Exercise solution 2, page 185 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) To put fractions in ascending order, arrange them from the smallest value to the largest value.

If two fractions do not have the same denominator, it is often easier to compare them by rewriting them with a common denominator, or by converting them to decimals.

B) A fraction represents a division. For example, \(\frac{7}{8}\) means \(7 \div 8\).

To compare simple fractions, you can sometimes use what you already know about their size:

• If the numerator is smaller than the denominator, the fraction is less than \(1\).
• If the numerator is equal to the denominator, the fraction is equal to \(1\).
• If the numerator is greater than the denominator, the fraction is greater than \(1\).

C) One reliable method is to write the fractions with a common denominator. To do this, find a common multiple of the denominators.

For example, to compare \(\frac{3}{4}\) and \(\frac{5}{6}\), we can use \(12\) as a common denominator:

\[ \begin{aligned} \frac{3}{4} &= \frac{3\times 3}{4\times 3} = \frac{9}{12} \\ \frac{5}{6} &= \frac{5\times 2}{6\times 2} = \frac{10}{12} \end{aligned} \]

Since \(9 < 10\), we have \(\frac{3}{4} < \frac{5}{6}\).

D) Another useful method is to convert fractions into decimals. This is done by dividing the numerator by the denominator.

For example:

\[ \begin{aligned} \frac{11}{20} &= 11 \div 20 \\ &= 0.55 \end{aligned} \]

Once all the numbers are written as decimals, it becomes easier to place them in order.

E) A mixed number is made of an integer and a fraction, such as \(3\frac{1}{4}\). It means:

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

So a mixed number is always greater than the whole number part alone. For instance, \(3\frac{1}{4}\) is greater than \(3\), but less than \(4\).

This helps when comparing mixed numbers with ordinary fractions: any fraction less than \(1\) will be smaller than a mixed number greater than \(1\).

F) You can also change a mixed number into an improper fraction. This can make comparison easier.

For example:

\[ \begin{aligned} 4\frac{2}{3} &= \frac{4\times 3 + 2}{3} \\ &= \frac{12+2}{3} \\ &= \frac{14}{3} \end{aligned} \]

After that, you can compare it with other fractions using a common denominator or decimals.

G) Always check whether the numbers are all of the same type:

• some may be ordinary fractions, such as \(\frac{5}{8}\),
• some may be mixed numbers, such as \(2\frac{3}{7}\).

Before ordering them, it is often helpful to rewrite them all in a similar form.

H) After comparing the values, write the final answer carefully from left to right, from the smallest to the largest. A good habit is to use inequality signs to show the order clearly, for example:

\[ \frac{2}{9} < \frac{1}{3} < \frac{5}{9} \]

This shows immediately that the fractions are in ascending order.