Exercise solution 3, page 343 - AQA GCSE MATHS HIGHER

⭐ What you need to know:

A) To multiply or divide fractions, it is important to know what the numerator and the denominator are. In a fraction such as \(\frac{7}{10}\), the top number is the numerator and the bottom number is the denominator. Fractions represent parts of a whole.

B) To multiply two fractions, multiply the numerators together and multiply the denominators together:

\[ \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd} \]

For example:

\[ \frac{4}{7}\times \frac{3}{5}=\frac{12}{35} \]

After multiplying, always check whether the answer can be simplified.

C) Before or after multiplying, you can sometimes simplify by cancelling common factors. This makes the calculation easier. You may only cancel between a numerator and a denominator.

For example:

\[ \frac{6}{11}\times \frac{22}{15} \]

Here, \(6\) and \(15\) have a common factor of \(3\), and \(22\) and \(11\) have a common factor of \(11\). This gives:

\[ \frac{6}{11}\times \frac{22}{15}=\frac{2}{1}\times \frac{2}{5}=\frac{4}{5} \]

D) To divide by a fraction, keep the first fraction the same, change the division sign to multiplication, and turn the second fraction upside down. This is called using the reciprocal.

\[ \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c} \]

For example:

\[ \frac{7}{12}\div \frac{14}{9}=\frac{7}{12}\times \frac{9}{14} \]

Then simplify and multiply.

E) A mixed number is a whole number and a fraction written together, such as \(3\frac{2}{5}\). Before multiplying or dividing mixed numbers, convert them into improper fractions.

To do this, multiply the whole number by the denominator, then add the numerator:

\[ 3\frac{2}{5}=\frac{3\times 5+2}{5}=\frac{17}{5} \]

Another example:

\[ 4\frac{1}{6}=\frac{4\times 6+1}{6}=\frac{25}{6} \]

F) Once a mixed number has been changed into an improper fraction, you can use the usual rules for multiplication or division of fractions.

For example, to multiply:

\[ 1\frac{3}{4}\times \frac{2}{9} \]

First convert the mixed number:

\[ 1\frac{3}{4}=\frac{7}{4} \]

Then multiply:

\[ \frac{7}{4}\times \frac{2}{9}=\frac{14}{36}=\frac{7}{18} \]

G) After every calculation, give the answer in its simplest form. A fraction is simplified when the numerator and denominator have no common factor greater than \(1\).

For example:

\[ \frac{18}{24}=\frac{3}{4} \]

because both \(18\) and \(24\) can be divided by \(6\).

H) If needed, you can also change an improper fraction back into a mixed number at the end. Divide the numerator by the denominator.

For example:

\[ \frac{19}{6}=3\frac{1}{6} \]

This is useful when the question expects a mixed-number answer, but if the instruction says to evaluate exactly, a simplified improper fraction is usually acceptable unless your teacher asks otherwise.