4.2 Multiply and divide fractions  (Page 2/7)

Simplify: $-\frac{56}{32}.$

Solution

	$-\frac{56}{32}$
Rewrite the numerator and denominator, showing the common factors, 8.
Remove common factors.
Simplify.	$-\frac{7}{4}$

So, the simplified form of $-\frac{56}{32}$ is $-\frac{7}{4}.$

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Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.

Simplify: $\frac{210}{385}.$

Solution

Use factor trees to factor the numerator and denominator.	$\frac{210}{385}$
Rewrite the numerator and denominator as the product of the primes.	$\frac{210}{385}=\frac{2\cdot 3\cdot 5\cdot 7}{5\cdot 7\cdot 11}$
Remove the common factors.
Simplify.	$\frac{2\cdot 3}{11}$
Multiply any remaining factors.	$\frac{6}{11}$

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We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator , we remove it just as we do with an integer factor.

Multiply fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\frac{1}{2}·\frac{3}{4}.$ To multiply $\frac{1}{2}$ and $\frac{3}{4},$ think $\frac{1}{2}$ of $\frac{3}{4}.$

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three $\frac{1}{4}$ tiles evenly into two parts, we exchange them for smaller tiles.

We see $\frac{6}{8}$ is equivalent to $\frac{3}{4}.$ Taking half of the six $\frac{1}{8}$ tiles gives us three $\frac{1}{8}$ tiles, which is $\frac{3}{8}.$

Therefore,

Use a diagram to model $\frac{1}{2}·\frac{3}{4}.$

Solution

First shade in $\frac{3}{4}$ of the rectangle.

We will take $\frac{1}{2}$ of this $\frac{3}{4},$ so we heavily shade $\frac{1}{2}$ of the shaded region.

Notice that $3$ out of the $8$ pieces are heavily shaded. This means that $\frac{3}{8}$ of the rectangle is heavily shaded.

Therefore, $\frac{1}{2}$ of $\frac{3}{4}$ is $\frac{3}{8},$ or $\frac{1}{2}·\frac{3}{4}=\frac{3}{8}.$

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Look at the result we got from the model in [link] . We found that $\frac{1}{2}·\frac{3}{4}=\frac{3}{8}.$ Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.