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Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth?

2

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Use fraction tiles to find equivalent fractions: How many twelfths equal one-fourth?

3

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Find equivalent fractions

We used fraction tiles to show that there are many fractions equivalent to 1 2 . For example, 2 4 , 3 6 , and 4 8 are all equivalent to 1 2 . When we lined up the fraction tiles, it took four of the 1 8 tiles to make the same length as a 1 2 tile. This showed that 4 8 = 1 2 . See [link] .

We can show this with pizzas, too. [link] (a) shows a single pizza, cut into two equal pieces with 1 2 shaded. [link] (b) shows a second pizza of the same size, cut into eight pieces with 4 8 shaded.

Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.

This is another way to show that 1 2 is equivalent to 4 8 .

How can we use mathematics to change 1 2 into 4 8 ? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

1 times 4 over 2 times 4 is written with the 4s in red. This is set equal to 4 over 8.

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Equivalent fractions property

If a , b , and c are numbers where b 0 and c 0 , then

a b = a · c b · c

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

The top line says that 1 times 3 over 2 times 3 equals 3 over 6, so one half equals 3 sixths. The next line says that 1 times 2 over 2 times 2 equals 2 over 4, so one half equals 2 fourths. The last line says that 1 times 10 over 2 times 10 equals 10 over 20, so one half equals 10 twentieths.

So, we say that 1 2 , 2 4 , 3 6 , and 10 20 are equivalent fractions.

Find three fractions equivalent to 2 5 .

Solution

To find a fraction equivalent to 2 5 , we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by 2 , 3 , and 5 .

On the left, we see that 2 times 2 over 5 times 2 equals 4 over 10. In the middle, we see that 2 times 3 over 5 times 3 equals 6 over 15. On the right, we see that 2 times 5 over 5 times 5 equals 10 over 25.

So, 4 10 , 6 15 , and 10 25 are equivalent to 2 5 .

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Find three fractions equivalent to 3 5 .

Correct answers include 6 10 , 9 15 , and 12 20 .

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Find three fractions equivalent to 4 5 .

Correct answers include 8 10 , 12 15 , and 16 20 .

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Find a fraction with a denominator of 21 that is equivalent to 2 7 .

Solution

To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in 21 .

Since we can multiply 7 by 3 to get 21 , we can find the equivalent fraction by multiplying both the numerator and denominator by 3 .

2 over 7 equals 2 time 3 over 7 times 3. The 3s are shown in red. This is set equal to 6 over 21.

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Find a fraction with a denominator of 21 that is equivalent to 6 7 .

18 21

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Find a fraction with a denominator of 100 that is equivalent to 3 10 .

30 100

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Locate fractions and mixed numbers on the number line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

Doing the Manipulative Mathematics activity "Number Line Part 3 " will help you develop a better understanding of the location of fractions on the number line.

Let us locate 1 5 , 4 5 , 3 , 3 1 3 , 7 4 , 9 2 , 5 , and 8 3 on the number line.

We will start with the whole numbers 3 and 5 because they are the easiest to plot.

A number line is shown with the numbers 3, 4, and 5. There are red dots at 3 and at 5.

The proper fractions listed are 1 5 and 4 5 . We know proper fractions have values less than one, so 1 5 and 4 5 are located between the whole numbers 0 and 1 . The denominators are both 5 , so we need to divide the segment of the number line between 0 and 1 into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as 1 5 , 2 5 , 3 5 , and 4 5 .

Practice Key Terms 4

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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