Adding Fractions Calculator

Adding fractions can be tricky because of one simple fact: Unless the pieces that make up a fraction are all the same size, you can't just add them like whole numbers. A third and a fourth represent different sizes, so they must first be converted to have the same denominator.

This calculator is designed for that purpose. Enter two fractions, select addition or subtraction and the calculator will find the least common denominator, convert each fraction to have this denominator then perform the calculation. The result is then shown as a reduced fraction. The calculator shows the steps involved and also gives the answer in mixed number form, decimal form and percentage form.

Each fraction has an optional integer field so mixed numbers like "two and a half" can be entered directly without first converting to improper fractions. If you need to add or subtract three fractions at once, just enable the third fraction field.

Fractions with same denominator and fractions with different denominators:

When the denominators of two fractions are already equal they are called "like" or "similar" fractions and addition is very simple. Since each part has the same size, only the numerators are added together while the denominator stays the same.

$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$

In most cases the denominators of two fractions will be different. These are called "unlike" fractions and that is why an extra step is required to add them. First both fractions must be converted into equivalent fractions with a common denominator before the numerators can be added together.

How to use least common denominator:

The required common denominator is called the least common denominator (LCD), often abbreviated as LCD. It is the smallest number that can be divided by all denominators, i.e., the least common multiple of these denominators.

For example, if you are adding half and a third, the denominators are 2 and 3. The smallest number divisible by both of those is 6, so that will be the common denominator. Multiply each fraction's numerator and denominator by the same factor to get a denominator of 6 in each case.

$\frac{1}{2} = \frac{3}{6} \qquad \frac{1}{3} = \frac{2}{6}$

Now that all of the fractions have the same denominator, you can just add the numerators and leave the denominators alone.

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

The rule for adding two fractions with a common denominator can be summarized into one formula: At the end, the result is simplified to a reduced fraction.

$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot \frac{L}{b} + c \cdot \frac{L}{d}}{L}, \quad L = \operatorname{lcm}(b, d)$

A faster method is to simply multiply the two denominators together to get a common denominator. This method always works, but it may result in larger numbers than necessary at the end. Using the least common denominator allows you to keep your intermediate calculations smaller and easier to manage. Both methods will give the same reduced fraction as the final answer. This calculator automatically performs the simplification for you.

Subtracting fractions

Subtraction is done in exactly the same way. First both fractions are converted to have a common denominator, and instead of adding the numerators, you subtract the numerator of the second fraction from the numerator of the first one. Subtracting a sixth from three quarters effectively gives nine twelfths minus two twelfths, which is seven twelfths.

$\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$

If the subtrahend is larger than the original fraction, then the result will be a negative number. This is completely normal and if you set the operator to subtraction, the calculator automatically processes the signs.

Addition of mixed numbers

Mixed numbers are whole numbers mixed with fractions, for example two and three-quarters. There are two simple ways to add mixed numbers: you can either add the whole numbers and the fractions separately or convert each mixed number to an improper fraction first.

To convert a mixed number to an improper fraction, first multiply the whole number by the denominator and then add the numerator. Two and three fourths becomes an improper fraction by multiplying two by four which is eight, and then adding three which gives eleven, so it's eleven fourths.

$2\tfrac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$

When a whole number is entered into the integer field, the calculator will automatically convert it to a fraction and display the result as both an improper fraction and mixed number.

Simultaneous calculation of three fractions:

By enabling a third fraction, the same method can be extended to three fractions. In this case, the least common denominator must contain all three denominators, so the least common multiple of all three denominators is computed. Each fraction is converted to have the common denominator and then the numerators are added or subtracted in their original order.

For example, if you have a half factor and a third factor plus a sixth factor, the least common denominator is 6. The three fractions are then three sixths, two sixths, and one sixth. When added together they give six sixths, which equals exactly 1.

$\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = 1$

Reduction of a result to a shortened fraction:

A fraction is reduced if the numerator and denominator have no common factors other than 1. To reduce a fraction, both the numerator and denominator are simultaneously divided by their greatest common divisor. Since the numerator and denominator of 22/24 share the common factor of 2, it can be reduced to 11/12 which is its most reduced form.

This calculator automatically reduces any answer to its lowest common denominator and shows the factors that were removed by reduction in a step-by-step calculation panel, making it easy to understand the reduction process.

Convert answers to decimals or percentages.

Since a fraction is a division problem, any fraction can be converted to a decimal by dividing the numerator (the top number) by the denominator (the bottom number). Five-sixths equals five divided by six which results in the answer of .8333. When you multiply a decimal by one hundred, it gives you a percentage so five-sixths is equal to approximately 83.3 percent.

Examples of calculations:

If you do the same addition and subtraction with two fractions, you will notice that the result changes depending on which operator is used while the method of calculation remains the same.

Commonly used least common denominator.

There are some very common combinations of denominators and it is useful to know their least common multiples.

Note that the least common multiple of 2 and 4 is 4, not 8. This is because 4 is a multiple of 2. Often, the least common multiple will be smaller than the product of the two denominators, which makes calculations easier.