What are simple, complex, and compound fractions?
How do you convert decimals and percentages to fractions?
Converting Fractions, Decimals, and Percents
I. Background
 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.
 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.
 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.
[Background]
[Decimal>Fraction]
[Fraction>Decimal]
[Percentage]
[Archives]
II. Changing a decimal to a fraction
 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:
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:
How about converting .7 or 47.6 to fractions?
.7 = 7/10 or seven tenths
47.6 = 476/10 or four hundred seventysix tenths
 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 10^{2} = 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 10^{2} 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
10^{3} = 1000.
F = 1.3481481481481...
1000 F = 1348.1481481481481...
999 F = 1346.8000000000000...
= 1346.8 = 6734/5,
F = 6734/(5*999) = 6734/4995 = 182/135
 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/[Fa(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 valuesofar fraction is
(1*5+1)/(8*5+7) = 6/47.
Here is a problem that lends itself to this technique:
"After fewer than 50 atbats, 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
atbats 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 atbats.
[Background]
[Decimal>Fraction]
[Fraction>Decimal]
[Percentage]
[Archives]
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

3488 is the answer before adding the decimal point
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

3125 is the answer before adding the decimal point
.03125 is the final answer  it has 5 decimal places
since that's how many decimal places there were
in the two original numbers.
[Background]
[Decimal>Fraction]
[Fraction>Decimal]
[Percentage]
[Archives]
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:
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