# An Intellectual Arithmetic Upon The Inductive Method

Below are multiple fraction thietkewebshop.vns capable of addition, subtraction, multiplication, division, simplification, và conversion between fractions and decimals. Fields above the solid black line represent the numerator, while fields below represent the denominator.

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 + - x / = ? ?

## Mixed Numbers thietkewebshop.vn

 + - x / = ?

 =?

 = ? ?

 = ?

## Big Number Fraction thietkewebshop.vn

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

 + - x / = ?

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 lớn eat 3 slices, the remaining fraction of the pie would therefore be
 5 8
as shown in the image to the right. Lưu ý 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 & subtracting integers such as 2 và 8, fractions require a common denominator lớn undergo these operations. One method for finding a common denominator involves multiplying the numerators và 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 khổng lồ ensure that the fractions have a common denominator. However, in most cases, the solutions khổng lồ these equations will not appear in simplified form (the provided thietkewebshop.vn computes the simplification automatically). Below is an example using this method.

 a b
+ c
d
= a×d
b×d
+ c×b
d×b
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 hàng hóa of the denominators of all the other fractions (not including its own respective denominator) in the problem.

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 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 lớn determine the least common multiple (LCM) for the denominators, then địa chỉ cửa hàng or subtract the numerators as one would an integer. Using the least common multiple can be more efficient và is more likely to result in a fraction in simplified form. In the example above, the denominators were 4, 6, và 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. Lớn complete an addition (or subtraction) problem, multiply the numerators & denominators of each fraction in the problem by whatever value will make the denominators 12, then địa chỉ 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 lớn occur. Refer khổng lồ the addition section as well as the equations below for clarification.

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 a b
– c
d
= a×d
b×d
– c×b
d×b
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 lớn compute a common denominator in order to lớn multiply fractions. Simply, the numerators and denominators of each fraction are multiplied, và the result forms a new numerator & denominator. If possible, the solution should be simplified. Refer lớn 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 lớn 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 & the denominator. The reciprocal of the fraction
 3 4
would therefore be
 4 3
. Refer khổng lồ the equations below for clarification.

 a b
/ c
d
= a
b
× d
c
bc
 EX: 3 4
/ 1
6
= 3
4
× 6
1
= 18
4
= 9
2

### Simplification:

It is often easier lớn 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 thietkewebshop.vn provided returns fraction inputs in both improper fraction khung as well as mixed number form. In both cases, fractions are presented in their lowest forms by dividing both numerator và denominator by their greatest common factor.

### Converting between fractions and decimals:

Converting from decimals lớn fractions is straightforward. It does, however, require the understanding that each decimal place to the right of the decimal point represents a power nguồn 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 khổng lồ the right of the decimal point as the numerator, & 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

 1234 10000
, which simplifies to
 617 5000
, since the greatest common factor between the numerator và denominator is 2.

Similarly, fractions with denominators that are powers of 10 (or can be converted to lớn powers of 10) can be translated to decimal form using the same principles. Take the fraction

 1 2
for example. Khổng lồ 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 form size of components such as pipes and bolts. The most common fractional and decimal equivalents are listed below.

 64th 32nd 16th 8th 4th 2nd Decimal Decimal(inch lớn 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