Fraction Multiplication and Division with Mental Math

Multiplication and division of simple fractions is more straightforward than addition and subtraction of fractions. However, when doing these calculations in your head, there are some challenges!

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.

Basic Multiplication by a Fraction

Multiplication by the fraction \(\frac{a}{b}\), means multiplication by \(a\) and division by \(b\). The result is typically a fraction:

\(7 \times \frac{2}{15} = \frac{14}{15}\)

Basic Multiplication of Fractions

When two or more fractions are multiplied together, the numerators are multiplied together, and the denominators are multiplied together:

\(\frac{a}{b} \times \frac{c}{d}\times \frac{e}{f} = \frac{a \times c \times e}{b \times d \times f}\)

\(\frac{8}{9}\times \frac{5}{7} = \frac{8 \times 5}{9 \times 7} = \frac{40}{63}\)

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 multiply or divide fractions that are already simplified, which is usual for mental math competitions. Otherwise it is usually easiest to simplify them first.

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

To multiply or divide mixed fractions, it is usually much easier to first convert them to improper fractions.

To do this, you can use the formula:

\(n \frac{a}{b} = \frac{b \times n + a}{b}\)

As an example:

\(2 \frac{3}{4} = \frac{4 \times 2 + 3}{4} = \frac{11}{4}\)

This is true, because if we split the integer part—\(n\)—into \(b\) equal pieces, there would be \(n \times b\) of these pieces. Add this to the \(a\) pieces that were already represented by the fraction, and there would be \(n \times b + a\) in total.

In mental calculation competitions, you must give all answers in mixed form. Improper fractions are marked incorrect! Therefore you should also know how to convert an improper fraction to a mixed fraction.

To do this, divide the numerator by the denominator and obtain the remainder. For example:

\(14 \div 3 = 4\) 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:

\(2 \frac{3}{4} \times 5 \frac{6}{7} = \frac{11}{4} \times \frac{41}{7}\)

\(= \frac{11 \times 41}{4 \times 7}\)

\(= \frac{451}{28} = 16 \frac{3}{28}\)

Simplification of the Final Fraction

In mental calculation competitions, you must give all answers in simplified form. Unsimplified fractions are marked incorrect! Even outside of formal competitions, it is better to present fractions in a simplified form.

For multiplications and divisions of fractions, you always need to check whether the result can be simplified. In the examples we have already seen, no simplification was available, so let’s look at an example with simplification. There are two methods you can choose:

Method 1: Simplify at the End

\(1 \frac{1}{15} \times 4 \frac{3}{8}\)

\(= \frac{16}{15} \times \frac{35}{8}\)

\(= \frac{16 \times 35}{15 \times 8}\)

\(= \frac{560}{120}\)

The numerator and denominator share various factors, including 10, 8, etc. The biggest shared factor is 40, so divide both halves of the fraction by 40:

\(\frac{560}{120} = \frac{14}{3} = 4 \frac{2}{3}\)

Method 2: Cancel factors from Improper Fractions

This method is better when you have large numbers—larger than the 560 and 120 above—and helps you avoid doing arithmetic with these large numbers. However, it is more difficult to follow the method.

Start the same way as before:

\(1 \frac{1}{15} \times 4 \frac{3}{8}\)

\(= \frac{16}{15} \times \frac{35}{8}\)

Next, notice that when these are multiplied together, the fraction will have a factor of 8 at the bottom (from the second denominator, 8), and also at the top of the fraction (from the first numerator, 16 = 8 × 2). Divide the relevant numbers by 8:

\(= \frac{2}{15} \times \frac{35}{1}\)

Can we do the same with any other numbers? In fact, in this case we can do the same thing again with 5, as 5 is a factor of the numerator (35) and the denominator (15). So divide both halves by 5, check that there is no more simplification possible, and complete:

\(= \frac{2}{3} \times \frac{7}{1} = \frac{14}{3} = 4 \frac{2}{3}\)

Note that if you spot all of the factors for simplification before the end, it is guaranteed that you won’t have to simplify the final fraction. But if you miss any—such as the 5, or if you had divided by 4 rather than 8—then you would need to simplify the final fraction.

Also note that often there is no available simplification. In this case, methods 1 and 2 are the same, with no simplification steps. You may encounter large numbers during the calculation, with no way to avoid them.

Final Summary for Mental Calculation (Multiplication)

When multiplying fractions:

Convert any mixed fractions to improper fractions.
Optionally perform some simplification at this stage (method 2) by finding numbers that are factors of a numerator and a denominator.
Multiply all the numerators to get the new numerator. Multiply all the denominators to get the new denominator.
Simplify the answer, if possible.
Convert to a mixed fraction.
Remember not to write down any intermediate steps if training for a competition!

Division of Fractions

Dividing by an improper fraction \(\frac{a}{b}\) is the opposite of multiplying by it. Therefore it means dividing by \(a\) and multiplying by \(b\).

Therefore \(\div \frac{a}{b}\) can be replaced by \(\times \frac{b}{a}\)

Simply turn the fraction “upside-down”, then proceed using what you know about multiplying fractions.

Example: (using method 2 for simplification)

\(4 \frac{5}{6} \div 1 \frac{2}{3}\)

\(= \frac{29}{6} \div \frac{5}{3}\)

\(= \frac{29}{6} \times\frac{3}{5}\)

\(= \frac{29}{2} \times\frac{1}{5}\)

\(= \frac{29}{10}\)

\(= 2 \frac{9}{10}\)

Remember, of course, that if you are training for a mental calculation event, you should be able to do all of these steps without writing anything except the final answer!