simplify the Result

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 15 in this case.\(\frac{-105 \div 15}{60 \div 15} = \frac{-7}{4}\).

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Key Concepts

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

fraction division
When dividing fractions, the key idea is to multiply by the reciprocal of the fraction you are dividing by.
The reciprocal of a fraction is simply the fraction with its numerator and denominator flipped.
For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
To divide fractions, follow these steps:
  • First, change the division sign to a multiplication sign.
  • Then, flip the second fraction to find its reciprocal.
  • Finally, multiply the first fraction by this reciprocal.

Applying this to our problem: We have: \(\frac{-\frac{21}{10}}{\frac{6}{5}}\).
So, we change the division to multiplication and find the reciprocal of \(\frac{6}{5}\), which is \(\frac{5}{6}\).
This turns our problem into: \(-\frac{21}{10} \times \frac{5}{6} \).
multiplication of fractions
Multiplying fractions involves a straightforward process.
Follow these simple steps:
  • Multiply the numerators (the top numbers) of the two fractions to get the new numerator.
  • Multiply the denominators (the bottom numbers) to get the new denominator.

In our problem: \(-\frac{21}{10} \times \frac{5}{6} \),
we multiply the numerators together: -21 times 5 = -105.
Then, we multiply the denominators: 10 times 6 = 60.
So, the product is \(\frac{-105}{60}\).
Remember to keep the sign (negative or positive) from each fraction.
finding greatest common divisor (GCD)
After you multiply the fractions, you may need to simplify the resulting fraction.
Simplify it by dividing the numerator and the denominator by their Greatest Common Divisor (GCD).
The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
To find the GCD of -105 and 60, note that the GCD is 15.
\(\frac{-105}{60} \div 15 = \frac{-7}{4}\).
We divide both -105 and 60 by 15, resulting in \(\frac{-7}{4}\), which is the simplified form of the original expression.

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