Fractions, Decimals, Percentages

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What are simple, complex, and compound fractions? How do you convert decimals and percentages to fractions?

### Converting Fractions, Decimals, and Percents

I. Background

1. Integers

Integers have no digits to the right of the decimal point. Examples of integers are 628, 3232542364357, and -54.

In the number 628,

8 stands for 8 ones
2 stands for 2 tens
6 stands for 6 hundreds

So 628 represents the sum of 6 hundreds, 2 tens, and 8 ones, or 600 + 20 + 8.

2. Base 10

When we write numbers, we use a system mathematicians call "base 10." In base 10, the first number to the left of the decimal point represents the number of 1's, the second number represents the number of 10's, the third number represents the number of 100's, the fourth number represents the number of 1000's, and so on.

Here are the place values:

Notice that as we move to the right, we divide by 10. We need to understand this to work with decimals.

3. Decimals

We have seen that to the left of the decimal place, the digits represent 1's, 10's, 100's, 1000's, and so on. To find the value of a decimal place, we divide the value of the decimal place to the left of it by 10.

Well, we do the same thing with digits to the right of the decimal place!

Let's look again at the sequence of numbers 1000, 100, 10, 1, and continue the pattern to get new terms by dividing previous terms by 10:

.1 = 1/10
.01 = 1/100
.001 = 1/1000

So just as the digits to the left of the decimal represent 1's, 10's, 100's, and so forth, digits to the right of the decimal point represent 1/10's, 1/100's, 1/1000's, and so forth.

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II. Changing a decimal to a fraction

1. Terminating Decimals

Now that we understand how decimal representation works, let's try to figure out the fractional equivalent of the decimal .345.

The 3 represents 3 tenths,
the 4 represents 4 hundredths
the 5 represents 5 thousandths

So, we can write .345 = 3/10 + 4/100 + 5/1000.

In order to add these fractions, we need to find the common denominator. In this case, it is 1000.

3/10 + 4/100 + 5/1000 = (300 + 40 + 5)/1000 = 345/1000.

The fraction 345/1000 needs some reducing. We can divide each side by 5: 345/1000 reduces to 69/200, so the final answer is:

.345 = 69/200.

Here's a shortcut for dealing with decimals: instead of writing out .345 = 3/10 + 4/100 + 5/1000, then finding the common denominator, adding, and reducing, just think:

.345 = 345/1000.

How about converting .7 or 47.6 to fractions?

.7 = 7/10 or seven tenths
47.6 = 476/10 or four hundred seventy-six tenths

2. Repeating Decimals

If the decimal is a repeating decimal instead of a terminating one, we can still convert it to a fraction. Let's try to figure out the fractional equivalent of 0.5757575757... .

Let F be this fraction. We see that the repeating group has length 2. This tells us to multiply F by 102 = 100. See what happens when we do:

```       F =  0.5757575757...
100 F = 57.5757575757....
```
Now we can subtract the first equation from the second, and the repeating part of the decimal is cancelled:
```    99 F = 57.0000000000... = 57
```
(Now you can see why we chose 102 as a multiplier. It was just to make this cancellation happen.) Now it is easy to find F. Remember to reduce it to lowest terms when you have it in the form of a fraction:
```       F = 57/99 = 19/33.
```
Sure enough, when we do long division, we find that 19/33 = 0.5757575757... .

As another example, let's convert F = 1.3481481481481... to a fraction. Since the repeating group has length 3, we should multiply F by 103 = 1000.

```        F =    1.3481481481481...
1000 F = 1348.1481481481481...
999 F = 1346.8000000000000...
= 1346.8 = 6734/5,
F = 6734/(5*999) = 6734/4995 = 182/135
```

3. Fractions with Small Denominators

If the decimal is a fraction with a small denominator, but the length of the repeating part is long enough that you cannot see a repeat, you can still recover the fraction by expanding the decimal as a simple continued fraction. This is a compound fraction of the form

```                        1
F = a(0) + -------------------------------
1
a(1) + ------------------------
1
a(2) + -----------------
1
a(3) + ----------
a(4) + ...
```
where all the a(n)'s are integers, and all but possibly a(0) are positive. If we start with the value of F, we can compute the integers a(0), a(1), a(2), and so on, by the following method:

a(0) is the largest integer less than or equal to F. Subtract a(0) from F and take the reciprocal, 1/[F-a(0)]. a(1) is the largest integer less than or equal to this quotient. Subtract a(1) from this and take the reciprocal. a(2) is the largest integer less than or equal to this new quotient. Continue this way as many steps as you wish.

If the decimal F is an exact fraction, this process will end with an attempt to take the reciprocal of 0, because one of the quotients will be an integer. If the decimal F is a close approximation to a fraction, this process will encounter a step where you are taking the reciprocal of a very small number, very close to zero. If you stop at this point, you will have a rational number that is very close to the decimal F.

As an example, let's find the fraction for F = 0.127659574... .

```n    Quotient    a(n)  Fraction      Reciprocal      Value So Far

0   0.127659574   0   0.127659574   7.833333362    0/1 = 0.000000000
1   7.833333362   7   0.833333362   1.199999959    1/7 = 0.142857143
2   1.199999959   1   0.199999959   5.000001034    1/8 = 0.125000000
3   5.000001034   5   0.000001034       Stop      6/47 = 0.127659574
```
Then
```             1           1      6
F = 0 + ----------- = ------- = --.
1           5    47
7 + -------   7 + ---
1         6
1 + ---
5
```
Sure enough, 6/47 = 0.127659574468... .

The numerators and denominators of the "Value So Far" column can be computed by starting with 0/1 and 1/0. Multiply the last numerator by a(n) and add it to the numerator before that to get the new numerator, and likewise for the denominators. Above, we had 1/7 and 1/8, and a(3) = 5, so the new value-so-far fraction is (1*5+1)/(8*5+7) = 6/47.

Here is a problem that lends itself to this technique:

"After fewer than 50 at-bats, a baseball player had a batting average of .297. How many hits did he have?"

The batting average is the number of hits H divided by the number of at-bats A, rounded to three decimal places. We need to find H and A with A < 50 and 0.2965 <= H/A < 0.2975. We write 0.297 as a simple continued fraction:

``` n   Quotient   a(n)   Fraction      Reciprocal          Value So Far

0  0.297000000   0   0.297000000   3.367003367      0/1 = 0.000000000 = 0.000
1  3.367003367   3   0.367003367   2.724770642      1/3 = 0.333333333 = 0.333
2  2.724770642   2   0.724770642   1.379746835      2/7 = 0.285714286 = 0.286
3  1.379746835   1   0.379746835   2.633333333     3/10 = 0.300000000 = 0.300
4  2.633333333   2   0.633333333   1.578947368     8/27 = 0.296296296 = 0.296
5  1.578947368   1   0.578947368   1.727272727    11/37 = 0.297297297 = 0.297
6  1.727272727   1   0.727272727   1.375000000    19/64 = 0.296875000 = 0.297
7  1.375000000   1   0.375000000   2.666666667   30/101 = 0.297029703 = 0.297
8  2.666666667   2   0.666666667   1.500000000   79/266 = 0.296992481 = 0.297
9  1.500000000   1   0.500000000   2.000000000  109/367 = 0.297002725 = 0.297
10  2.000000000   2   0.000000000      Stop     297/1000 = 0.297000000 = 0.297
```
Then we have that the fraction that rounds to 0.297 with the smallest denominator is 11/37, and the next one is 19/64. 64 > 50, so H/A must have been 11/37, and H = 11, A = 37 < 50. The player had 11 hits in 37 at-bats.

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III. Changing a fraction to a decimal:

Divide the numerator by the denominator

A. 5/10 (five tenths) = five divided by ten:
```          .5
-----
10 ) 5.0
5 0
----```
So 5/10 (five tenths) = .5 (five tenths).

B. How about 1/2 (one half) or 1 divided by 2 ?

```          .5
-----
2 ) 1.0
1 0
----```
So 1/2 (one half) = .5 (five tenths)

Notice that equivalent fractions convert to
the same decimal representation.

C. What about 2/3 (two thirds) = two divided by three?

```         .666
------
3 )2.000
1 8
---
20
18
--
20
18   etc.```
Here the answer is a repeating decimal.

Multiplying decimals

To multiply two numbers that have decimal points in them, first multiply as if they had no decimal points. Then count up the total number of digits to the right of the decimal point in the two numbers you multiplied, and put a decimal point in the answer so that there are that many digits to the right of the decimal point. (Add zeros to the left of the answer if necessary).

For example:

```      4.36
x  0.8
------
3.488 is the final answer - it has 3 decimal places
since that's how many decimal places there were
in the two original numbers.

0.25
x 0.125
-------
.03125 is the final answer - it has 5 decimal places
since that's how many decimal places there were
in the two original numbers.
```

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IV. Percentage

Percent means per hundred. 10 percent is just another way of saying "ten out of a hundred," or ten hundredths. If exactly 10 percent of your 100 M&Ms are yellow, then 90 of them are some other color and 10 of them are yellow.

To convert a fraction to a percentage, divide the numerator by the denominator. Then move the decimal point two places to the right (which is the same as multiplying by 100) and add a percent sign.

For example: Given the fraction  5/8  what is the percentage?

```         .625     .625 x 100 = 62.5 or 62.5%
------
8 )5.000
4 8
---
20
16
--
40
40```
So 5/8 of your M&Ms would be 62 1/2 of every 100.

To change a percentage to a fraction, divide it by 100 and reduce the fraction or move the decimal point to the right until you have only integers:

10% = 10/100 = 1/10

62.5% = 62.5/100 = 625/1000
625/1000 = 125/200 = 25/40 = 5/8

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