Thursday, June 21, 2012

Changing a repeating decimal to a fraction in lowest terms.

Using Blogger editing, it's a little tricky to put a bar over a group of numbers. Instead I am going to put brackets [] around the repeating part. To write .16161616...., I'll type .[16] and to type .1666666..., I'll type .1[6].

Sorry if it's a little hard to read.

Method for changing from repeating decimal to fraction in lowest terms.

1. Call the repeating decimal x.
2. Multiply x by a power of ten that has as many zeros as there are digits in the repeating part.
3. Subtract x from the bigger number, which will cancel out the repeating part.
4*. IF the subtraction gives you a decimal number, multiply by some power of ten so you get (whole number times) x = (some other whole number)
5. Divide both sides of the equation by the number multiplying x.
6. Reduce the fraction to lowest terms.

Examples.
Example #1: .[16] = 0.1616161616....
1. x = 0.16161616...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 16.16161616....
3. 100x - x = 16.1616161616... - 0.1616161616...
which reduces to 99x = 16.
step 4 isn't needed.
5. x = 16/99, which is reduced to lowest terms.

Example #2: .1[6] = 0.166666....
1. x = 0.16666...
2. Because the repeating part has one digit, multiply x by 10 to get 10x = 1.6666....
3. 10x - x = 1.66666... - 0.166666...
which reduces to 9x = 1.5
4. 1.5 isn't a whole number, so multiply by 10 on both sides to get 90x = 15.
5. x = 15/90, which is not in lowest terms.
6. 15/90 = 5/30 = 1/6.

Practice problems.

a) Find the fraction for .[35]
b) Find the fraction for .3[5]
c) Find the fraction for .3[54]

Answers in the comments.

Click on this link for more practice problems with solutions.

1 comment:

Prof. Hubbard said...

a) Find the fraction for .[35]

1. x = 0.35353535...

2. Because the repeating part has two digits, we multiply x by 100 to get 100x = 35.353535...

3. 100x - x = 35.353535... - 0.353535...
which reduces to 99x = 35.

Step 4 isn't needed because we have no decimals.

5. x = 35/99, which is reduced to lowest terms.

==

b) Find the fraction for .3[5]

1. x = 0.35555...

2. Because the repeating part has one digit, we multiply x by 10 to get 10x = 3.55555...

3. 10x - x = 3.55555... - 0.355555...
which reduces to 9x = 3.2.

4. Because 3.2 is a decimal, we multiply both sides of the equation by 10 to get 90x = 32.


5. x = 32/90, which is not reduced to lowest terms.

6. x = 16/45.


==

c) Find the fraction for .3[54]


1. x = 0.3545454...

2. Because the repeating part has two digits, we multiply x by 100 to get 10x = 35.4545454

3. 100x - x = 35.4545454... - 0.3545454...
which reduces to 99x = 35.1.

4. Because 35.1 is a decimal, we multiply both sides of the equation by 10 to get 990x = 351.


5. x = 351/990, which is not reduced to lowest terms because both numbers are divisible by 3.

6. x = 117/330 = 39/110. The last answer is lowest terms.