How to Add and Subtract Fractions with Mental Maths

While addition and subtraction of integers is usually straightforward, there are more steps involved when you add or subtract fractions.

A fraction consists of a number—the numerator—divided by another number—called the denominator. It is usual for both these numbers to be positive integers (whole numbers).

For example, in \(\frac{4}{15}\), the numerator is 4, and the denominator is 15.

General Formula

The basic formula for adding fractions is:

\(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)

In words, this means that you multiply each numerator by the opposite demonimator, and add these results to get the new numerator. The new denominator is the product of the original denominators.

Subtraction uses the same calculation, except using a minus rather than a plus:

\(\frac{a}{b} – \frac{c}{d} = \frac{ad – bc}{bd}\)

On this page, I’ll use examples using addition calculations only.

As an example:

\(\frac{2}{5} + \frac{3}{8} = \frac{2 \times 8 + 3 \times 5}{5 \times 8} = \frac{31}{40}\)

Explanation of Formula

You can use this formula without understanding it, but helps you memorize it, and be creative with it, if you do understand it.

Imagine you have two pizzas of equal size:

Pizza 1 is cut into \(b\) equal slices, of which you’ll eat \(a\) of them.
Pizza 2 is cut into \(d\) equal slices, of which you’ll eat \(c\) of them.

In total you will eat \(\frac{a}{b} + \frac{c}{d}\) pizza. How much is this?

Imagine that you would carefully cut each slice of the first pizza into \(d\) slices. The slices are much smaller now—the pizza is divided into \(bd\) slices—and you’ll eat \(ad\) of them.

Then cut each slice of the second pizza into \(b\) slices. This pizza is also now divided into \(bd\) slices—and you’ll eat \(bc\) of them.

In total, you ate \(ad + bc\) slices, and each one was \(\frac{1}{bd}\) of a whole pizza.

Simplified Fractions

A fraction is simplified if there are no prime numbers that divide into both the numerator and denominator. For example, \(\frac{40}{60}\) is not simplified, because \(2\) divides into both \(40\) and \(60\). In fact, so does \(5\), and even some larger non-prime numbers, like \(20\). If you divide the top and bottom of the fraction by \(20\), the fraction becomes \(\frac{2}{3}\), which is the simplified form.

On this page, I’ll assume that you need to add or subtract fractions that are already simplified, which is usual for mental math competitions. There is a note at the end to describe what you should do if they are not simplified.

Mixed Fractions

A fraction is improper, if the numerator is larger than the denominator. For example, \(\frac{14}{3}\) is an improper fraction. Improper fractions can be written as mixed fractions—with an integer part and a proper fraction part. For example, \(\frac{14}{3} = 4 \frac{2}{3}\)

In mental calculation competitions, you must give all answers in mixed form. Improper fractions are marked incorrect!

To convert an improper fraction to a mixed fraction, first divide the numerator by the denominator and obtain the remainder:

\(14 \div 3 = 4\) rem. \(2\)

The integer part is the result of the division—\(4\)—and the remainder—\(2\)—is the numerator for the mixed fraction.

\(\frac{14}{3} = 4 \frac{2}{3}\)

As a complete example:

\(\frac{2}{3} + \frac{4}{5} = \frac{22}{15} = 1 \frac{7}{15}\)

Cases that Require Simplification

If two numbers—\(b\) and \(d\)—don’t share any factors, they are called co-prime. This is the same as saying that \(\frac{b}{d}\) would be a simplified fraction.

If the two denominators—\(b\) and \(d\)—are co-prime, then it is guaranteed that the resulting fraction will not need simplification. But otherwise, you will also need to try to simplify the final fraction.

In mental calculation competitions, you must give all answers in simplified form. Unsimplified fractions are marked incorrect!

\(8\) and \(24\) are not co-prime, as both are even, so we must simplify at the end:

\(\frac{3}{8} + \frac{7}{24} = \frac{3 \times 24 + 7 \times 8}{8 \times 24} = \frac{128}{192} = \frac{2}{3}\)

This calculation involved some fairly large numbers, and it could be worse if the original fractions had larger denominators! Luckily, there is a shortcut:

Find any number that the denominators both divide into—the smaller the better. In the example above, you could use \(48\), but the best would be to use \(24\).
Express both fractions using this new denominator: \(\frac{9}{24} + \frac{7}{24}\). In this case, the second fraction did not need to change, but in the first one, the deminomator had been multiplied by \(3\) to get from \(8\) to \(24\). So it was necessary to multiply its numerator the same way: \(3 \times 3 = 9\).
Simply add the numerators, and place that above the new denominator.

\(\frac{3}{8} + \frac{7}{24} = \frac{9}{24} + \frac{7}{24} = \frac{16}{24} = \frac{2}{3}\)

Notice that sometimes—like here—we need to do a simplification step at the end, even though we already simplified at an earlier stage.

Proof that Addition of Simplified Fractions with Co-prime Denominators Never Requires Simplification

You can skip this paragraph if you are not currently curious about the mathematics behind the method.

Does the result from the formula, \(\frac{ad + bc}{bd}\), require simplification?

Assume that \(b\) and \(d\) share no prime factors—meaning they are co-prime. Does any factor of \(b\) divide into the numerator \(ad + bc\)?

Certainly it will divide into \(bc\). But it doesn’t divide into \(ad\), because \(b\) shares no prime factors with \(d\), and nor with \(a\) (as \(\frac{a}{b}\) was already simplified). Since it does divide into \(bc\) but not into \(ad\), it can’t divide into their sum.

By the same argument, the numerator also shares no prime factors with \(d\).

Therefore there are no prime numbers—and thus no integers of any type—that we can divide by to simplify \(\frac{ad + bc}{bd}\).

Final Summary for Mental Calculation

When adding or subtracting fractions using mental math:

Check—or assume—that the fractions can’t be simplified.
Check whether the denominators share any factors.
If they don’t share any factors, use the general formula, and leave your answer as a mixed fraction.
If they do share factors, you can either use the general method (simpler) or just change the fractions manually to have the same denominator (easier arithmetic). Then simplify the final answer, if necessary.
Remember not to write down any intermediate steps if training for a competition!

As a final example:

Using the general formula:

\(3 \frac{1}{4} – \frac{5}{6} = 3 \frac{6 – 20}{24} = 3 – \frac{14}{24} = 2 \frac{10}{24}= 2 \frac{5}{12}\)

Alternatively, by changing fractions manually:

\(3 \frac{1}{4} – \frac{5}{6} = 3 \frac{3}{12} – \frac{10}{12}= 3 – \frac{7}{12} = 2 \frac{5}{12}\)

Here, we must check whether \(2 \frac{5}{12}\) can be simplified further, but it cannot. So it is the final answer.