1.5 Visualize fractions (Page 2/12)

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Equivalent fractions property

If $a, b, c$ are numbers where $b \neq 0, c \neq 0,$ then

\frac{a}{b} = \frac{a \cdot c}{b \cdot c}

If we had cut the pizza differently, we could get

$An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.$

So, we say $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, and \frac{10}{20}$ are equivalent fractions.

Doing the Manipulative Mathematics activity “Equivalent Fractions” will help you develop a better understanding of what it means when two fractions are equivalent.

Find three fractions equivalent to $\frac{2}{5} .$

Solution

To find a fraction equivalent to $\frac{2}{5},$ we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5.
$A row of fractions reads “2 times 2, divided by 5 times 2, equals four tenths”. Next to this is “2, times 3, divided by 5 times 3, equals six fifteenths”. Next to this is “2 times 5, divided by 5 times 5, equals ten twenty-fifths”.$
So, $\frac{4}{10}, \frac{6}{15}, and \frac{10}{25}$ are equivalent to $\frac{2}{5} .$

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Find three fractions equivalent to $\frac{3}{5} .$

$\frac{6}{10}, \frac{9}{15}, \frac{12}{20};$ answers may vary

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Find three fractions equivalent to $\frac{4}{5} .$

$\frac{8}{10}, \frac{12}{15}, \frac{16}{20};$ answers may vary

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Simplify fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

$\frac{2}{3}$ is simplified because there are no common factors of 2 and 3.
$\frac{10}{15}$ is not simplified because $5$ is a common factor of 10 and 15.

Simplified fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In [link] , we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent fractions property

If $a, b, c$ are numbers where $b \neq 0, c \neq 0,$

then \frac{a}{b} = \frac{a \cdot c}{b \cdot c} and \frac{a \cdot c}{b \cdot c} = \frac{a}{b}

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

Solution

	$- \frac{32}{56}$
Rewrite the numerator and denominator showing the common factors.
Simplify using the equivalent fractions property.	$- \frac{4}{7}$

Notice that the fraction

- \frac{4}{7}

is simplified because there are no more common factors.

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Simplify: $- \frac{42}{54} .$

$- \frac{7}{9}$

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Simplify: $- \frac{45}{81} .$

$- \frac{5}{9}$

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Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime number s. Then divide out the common factors using the equivalent fractions property.

How to simplify a fraction

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

Solution

A table is shown with three columns and three rows. The first row of the left column reads “Step 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree”. Next to this in the middle column, it reads “rewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads “negative 210 divided by 385.” Under this, is the equation “two times three times five times seven.” The five and 7 are blue and red respectively.

$The next row down reads “Step 2. Simplify using the equivalent fractions property by dividing out common factors.” Next to this in the middle column, it reads, “Mark the common factors 5 and 7.” Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation “negative two times 3 divided by 11.”$ The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.

The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.

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Simplify: $- \frac{69}{120} .$

$- \frac{23}{40}$

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Simplify: $- \frac{120}{192} .$

$- \frac{5}{8}$

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We now summarize the steps you should follow to simplify fractions.

Simplify a fraction.

Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first.
Simplify using the equivalent fractions property by dividing out common factors.
Multiply any remaining factors, if needed.

Simplify: $\frac{5 x}{5 y} .$

Solution

	$\frac{5 x}{5 y}$
Rewrite showing the common factors, then divide out the common factors.
Simplify.	$\frac{x}{y}$

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Simplify: $\frac{7 x}{7 y} .$

$\frac{x}{y}$

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Simplify: $\frac{3 a}{3 b} .$

$\frac{a}{b}$

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Multiply fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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