Fraction Calculator

A fraction is a number that represents a value when a whole is divided into equal parts. The top part (numerator) indicates how many of those parts there are while the bottom part (denominator) specifies how many parts are needed to make up the whole. This calculator allows you to input two fractions and perform addition, subtraction, multiplication or division operations on them, displaying the result in its simplest form as a mixed number or decimal.

Every fraction has an optional integer input field so you can directly enter mixed numbers like "two and three quarters" without having to convert them first. If this field is left at zero, normal fractions are treated.

The individual parts of a fraction:

Imagine a pie cut into 8 equal slices. If you take three of those slices, then you have three-eighths of the pie. The numerator (3) tells how many slices were taken. The denominator (8) tells how many slices there are in total.

Symbolically this same thing means that the fraction bar just indicates that the top number is being divided by the bottom number.

The numerator and denominator are represented as follows: $\frac{\text{numerator}}{\text{denominator}} = \frac{3}{8}$

The only rule that must be followed is that the denominator cannot be zero. A denominator of zero means there is nothing to divide into and so fractions with a denominator of zero are undefined.

Addition and subtraction of fractions

Fractions cannot be added like whole numbers because the size of a part is different in thirds and fourths. To add two fractions, you rewrite them with a common denominator. The easiest common denominator to use is the product of the two denominators. Then you just add or subtract the numerators and use that same denominator.

The formula for adding fractions is: $\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}$

For example, if we were adding three quarters and one sixth together, the lowest common denominator would be 24, so our sum would be 22 over 24, which can then be simplified to 11 over 12.

Here's a calculation example: $\frac{3}{4} + \frac{1}{6} = \frac{18}{24} + \frac{4}{24} = \frac{22}{24} = \frac{11}{12}$

Subtraction works the same way as addition except you subtract the numerator instead of adding it. If we take one sixth away from three fourths, we get seven twelfths.

Multiplying the denominators together will always give you a common denominator, but it might not be the least common denominator. Using the least common multiple allows you to keep numbers smaller while you're calculating. No matter which method you use, you'll end up with the same simplified answer in the end. This calculator tool simplifies the answer at the end.

Multiplying fractions

Multiplying is very simple. You don't have to find a common denominator. You just multiply the two numerators to get the new numerator and you multiply the two denominators to get the new denominator, then simplify if necessary.

The formula for multiplying fractions is: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

When you multiply three quarters by one sixth, you get three twenty-fourths, which can be simplified to one eighth. As a practical shortcut, before multiplying, you can cancel out common factors between the numerator and denominator to keep the numbers small.

Division of fractions

Dividing fractions is the same as multiplying by the reciprocal. The reciprocal of a fraction is found by switching the numerator and denominator. You flip the second fraction, change the division to multiplication, then proceed with the calculation like you would for regular multiplication.

The formula for dividing fractions is: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

For example, if you divide three quarters by one sixth, it is the same as multiplying three quarters by six, which gives eighteen quarters or nine halves. The reciprocal of a number is obtained by dividing that number by one. In fractions, this simply means swapping the numerator and denominator.

Equivalent fractions and simplifying fractions

If two fractions represent the same point on the number line, they are equivalent fractions. For example, one-half and two-hundred-twenty over four-hundred-forty are equivalent. A fraction in simplest form has a numerator and denominator that are as small as possible. You can get this by dividing both the numerator and denominator by their greatest common divisor.

Since both the numerator and denominator of 22/24 are divisible by 2, we can simplify it by dividing them by 2 to get 11/12. This fraction cannot be simplified further. All answers will automatically be shown in their simplest form here.

Mixed and improper fractions.

An improper fraction has a numerator that is greater than or equal to the denominator, such as 17/12. A mixed number represents the same value using a whole number and a proper fraction, such as 1 5/12. Both are different ways of writing the same number.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole part and the remainder becomes the numerator over the original denominator. To go the other way, multiply the whole part by the denominator then add the numerator. This calculator allows you to enter mixed numbers directly in the whole number field.

Convert fractions to decimals or percentages.

Since a fraction represents division, any fraction can be converted into a decimal by performing the division. Three-eighths is the result of dividing three by eight, which is 0.375. To convert a decimal to a percentage, multiply it by one hundred. Therefore, the percentage of three-eighths is 37.5.

Some fractions can be represented as finite decimals. For example 0.375. On the other hand there are also fractions where a certain sequence of digits is repeated infinitely, such as one third (0.333). A fraction can only be exactly represented by a finite decimal if the denominator of the reduced fraction contains only factors two and five.

Overview of examples for the four basic arithmetic operations:

The four operations are each performed on the same pair of numbers to show how different the results can be.

Note that the result of a division is larger than both original fractions. This is because you are dividing by a number less than one, which results in a larger answer.

Common fractions and their equivalent decimal values:

There are some very common fractions whose decimal equivalents are worth knowing.