Below are multiple fraction calculators capable of addition, subtraction, multiplication, division, simplification, and conversion between fractions and decimals. Fields above the solid black line represent the numerator, while fields below represent the denominator.
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Mixed Numbers Calculator
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Simplify Fractions Calculator
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Decimal to Fraction Calculator
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Fraction to Decimal Calculator
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In mathematics, a fraction is a number that represents a part of a whole. It consists of a numerator and a denominator. The numerator represents the number of equal parts of a whole, while the denominator is the total number of parts that make up said whole. For example, in the fraction of
3
8
, the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore be
5
8
as shown in the image to the right. Note that the denominator of a fraction cannot be 0, as it would make the fraction undefined. Fractions can undergo many different operations, some of which are mentioned below.
Addition:
Unlike adding and subtracting integers such as 2 and 8, fractions require a common denominator to undergo these operations. One method for finding a common denominator involves multiplying the numerators and denominators of all of the fractions involved by the product of the denominators of each fraction. Multiplying all of the denominators ensures that the new denominator is certain to be a multiple of each individual denominator. The numerators also need to be multiplied by the appropriate factors to preserve the value of the fraction as a whole. This is arguably the simplest way to ensure that the fractions have a common denominator. However, in most cases, the solutions to these equations will not appear in simplified form (the provided calculator computes the simplification automatically). Below is an example using this method.
a
b
+
c
d
=
a×d
b×d
+
c×b
d×b
=
ad + bc
bd
EX:
3
4
+
1
6
=
3×6
4×6
+
1×4
6×4
=
22
24
=
11
12
This process can be used for any number of fractions. Just multiply the numerators and denominators of each fraction in the problem by the product of the denominators of all the other fractions (not including its own respective denominator) in the problem.
EX:
1
4
+
1
6
+
1
2
=
1×6×2
4×6×2
+
1×4×2
6×4×2
+
1×4×6
2×4×6
=
12
48
+
8
48
+
24
48
=
44
48
=
11
12
An alternative method for finding a common denominator is to determine the least common multiple (LCM) for the denominators, then add or subtract the numerators as one would an integer. Using the least common multiple can be more efficient and is more likely to result in a fraction in simplified form. In the example above, the denominators were 4, 6, and 2. The least common multiple is the first shared multiple of these three numbers.
Multiples of 2: 2, 4, 6, 8 10, 12
Multiples of 4: 4, 8, 12
Multiples of 6: 6, 12
The first multiple they all share is 12, so this is the least common multiple. To complete an addition (or subtraction) problem, multiply the numerators and denominators of each fraction in the problem by whatever value will make the denominators 12, then add the numerators.
EX:
1
4
+
1
6
+
1
2
=
1×3
4×3
+
1×2
6×2
+
1×6
2×6
=
3
12
+
2
12
+
6
12
=
11
12
Subtraction:
Fraction subtraction is essentially the same as fraction addition. A common denominator is required for the operation to occur. Refer to the addition section as well as the equations below for clarification.
a
b
–
c
d
=
a×d
b×d
–
c×b
d×b
=
ad – bc
bd
EX:
3
4
–
1
6
=
3×6
4×6
–
1×4
6×4
=
14
24
=
7
12
Multiplication:
Multiplying fractions is fairly straightforward. Unlike adding and subtracting, it is not necessary to compute a common denominator in order to multiply fractions. Simply, the numerators and denominators of each fraction are multiplied, and the result forms a new numerator and denominator. If possible, the solution should be simplified. Refer to the equations below for clarification.
a
b
×
c
d
=
ac
bd
EX:
3
4
×
1
6
=
3
24
=
1
8
Division:
The process for dividing fractions is similar to that for multiplying fractions. In order to divide fractions, the fraction in the numerator is multiplied by the reciprocal of the fraction in the denominator. The reciprocal of a number a is simply
1
a
. When a is a fraction, this essentially involves exchanging the position of the numerator and the denominator. The reciprocal of the fraction
3
4
would therefore be
4
3
. Refer to the equations below for clarification.
a
b
/
c
d
=
a
b
×
d
c
=
ad
bc
EX:
3
4
/
1
6
=
3
4
×
6
1
=
18
4
=
9
2
Simplification:
It is often easier to work with simplified fractions. As such, fraction solutions are commonly expressed in their simplified forms.
220
440
for example, is more cumbersome than
1
2
. The calculator provided returns fraction inputs in both improper fraction form as well as mixed number form. In both cases, fractions are presented in their lowest forms by dividing both numerator and denominator by their greatest common factor.
Converting between fractions and decimals:
Converting from decimals to fractions is straightforward. It does, however, require the understanding that each decimal place to the right of the decimal point represents a power of 10; the first decimal place being 10^{1}, the second 10^{2}, the third 10^{3}, and so on. Simply determine what power of 10 the decimal extends to, use that power of 10 as the denominator, enter each number to the right of the decimal point as the numerator, and simplify. For example, looking at the number 0.1234, the number 4 is in the fourth decimal place, which constitutes 10^{4}, or 10,000. This would make the fraction
1234
10000
, which simplifies to
617
5000
, since the greatest common factor between the numerator and denominator is 2.
Similarly, fractions with denominators that are powers of 10 (or can be converted to powers of 10) can be translated to decimal form using the same principles. Take the fraction
1
2
for example. To convert this fraction into a decimal, first convert it into the fraction of
5
10
. Knowing that the first decimal place represents 10^{-1},
5
10
can be converted to 0.5. If the fraction were instead
5
100
, the decimal would then be 0.05, and so on. Beyond this, converting fractions into decimals requires the operation of long division.
Common Engineering Fraction to Decimal Conversions
In engineering, fractions are widely used to describe the size of components such as pipes and bolts. The most common fractional and decimal equivalents are listed below.
64^{th}
32^{nd}
16^{th}
8^{th}
4^{th}
2^{nd}
Decimal
Decimal (inch to mm)
1/64
0.015625
0.396875
2/64
1/32
0.03125
0.79375
3/64
0.046875
1.190625
4/64
2/32
1/16
0.0625
1.5875
5/64
0.078125
1.984375
6/64
3/32
0.09375
2.38125
7/64
0.109375
2.778125
8/64
4/32
2/16
1/8
0.125
3.175
9/64
0.140625
3.571875
10/64
5/32
0.15625
3.96875
11/64
0.171875
4.365625
12/64
6/32
3/16
0.1875
4.7625
13/64
0.203125
5.159375
14/64
7/32
0.21875
5.55625
15/64
0.234375
5.953125
16/64
8/32
4/16
2/8
1/4
0.25
6.35
17/64
0.265625
6.746875
18/64
9/32
0.28125
7.14375
19/64
0.296875
7.540625
20/64
10/32
5/16
0.3125
7.9375
21/64
0.328125
8.334375
22/64
11/32
0.34375
8.73125
23/64
0.359375
9.128125
24/64
12/32
6/16
3/8
0.375
9.525
25/64
0.390625
9.921875
26/64
13/32
0.40625
10.31875
27/64
0.421875
10.715625
28/64
14/32
7/16
0.4375
11.1125
29/64
0.453125
11.509375
30/64
15/32
0.46875
11.90625
31/64
0.484375
12.303125
32/64
16/32
8/16
4/8
2/4
1/2
0.5
12.7
33/64
0.515625
13.096875
34/64
17/32
0.53125
13.49375
35/64
0.546875
13.890625
36/64
18/32
9/16
0.5625
14.2875
37/64
0.578125
14.684375
38/64
19/32
0.59375
15.08125
39/64
0.609375
15.478125
40/64
20/32
10/16
5/8
0.625
15.875
41/64
0.640625
16.271875
42/64
21/32
0.65625
16.66875
43/64
0.671875
17.065625
44/64
22/32
11/16
0.6875
17.4625
45/64
0.703125
17.859375
46/64
23/32
0.71875
18.25625
47/64
0.734375
18.653125
48/64
24/32
12/16
6/8
3/4
0.75
19.05
49/64
0.765625
19.446875
50/64
25/32
0.78125
19.84375
51/64
0.796875
20.240625
52/64
26/32
13/16
0.8125
20.6375
53/64
0.828125
21.034375
54/64
27/32
0.84375
21.43125
55/64
0.859375
21.828125
56/64
28/32
14/16
7/8
0.875
22.225
57/64
0.890625
22.621875
58/64
29/32
0.90625
23.01875
59/64
0.921875
23.415625
60/64
30/32
15/16
0.9375
23.8125
61/64
0.953125
24.209375
62/64
31/32
0.96875
24.60625
63/64
0.984375
25.003125
64/64
32/32
16/16
8/8
4/4
2/2
1
25.4
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