Fraction Calculator

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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Big Number Fraction Calculator

Use this calculator if the numerators or denominators are very big integers.

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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

38, 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 58 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:

, 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 beas 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.

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.

ab
+
cd
=
a×db×d
+
c×bd×b
=
ad + bcbd

EX:
34
+
16
=
3×64×6
+
1×46×4
=
2224
=
1112

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:

14
+
16
+
12
=
1×6×24×6×2
+
1×4×26×4×2
+
1×4×62×4×6

=

1248
+
848
+
2448
=
4448
=
1112

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:

14
+
16
+
12
=
1×34×3
+
1×26×2
+
1×62×6

=

312
+
212
+
612
=
1112

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.

ab
–
cd
=
a×db×d
–
c×bd×b
=
ad – bcbd

EX:
34
–
16
=
3×64×6
–
1×46×4
=
1424
=
712

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.

ab
×
cd
=
acbd

EX:
34
×
16
=
324
=
18

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

1a . When a is a fraction, this essentially involves exchanging the position of the numerator and the denominator. The reciprocal of the fraction 34 would therefore be 43. Refer to the equations below for clarification.

ab
/
cd
=
ab
×
dc
=
adbc

EX:
34
/
16
=
34
×
61
=
184
=
92

Simplification:

. When a is a fraction, this essentially involves exchanging the position of the numerator and the denominator. The reciprocal of the fractionwould therefore be. Refer to the equations below for clarification.

It is often easier to work with simplified fractions. As such, fraction solutions are commonly expressed in their simplified forms.

220440 for example, is more cumbersome than 12. 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:

for example, is more cumbersome than. 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 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 101, the second 102, the third 103, 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 104, or 10,000. This would make the fraction

123410000, which simplifies to 6175000, since the greatest common factor between the numerator and denominator is 2.

, which simplifies to, 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

12 for example. To convert this fraction into a decimal, first convert it into the fraction of 510. Knowing that the first decimal place represents 10-1, 510 can be converted to 0.5. If the fraction were instead 5100, 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

for example. To convert this fraction into a decimal, first convert it into the fraction of. Knowing that the first decimal place represents 10can be converted to 0.5. If the fraction were instead, the decimal would then be 0.05, and so on. Beyond this, converting fractions into decimals requires the operation of long division.

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.

64th32nd16th8th4th2ndDecimalDecimal
(inch to mm)

1/64     0.0156250.396875
2/641/32    0.031250.79375
3/64     0.0468751.190625
4/642/321/16   0.06251.5875
5/64     0.0781251.984375
6/643/32    0.093752.38125
7/64     0.1093752.778125
8/644/322/161/8  0.1253.175
9/64     0.1406253.571875
10/645/32    0.156253.96875
11/64     0.1718754.365625
12/646/323/16   0.18754.7625
13/64     0.2031255.159375
14/647/32    0.218755.55625
15/64     0.2343755.953125
16/648/324/162/81/4 0.256.35
17/64     0.2656256.746875
18/649/32    0.281257.14375
19/64     0.2968757.540625
20/6410/325/16   0.31257.9375
21/64     0.3281258.334375
22/6411/32    0.343758.73125
23/64     0.3593759.128125
24/6412/326/163/8  0.3759.525
25/64     0.3906259.921875
26/6413/32    0.4062510.31875
27/64     0.42187510.715625
28/6414/327/16   0.437511.1125
29/64     0.45312511.509375
30/6415/32    0.4687511.90625
31/64     0.48437512.303125
32/6416/328/164/82/41/20.512.7
33/64     0.51562513.096875
34/6417/32    0.5312513.49375
35/64     0.54687513.890625
36/6418/329/16   0.562514.2875
37/64     0.57812514.684375
38/6419/32    0.5937515.08125
39/64     0.60937515.478125
40/6420/3210/165/8  0.62515.875
41/64     0.64062516.271875
42/6421/32    0.6562516.66875
43/64     0.67187517.065625
44/6422/3211/16   0.687517.4625
45/64     0.70312517.859375
46/6423/32    0.7187518.25625
47/64     0.73437518.653125
48/6424/3212/166/83/4 0.7519.05
49/64     0.76562519.446875
50/6425/32    0.7812519.84375
51/64     0.79687520.240625
52/6426/3213/16   0.812520.6375
53/64     0.82812521.034375
54/6427/32    0.8437521.43125
55/64     0.85937521.828125
56/6428/3214/167/8  0.87522.225
57/64     0.89062522.621875
58/6429/32    0.9062523.01875
59/64     0.92187523.415625
60/6430/3215/16   0.937523.8125
61/64     0.95312524.209375
62/6431/32    0.9687524.60625
63/64     0.98437525.003125
64/6432/3216/168/84/42/2125.4

Transcript

Ex 1.3, 3
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A

Here, A = {1, 2, {3, 4,} 5}

Let {3, 4} = x
So, A = {1, 2, x, 5}
3, 4 is in {3, 4} but not in A

So, {3, 4} is not a subset of A

So, the given statement is incorrect.

Ex 1.3, 3
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A

Here, A = {1, 2, {3, 4,} 5}

Let {3, 4} = x
So, A = {1, 2, x, 5}
3, 4 is in {3, 4} but not in A

So, {3, 4} is not a subset of A

So, the given statement is incorrect.

⊂ – is a subset
A ⊂ B if all elements of A are in B
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(ii) {3, 4} ∈ A

Let {3, 4} = x
A = {1, 2, {3, 4,}, 5}
So, A = {1, 2, x, 5}

Now, x is in set A
So, x ∈ A

∴ The statement is correct
∈ – belongs to
relation between element & set
eg: A = {1, 2, 3, 4}
2 is in set A
So, 2 ∈ A
Ex 1.3, 3
Let A= {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(iii) {{3, 4}}⊂ A

{{3, 4}} is a set containing {3, 4}
as an element
Let {3, 4} = x
So, {{3, 4}} = {x}

Also, A = {1, 2, {3, 4}, 5}
So, A = {1, 2, x, 5}

So, x is in {x} and also in A
⊂ – is a subset
A ⊂ B if all elements of A are in B
So, {x} is a subset of A
So, the given statement is correct

Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(iv) 1 ∈ A

1 is an element of A.

The statement is correct

∈ – belongs to
Relation b/w element & set
Example: A = {1,2,3,4}
2 is in set A
So, 2 ∈ A
⊂ – is a subset
A ⊂ B if all elements of A are in B
(v) 1 ⊂ A

Here, we are given 1
1 is an element, not a set.

For subset, we should have two sets
Element cannot be subset of a set.

Therefore, the statement is incorrect.
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(vi) {1, 2, 5} ⊂ A

Each element of the set {1, 2, 5}
i.e. 1,2,5 is also an element of A.

Therefore, statement is correct
⊂ – is a subset
A ⊂ B if all elements of A are in B
∈ – belongs to
Relation b/w element & set
Example: A = {1,2,3,4}
2 is in set A
So, 2 ∈ A
(vii) {1, 2, 5} ∈ A

Now, {1, 2, 5} is not an element
of A.
It will be if
A = {1, 2, {3, 4}, 5, {1, 2, 5} }

∴ The statement is Incorrect
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(viii) {1, 2, 3} ⊂ A

Here, A = {1, 2, {3, 4,}, 5}

Let {3, 4} = x
So, A = {1, 2, x, 5}

3 is in {1, 2, 3} but not in A
So, {1, 2, 3} is not a subset of A

So, the given statement is incorrect.
⊂ – is a subset
A ⊂ B if all elements of A are in B
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(ix) ϕ ∈ A

ϕ is not an element of A.
(It will be If A= {1, ϕ, 2, {3, 4,}, 5} )

The statement is incorrect
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(ix) ϕ ∈ A

ϕ is not an element of A.
(It will be If A= {1, ϕ, 2, {3, 4,}, 5} )

The statement is incorrect
∈ – (belongs to) element in set
2 ∈ A if 2 is in set A
Ex 1.3, 3
Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(x) ϕ ⊂ A

Null set is a subset
of each set.
So, the statement is correct.
⊂ – is a subset
A ⊂ B if all elements of A are in B
(xi) {ϕ} ⊂ A

{ϕ} is a set containing null set
{ϕ} ⊂ A is possible only if
ϕ is in set A
(Eg: If A= {1, ϕ, 2, {3, 4,}, 5} )
But it is not there.
So, {ϕ} is not a subset of A.

Therefore the statement is incorrect.