M 302, Spring 2003 (Smith)

More on expressing repeating decimals as rational numbers

Warm-ups:

1. What do you get when you multiply 5.23 by 10?
    By 100?
    By 1000?

2. What's the difference between 5.232323 and 5.232323...?

3. What do you get when you multiply 5.232323... by 10?
    By 100?
    By 1000?

4. What do you have to multiply 2.343434 ... by to get 234.343434 ... ?

Answers to warm-ups

The Idea:

We'd like to take a repeating decimal and express it as a quotient of two whole numbers. The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out.

Examples:

1. x = .444...

10 x = 4.444 ...
x = .444...

So when we subtract, the .444... will cancel out on the right, and we'll be left with

9x = 4,

which we can solve to get

x = 3/4

2. x = .232323...

If we multiply by 10, we get 10x = 2.32323...
If we try to subtract x, we have

10x = 2.32323...
x = .232323...

This isn't any good -- the two's are lined up under the 3's and the 3's under the 2's, so when we subtract, we won't have things cancelling out. What's the problem? The pattern that repeats is two digits, so we need to move the decimal point two digits to make the repeating part cancel out. That means we need to multiply by 100:

100x = 23 .232323...
x = .232323...

Now things line up, so we can subtract and get

99x = 23,

then solve to get

x = 23/99

3. Here's a variation: x = 2.4232323 ...
The repeating pattern is two digits, so we try multiplying by 100:

100x = 242.32323 ...
x = 2.42323 ...

Now the repeating patterns 23line up, but only after the first decimal place. If we subtract, we get

99x =239 .9 (go through the details of the subtraction carefully yourself)

then solve to get

x = 239.9/99

But we're not done, because we want a whole number over a whole number, and this doesn't have a whole number on top. But if we multiply the top by 10 we will get a whole number. If we multiply both numerator and denominator by ten, we won't change the value of the number. Doing this gives our final answer:

x = 2399/990

Exercises to do:

For the given number x, figure out what you need to multiply by to get the desired cancelling, and why. Then express the number as a rational number.

1. x = 3.777...

2. x = 3.7666 ...

3. x = 3. 464646 ...

4. x = 3.5464646 ...

5. x = 4.123123...

6. x = 4.5123123123 ...

Answers to exercies