Rules of calculation for fractions

Reduce a fraction

If we take our fraction from earlier = \(\large \frac{2}{8}\)

We calculated that it is the same as 25%, which we normally call a quarter = \(\large \frac{1}{4}\)

The two fractions are therefore equal, even though they do not look the same. This is because you can reduce a fraction.

 

You reduce a fraction by dividing both numerator and denominator by the same number.

In our case, it is divided by 2:

 

$$ \large \frac{2}{8} \Leftrightarrow \frac{2:2}{8:2}\;\Leftrightarrow \frac{1}{4} $$

 

Not all fractions can be reduced. For example, it is not possible to find a common divisor for \(\large \frac{7}{16}\)

 

An example with larger numbers:

 

$$ \large \frac{15}{25} \Leftrightarrow \frac{15:5}{25:5}\;\Leftrightarrow \frac{3}{5} $$

 

 

Expand a fraction

Just as you can reduce a fraction, you can also expand it.

The rule is the same, but instead of dividing by a number, you multiply by a number.

 

If we want to expand our fraction to twenty-fourths, we must multiply both numerator and denominator by 3 (because 8 times 3 is 24).

 

$$ \large \frac{2}{8} \Leftrightarrow \frac{2 \cdot 3}{8 \cdot 3}\;\Leftrightarrow \frac{6}{24} $$

 

All fractions can be expanded, but not to just anything. Sixteenths cannot become twentieths. The closest is to multiply by 2, which makes thirty-seconds. With 3 it becomes forty-eighths.

 

$$ \large \frac{7}{16} \Leftrightarrow \frac{7 \cdot 3}{16 \cdot 3}\;\Leftrightarrow \frac{21}{48} $$

 

Expanding a fraction corresponds to making the denominator a multiple of the original number. This is often used to find a common denominator.

 

 

Common denominator

If you want to add or subtract, you must always find a common denominator before you start. For example, if you want to calculate the following, the common denominator can be 12, because both 3 and 4 divide into 12.

 

$$ \large \frac{2}{3} + \frac{1}{4} $$

 

If you find it difficult to find a common denominator, you can multiply the two denominators together. The result can be used as a common denominator \(\large 3 \cdot 4 = 12\)

The fractions are expanded by multiplying both numerator and denominator by the same number. The first fraction is multiplied by 4 and the second fraction is multiplied by 3. You then have two fractions with the same denominator.

 

$$ \large \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12} $$

$$ \large \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12} $$

 

Now you can add the two fractions together.

 

$$ \large \frac{8}{12} + \frac{3}{12} = \frac{8+3}{12} = \frac{11}{12} $$

 

Often the lowest common denominator is chosen, as it gives the simplest fractions to work with.

 

 

Addition and subtraction

Always remember the common denominator when adding and subtracting

 

Addition:

 

$$ \large \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} $$

 

Subtraction:

 

$$ \large \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c} $$

 

When adding, the order does not matter. \( \large a + c\) is the same as \( \large c + a\)

 

But when subtracting, it follows the same rule as all other subtraction. The order must be correct!

 

 

Multiplication and division

You do not need a common denominator when multiplying and dividing

 

Multiplying fractions:

 

$$ \large \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} $$

 

Example: \(\large \frac{2}{3} \cdot \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\)

 

Dividing fractions:

 

$$ \large \frac{a}{b} : \frac{c}{d} = \frac{a \cdot d}{b \cdot c} $$

 

Example: \(\large \frac{2}{3} : \frac{3}{4} = \frac{2 \cdot 4}{3 \cdot 3} = \frac{8}{9}\)

 

Note that in division the second fraction is inverted and multiplied with the first fraction.

When multiplying, the order does not matter. \(\large a \cdot c\) is the same as \(\large c \cdot a\)

 

But when dividing, the order is important!

It is the second fraction that is inverted. You must never invert the first fraction and therefore it is important that you write the calculation in the correct order.