Math 6 H Periods 1, 6 & 7 (Monday)

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




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.


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.


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..


Changing a Decimal to a Fraction 6-6
As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.
To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.

Change 0.385 to a fraction in lowest terms

.385 = 385/1000 = 77/200

Change 3.64 to a mixed number in simple form

3.64 = 3 64/100 = 3 16/25

To change a repeating decimal into a fraction follow these examples
th__
0.54

tththththh__
Let n = 0.54 = 0.54545454….

[How many numbers are under the vinculum?] 2
Multiple both sides by 102

So then, 100n = 54.54545454…


100n = 54.54545454…
n = .54545454….
We can subtract n from 100n to get 99n

100n = 54.54545454…
- n = .54545454….
99n = 54

Divide both sides by 99

99n = 54
99 99

n = 54/99 = 6/11


Let’s try
th___
0.243
theitheith___
Let n = 0.243 = .243243243243….

How many numbers are under the vinculum? 3

So multiply both sides by 103
1000n = 243.243243243243….

1000n = 243.243243…
n = 243.243243

999n = 243

Divide both sides by 999

n = 243/999 = 27/111 = 9/37

Let’s try one that is a bit more complicated
thethehtett__
Change 0.318 the vinculym is over just the 18.
[Notice this isn’t 0.318 nor is it 0.318
the__
0.318 so that means it is 03.1818181818....
How many numbers are under the vinculum? 2
So, we multiply by 102

Let n = 0.318181818…

100n = 31.818181818…
n= .318181818…

99n = 31.500000…
Divide both sides by 99

99n/99 = 31.5/99

n = 31.5/99 but that isn’t a proper fraction. What can I do to change this?

Multiply by 10

315/990 = 63/198 = 7/22

HERE ARE SOME STEPS TO FOLLOW:
Step 1 set up “ n= the repeating decimal” n = .515151…
Step 2 determine how many numbers are under the bar in this case = 2
Step 3 Use that number as a power of 10 102 = 100
Step 4 Multiply both sides of the equation in step 1 by that
power of 10 100n = 51.515151…
Step 5 Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation 100n = 51.5151…
- 00n= 51.5151…
Step 6 Solve as a 1-step equation 99n = 51 so n = 51/99
Step 7 Simplify 51/99 = 17/33
**** REMEMBER- sometimes you need to get the decimal out of the numerator—so multiply by a power of 10