Exercise solution 3, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) To add or subtract fractions, the denominators must be the same. If they are different, first find a common denominator. Then rewrite each fraction as an equivalent fraction.

\[ \frac{2}{3}+\frac{5}{12} = \frac{8}{12}+\frac{5}{12} = \frac{13}{12} \]

Here, \(12\) is a common denominator because \(\frac{2}{3}=\frac{8}{12}\).

B) When adding or subtracting fractions, only the numerators change once the denominators are the same. The denominator stays unchanged.

\[ \begin{aligned} \frac{7}{10}-\frac{1}{5} &= \frac{7}{10}-\frac{2}{10} \\ &= \frac{5}{10} \\ &= \frac{1}{2} \end{aligned} \]

Always simplify the final answer if possible.

C) To multiply fractions, multiply the numerators together and multiply the denominators together.

\[ \begin{aligned} \frac{4}{9}\times \frac{3}{7} &= \frac{4\times 3}{9\times 7} \\ &= \frac{12}{63} \\ &= \frac{4}{21} \end{aligned} \]

If possible, simplify the fraction at the end. You can also simplify before multiplying.

D) Dividing by a fraction means multiplying by its reciprocal. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

\[ \begin{aligned} \frac{5}{6}\div \frac{2}{11} &= \frac{5}{6}\times \frac{11}{2} \\ &= \frac{55}{12} \end{aligned} \]

This rule is very important when a question contains the division sign with fractions.

E) In calculations with several operations, use the correct order of operations. Multiplication and division are done before addition and subtraction, unless brackets change the order.

\[ \begin{aligned} \frac{3}{4}+\frac{2}{5}\times 6 &= \frac{3}{4}+\frac{12}{5} \end{aligned} \]

You should not add first. Work from left to right for multiplication and division when they appear in the same expression.

F) The word of in a fraction question usually means multiply.

\[ \begin{aligned} \frac{3}{8}\text{ of }\frac{4}{15} &= \frac{3}{8}\times \frac{4}{15} \\ &= \frac{12}{120} \\ &= \frac{1}{10} \end{aligned} \]

So whenever you see “of”, replace it mentally with a multiplication.

G) A mixed number is a whole number and a fraction together, such as \(2\frac{3}{4}\). To calculate with mixed numbers, it is usually easier to convert them into improper fractions first.

\[ \begin{aligned} 2\frac{3}{4} &= \frac{2\times 4+3}{4} \\ &= \frac{11}{4} \end{aligned} \]

After converting, you can add, subtract, multiply or divide as usual.

H) To add or subtract mixed numbers, convert each one to an improper fraction, then use a common denominator if needed.

\[ \begin{aligned} 1\frac{2}{3}+2\frac{1}{6} &= \frac{5}{3}+\frac{13}{6} \\ &= \frac{10}{6}+\frac{13}{6} \\ &= \frac{23}{6} \\ &= 3\frac{5}{6} \end{aligned} \]

This method helps avoid mistakes.

I) To simplify a fraction, divide the numerator and denominator by the same common factor.

\[ \begin{aligned} \frac{18}{24} &= \frac{18\div 6}{24\div 6} \\ &= \frac{3}{4} \end{aligned} \]

A fraction is in its simplest form when the numerator and denominator have no common factor greater than \(1\).

J) An improper fraction can be changed into a mixed number by dividing the numerator by the denominator.

\[ \begin{aligned} \frac{17}{5} &= 3\frac{2}{5} \end{aligned} \]

This is often useful when giving a final answer, especially if the question expects a mixed number.