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1.3 Fractions: equivalent fractions

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses equivalent fractions, reducing fractions to lowest terms, and raising fractions to higher terms. By the end of the module students should be able to recognize equivalent fractions, reduce a fraction to lowest terms and be able to raise a fraction to higher terms.

Section overview

  • Equivalent Fractions
  • Reducing Fractions to Lowest Terms
  • Raising Fractions to Higher Terms

Equivalent fractions

Let's examine the following two diagrams.

A rectangle divided equally into three parts, each marked one-third. The left two parts are shaded. To the right of the box is the caption, two-thirds of the whole is shaded. Below this is a rectangle equally divided into six part, with the leftmost four part shaded. to the right of this rectangle is the caption, four-sixths of the whole is shaded.

Notice that both 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} represent the same part of the whole, that is, they represent the same number.

Equivalent fractions

Fractions that have the same value are called equivalent fractions . Equiva­lent fractions may look different, but they are still the same point on the number line.

There is an interesting property that equivalent fractions satisfy.

two-thirds and four-sixths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

A test for equivalent fractions using the cross product

These pairs of products are called cross products .

Is two time six equal to three times four? Yes.

If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.

Thus, 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} are equivalent, that is, 2 3 = 4 6 size 12{ { {2} over {3} } = { {4} over {6} } } {} .

Sample set a

Determine if the following pairs of fractions are equivalent.

3 4 and 6 8 size 12{ { {3} over {4} } `"and " { {6} over {8} } } {} . Test for equality of the cross products.

three-fourths and six-eigths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

Is three times eight equal to six times four? yes. The cross products are equals.

The fractions 3 4 and 6 8 are equivalent, so 3 4 = 6 8 .

3 8 and 9 16 size 12{ { {3} over {8} } " and " { {9} over {"16"} } } {} . Test for equality of the cross products.

Three-eights and nine-sixteenths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

is three times sixteen equal to nine times eight? No. forty-eight does not equal seventy-two. The cross products are not equal.

The fractions 3 8 size 12{ { {3} over {8} } } {} and 9 16 size 12{ { {9} over {"16"} } } {} are not equivalent.

Practice set a

Determine if the pairs of fractions are equivalent.

1 2 size 12{ { {1} over {2} } } {} , 3 6 size 12{ { {3} over {6} } } {}

Six equals six. , yes

4 5 size 12{ { {4} over {5} } } {} , 12 15 size 12{ { {"12"} over {"15"} } } {}

Sixty equals sixty. , yes

2 3 size 12{ { {2} over {3} } } {} , 8 15 size 12{ { {8} over {"15"} } } {}

30 ≠ 24 , no

1 8 size 12{ { {1} over {8} } } {} , 5 40 size 12{ { {5} over {"40"} } } {}

Forty equals forty. , yes

3 12 size 12{ { {3} over {"12"} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Twelve equals twelve. , yes

Reducing fractions to lowest terms

It is often very useful to conver t one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} . First, divide both the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3. The fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} are equivalent.

(Can you prove this?) So, 9 15 = 3 5 size 12{ { {9} over {"15"} } = { {3} over {5} } } {} . We wish to convert 9 15 size 12{ { {9} over {"15"} } } {} to 3 5 size 12{ { {3} over {5} } } {} . Now divide the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3, and see what happens.

9 ÷ 3 15 ÷ 3 = 3 5 size 12{ { {9 div 3} over {"15" div 3} } = { {3} over {5} } } {}

The fraction 9 15 size 12{ { {9} over {"15"} } } {} is converted to 3 5 size 12{ { {3} over {5} } } {} .

A natural question is "Why did we choose to divide by 3?" Notice that

9 15 = 3 ⋅ 3 5 ⋅ 3 size 12{ { {9} over {"15"} } = { {3 cdot 3} over {5 cdot 3} } } {}

We can see that the factor 3 is common to both the numerator and denominator.

Reducing a fraction

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has reduced values in the numerator and denominator. The method is called reducing a fraction .

A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.

Nine-twelfths is equal to nine divided by three, over nine divided by three, which is equal to three-fourths. Sixteen thirtieths is equal to sixteen divided by two, over thirty divided by 2, which is equal to eight-fifteenths. Notice that three over three and two over two are both equal to 1.

Consider the collection of equivalent fractions

5 20 size 12{ { {5} over {"20"} } } {} , 4 16 size 12{ { {4} over {"16"} } } {} , 3 12 size 12{ { {3} over {"12"} } } {} , 2 8 size 12{ { {2} over {8} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Reduced to lowest terms

Notice that each of the first four fractions can be reduced to the last fraction, 1 4 size 12{ { {1} over {4} } } {} , by dividing both the numerator and denominator by, respectively, 5, 4, 3, and 2. When a fraction is converted to the fraction that has the smallest numerator and denomi­nator in its collection of equivalent fractions, it is said to be reduced to lowest terms . The fractions 1 4 size 12{ { {1} over {4} } } {} , 3 8 size 12{ { {3} over {8} } } {} , 2 5 size 12{ { {2} over {5} } } {} , and 7 10 size 12{ { {7} over {"10"} } } {} are all reduced to lowest terms.
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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