Exercise solution 1, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) To simplify a fraction, divide the numerator and the denominator by the same whole number greater than 1. This keeps the value of the fraction the same, but writes it in a simpler form.

For example, to simplify \(\frac{24}{36}\), notice that both numbers are divisible by \(12\):

\[\frac{24}{36}=\frac{24\div 12}{36\div 12}=\frac{2}{3}\]

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

B) A useful method is to find the highest common factor (HCF) of the numerator and denominator. The HCF is the greatest number that divides both exactly.

For example, for \(\frac{42}{56}\):

\[ \text{HCF of }42\text{ and }56 = 14 \]

So:

\[\frac{42}{56}=\frac{42\div 14}{56\div 14}=\frac{3}{4}\]

If you do not spot the HCF straight away, you can simplify in more than one step, as long as you divide top and bottom by the same number each time.

C) A mixed number is made of a whole number and a fraction, such as \(5\frac{7}{9}\). When simplifying a mixed number, only the fractional part needs to be simplified if the whole number stays the same.

For example:

\[ 5\frac{12}{18}=5\frac{2}{3} \]

This is because \(\frac{12}{18}\) simplifies to \(\frac{2}{3}\).

D) It is important to check whether the fractional part can be simplified by using common factors. You can often test small factors first, such as \(2\), \(3\), \(5\), or \(10\).

For example, in \(\frac{27}{45}\), both numbers are divisible by \(9\), so:

\[\frac{27}{45}=\frac{3}{5}\]

Checking divisibility helps you simplify accurately and efficiently.

E) Always make sure you divide both parts of the fraction by the same number. If you divide only the numerator or only the denominator, the value changes and the fraction is no longer equivalent.

Correct method:

\[\frac{18}{30}=\frac{18\div 6}{30\div 6}=\frac{3}{5}\]

This shows how equivalent fractions work: different-looking fractions can represent the same quantity.