Math 6 Honors Periods 6 & 7

Changing a Fraction to a Decimal 6-5

There are two methods that can be used to change a fraction into a decimal.

The first one, we try to find an equivalent fraction whose denominator is a power of 10.

13/25 is a great example because we can easily change the denominator into 100 : multiplying 24 by 4.

So

13/25 ( 4/4) = 52/100 = .52

In the second method of changing a fraction into a decimal, we divide the numerator by the denominator.

Change 3/8 divide 3 by 8 carefully

When the remainder is 0, as above, the decimal is referred to as a terminating decimal. By examining the denominator of a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal. If the denominator has no prime factors of then 2 or 5, the decimal representation will terminate.. (This is so since the fraction can be written as an equivalent fraction whose denominator is a power of ten)

7/40

40 = 2³ ∙ 5; since the only prime factors of the denominator are 2 and 5, the fraction can be expressed as a terminating decimal

5/12

12 = 2² ∙ 3 since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal

9/12 = 3/4

4 = 2². Since 4 has no prime factors other than 2, this fraction can be expressed as a terminating decimal

Now, what happens if the denominator of a fraction has prime factors other than 2 or 5

Change 15/22 to a decimal I know that this cannot be expressed as a terminating decimal because the denominator (22) has the prime factorization of 2 ∙ 11.

15 divided by 22 ? divide carefully and you will get 0.6818181….

Notice the pattern of repeating remainders of 18 and 4. They produce a repeating block of digits 81, in the quotient.

we write 15/22 = 0.681818181…. or 0.681 with a bar over the 81 where the bar, also know as the vinculum, means that the block 81 repeats without ending.

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a decimal such as 0.681 , in which a block of digits continues to repeat indefinitely is called a repeating decimal.

Property

Every fraction can be expressed as either a terminating decimal or a repeating decimal..