Wednesday, June 22, 2011

Changing repeating decimals into fractions in lowest terms.

Using Blogger editing, it's a little tricky to put a bar over a group of numbers, but there is a strikethrough option, so I'm going to use that. To write .16161616...., I'll type .16 and to type .1666666..., I'll type .16.

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: .16 = 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 .23
b) Find the fraction for .23
c) Find the fraction for .234

1 comment:

Prof. Hubbard said...

a) x = .23232323
1. x = 0.23232323...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 23.23232323....
3. 100x - x = 23.23232323... - 0.23232323...
which reduces to 99x = 23.
step 4 isn't needed.
5. x = 23/99, which is reduced to lowest terms.

b) x = .23333333
1. x = 0.23333...
2. Because the repeating part has one digit, multiply x by 10 to get 10x = 2.3333....
3. 10x - x = 2.33333... - 0.23333...
which reduces to 9x = 2.1.
4. Multiply by 10 to get 90x = 21
5. x = 21/90, which is not reduced to lowest terms.
6. x = 7/30.

c) x = .234343434...
1. x = 0.2343434...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 23.4343434....
3. 100x - x = 23.4343434... - 0.2343434...
which reduces to 99x = 23.2.
4. Multiply by 10 to get 990x = 232
5. x = 232/990, which is not reduced to lowest terms.
6. x = 116/495.