4.2 Multiply and divide fractions  (Page 3/7)

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive.

Multiply, and write the answer in simplified form: $-\frac{5}{8}\left(-\frac{2}{3}\right).$

Solution

	$-\frac{5}{8}\left(-\frac{2}{3}\right)$
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.	$\frac{5\cdot 2}{8\cdot 3}$
Simplify.	$\frac{10}{24}$
Look for common factors in the numerator and denominator. Rewrite showing common factors.
Remove common factors.	$\frac{5}{12}$

Another way to find this product involves removing common factors earlier.

	$-\frac{5}{8}\left(-\frac{2}{3}\right)$
Determine the sign of the product. Multiply.	$\frac{5\cdot 2}{8\cdot 3}$
Show common factors and then remove them.
Multiply remaining factors.	$\frac{5}{12}$

We get the same result.

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Multiply, and write the answer in simplified form: $-\frac{14}{15}·\frac{20}{21}.$

Solution

	$-\frac{14}{15}·\frac{20}{21}$
Determine the sign of the product; multiply.	$-\frac{14}{15}·\frac{20}{21}$
Are there any common factors in the numerator and the denominator? $$ We know that 7 is a factor of 14 and 21, and 5 is a factor of 20 and 15.
Rewrite showing common factors.
Remove the common factors.	$-\frac{2·4}{3·3}$
Multiply the remaining factors.	$-\frac{8}{9}$

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When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, $a,$ can be written as $\frac{a}{1}.$ So, $3=\frac{3}{1},$ for example.

Multiply, and write the answer in simplified form:

ⓐ $\frac{1}{7}·56$

ⓑ $\frac{12}{5}\left(\mathrm{-20}x\right)$

Solution

ⓐ
	$\frac{1}{7}·56$
Write 56 as a fraction.	$\frac{1}{7}·\frac{56}{1}$
Determine the sign of the product; multiply.	$\frac{56}{7}$
Simplify.	$8$

ⓑ
	$\frac{12}{5}\left(\mathrm{-20}x\right)$
Write −20x as a fraction.	$\frac{12}{5}\left(\frac{\mathrm{-20}x}{1}\right)$
Determine the sign of the product; multiply.	$-\frac{12·20·x}{5·1}$
Show common factors and then remove them.
Multiply remaining factors; simplify.	−48x

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Find reciprocals

The fractions $\frac{2}{3}$ and $\frac{3}{2}$ are related to each other in a special way. So are $-\frac{10}{7}$ and $-\frac{7}{10}.$ Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be $1.$

Such pairs of numbers are called reciprocals.

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to $1.$ Is there any number $r$ so that $0·r=1?$ No. So, the number $0$ does not have a reciprocal.