Exercise solution 4, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) When a calculation contains fractions, it is often easier to work carefully one step at a time. Remember that a fraction has a numerator on top and a denominator on the bottom. For example, in \(\frac{7}{12}\), 7 is the numerator and 12 is the denominator.

To simplify a fraction, divide the numerator and denominator by the same common factor. For example:

\[ \frac{18}{24}=\frac{18\div 6}{24\div 6}=\frac{3}{4} \]

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

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

Example:

\[ \frac{4}{7}\times \frac{3}{10}=\frac{12}{70}=\frac{6}{35} \]

If possible, simplify the answer at the end.

C) To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\).

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

Example:

\[ \frac{5}{6}\div \frac{2}{9}=\frac{5}{6}\times \frac{9}{2}=\frac{45}{12}=\frac{15}{4} \]

This is a very important rule for questions involving fraction division.

D) When adding or subtracting fractions, the denominators must be the same. If they are different, first rewrite the fractions with a common denominator.

Example:

\[ \frac{2}{3}+\frac{5}{8}=\frac{16}{24}+\frac{15}{24}=\frac{31}{24} \]

For subtraction:

\[ \frac{7}{10}-\frac{1}{4}=\frac{14}{20}-\frac{5}{20}=\frac{9}{20} \]

A common denominator is often the lowest common multiple of the two denominators.

E) A mixed number is a whole number and a fraction together, such as \(3\frac{4}{5}\). Before multiplying or dividing, it is usually best to convert it into an improper fraction.

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

\[ 3\frac{4}{5}=\frac{3\times 5+4}{5}=\frac{19}{5} \]

Another example:

\[ 2\frac{3}{7}=\frac{2\times 7+3}{7}=\frac{17}{7} \]

F) Use the order of operations carefully. Brackets should be worked out first. Inside brackets, follow the operations in a sensible order, especially when fractions are involved.

For example, in an expression such as

\[ \left(\frac{3}{4}+\frac{5}{6}\right)\times \frac{2}{3} \]

you should first calculate what is inside the brackets, then multiply by \(\frac{2}{3}\).

If there is a division inside brackets, remember to turn it into multiplication by the reciprocal before continuing.

G) When several steps are needed, lay out the calculation clearly so that each line follows from the previous one. This helps avoid mistakes with signs and denominators.

Example:

\[\begin{aligned} \frac{3}{5}+\frac{1}{2} &= \frac{6}{10}+\frac{5}{10} \\ &= \frac{11}{10} \\ &= 1\frac{1}{10} \end{aligned}\]

Writing the method clearly is especially useful when a question mixes addition, subtraction, multiplication and division of fractions.

H) After finding an answer, always check whether it can be simplified or written as a mixed number if needed.

Example:

\[\begin{aligned} \frac{28}{6} &= \frac{14}{3} \\ &= 4\frac{2}{3} \end{aligned}\]

A final answer should usually be given in its simplest form.