#### 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