Reduce and expand fractions calculator

With this calculator fractions can be reduced or expanded. For reducing, you can select whether the fraction should be reduced by dividing the numerator and denominator by a certain number or whether it should be reduced to its lowest terms.

Expand fractions

To expand a fraction, both the numerator and denominator must be multiplied by the same number.

As an example, the fraction

should be expanded by multiplying the numerator and denominator by 5. Then one calculates:

4∙5

7∙5

Reduce fractions

To reduce a fraction, both the numerator and denominator must be divided by the same number.

For example, if the fraction

should be reduced by dividing by 8, then one calculates:

16:8

56:8

It is important that both the numerator and denominator must be divisible by the number by which is divided.

can, for example, be reduced by dividing by 8, 4 or 2, but not by dividing by 7, because 16 is not divisible by 7.

Reducing fractions is often used to make it easier to continue calculating with the fractions. For example, it is usually much easier to continue calculating with

than with

314

1256

. Also, reduced fractions are often more meaningful to people than fractions with large numbers.

Find common divisors:

Divisibility rule:

Finding common denominators can be very tedious and sometimes (apart from 1) there is no common divisor at all. So-called divisibility rules can help to find common divisors. The following list contains some important divisibility rules. But there are more:

a number is divisible by 2 exactly when the last digit is even
a number is divisible by 3 exactly when the sum of its digits is divisible by 3
a number is divisible by 4 exactly when the number formed by the last 2 digits is divisible by 4
a number is divisible by 5 exactly if the last digit is a 0 or a 5
a number is divisible by 6 exactly when the number is divisible by 2 and by 3
a number is divisible by 8 exactly when the number formed by the last 3 digits is divisible by 8
a number is divisible by 9 exactly when the sum of the digits is divisible by 9
a number is divisible by 10 exactly when the last digit is a 0

As an example, divisors of 1428 are to be found:

divisibility by 2: The last digit (8) is even. Thus, 1428 is divisible by 2.

divisibility by 3: The sum of the digits of 1428 is 1+4+2+8 = 15. 15 is divisible by 3 and therefore 1428 is also divisible by 3.

divisibility by 4: The number formed by the last two digits is 28, and this is divisible by 4. Therefore, 1428 is also divisible by 4.

divisibility by 5: 1428 does not end with a 0 or a 5 and is therefore not divisible by 5.

divisibility by 6: 1428 is divisible by both 2 and 3, so it is also divisible by 6.

divisibility by 8: The number formed from the last 3 digits is 428. 400 is divisible by 8, but 28 is not. Thus, 428 is not divisible by 8 and this means that 1428 is not divisible by 8 either.

divisibility by 9: The sum of the digits is 15 and 15 is not divisible by 9. Therefore, 1428 is also not divisible by 9.

divisibility by 10: 1428 does not end with a 0 and is therefore not divisible by 10.

1428 is therefore divisible by 2, 3, 4, and 6, but there may be other divisors as well.

Suppose you want to simplify the fraction

927

1428

. Then you can apply the divisibility rules to the 927 as well. Then you get 3 and 9 as divisors. Again, there could be more divisors for 927, but 3 and 9 are the divisors found using above rules. Now we compare the found divisors of the two numbers and find out that 3 is a common divisor. So it is a good idea to reduce the fraction by dividing the numerator and denominator by 3. This results in

309

476

. Now you could apply the rules again to the new numerator and the new denominator. Then for 309 you get 3 as divisor and for 476 you get 2 and 4. Thus, with the rules applied, no further common divisor could be found.

Advantages and disadvantages: When searching for common divisors using divisibility rules, one does not always get the greatest common divisor, and this results in fractions not always being reduced to its lowest terms. However, the application of the divisibility rules is relatively simple and thus they are practical.

Prime factorization:

Any integer greater than 0 can be represented as a product of primes. The individual factors of the product are called prime factors.

For example, 140 can be represented as follows: 2∙2∙5∙7
And 84 as follows: 84 = 2∙2∙3∙7
The prime factors of 140 are thus: 2, 2, 5, 7
And prime factors of 84: 2, 2, 3, 7

The greatest common divisor (GCD) can now be determined from the two prime factorizations. To do this, one must form the product of all prime factors that occur in both prime factorizations. If a number appears multiple times in both prime factorizations, this number is also written multiple times in the product for calculating the greatest common divisor.

In the example, the common prime factors are 2, 2 and 7. Thus one calculates 2∙2∙7 = 28. Thus, 28 is the greatest common divisor of 140 and 84.

If you now shorten the fraction

140

by dividing the numerator and denominator by 28, you get

and that is reduced to its lowest terms.

Advantages and disadvantages: With the help of the prime factorization the greatest common divisor of the numerator and the denominator can be determined. When the numerator and denominator are divided by the greatest common divisor, the resulting fraction is reduced to its simplest terms. With small numbers, prime factorizations are often relatively easy to perform. For larger numbers, on the other hand, it can require a lot of effort and be impractical.

Euclidean algorithm:

Using the Euclidean algorithm, the greatest common divisor of 2 numbers can be calculated. In each step of the algorithm, the larger number is divided by the smaller number with remainder. In the next step, the smaller number becomes the new larger number and the remainder becomes the new smaller number. This is repeated until the remainder is 0. The GCD is the smaller number in the last row.

As an example, the GCD of 140 and 84 shall be determined again. So in the first step you divide 140 by 84 with remainder.

You get: 14:84 = 1 R 56

In the next step, 84 becomes the new larger number and 56 becomes the new smaller number.
So one calculates: 84:56 = 1 R 28

The next step is to divide the 56 by the 28:
56:28 = 2 R 0

The remainder is 0 and therefore the algorithm can be terminated. The greatest common divisor of 140 and 84 is the smaller number in the last step. So the 28.

Advantages and disadvantages: Like the prime factorization, the Euclidean algorithm results in the greatest common divisor. After the the numerator and denominator of the fraction are divided by the GCD, the fraction will be reduced to its lowest terms. With the Euclidean algorithm, however, there is a very clear procedure and there is no need to "try around". However, you often have to do multiple divisions for the Euclidean algorithm, which can be time-consuming if you do it by hand.