Exercise solution 2, page 101 - Mathematics Higher Student Book

⭐ What you need to know:

A) A ratio compares two quantities. Writing a ratio in the form \(1:n\) means making the first part equal to \(1\). To do this, divide both parts of the ratio by the first number.

B) Ratios can be simplified in the same way as fractions. If both parts of the ratio have a common factor, divide both parts by that factor first. For example, for the ratio \(18:6\):

\[\begin{aligned} 18:6 &= (18 \div 6):(6 \div 6) \\ &= 3:1 \end{aligned}\]

This is simplified, but it is not yet in the form \(1:n\). To write it as \(1:n\), divide both parts by \(3\):

\[\begin{aligned} 3:1 &= (3 \div 3):(1 \div 3) \\ &= 1:\frac{1}{3} \end{aligned}\]

C) If the first term is a whole number, you can turn it into \(1\) by dividing both parts by that number. For example, with \(5:35\):

\[\begin{aligned} 5:35 &= (5 \div 5):(35 \div 5) \\ &= 1:7 \end{aligned}\]

This gives the ratio directly in the form \(1:n\).

D) Ratios can also contain fractions. The method stays the same: divide both parts by the first term. For example, with \(\frac{2}{5}:4\):

\[\begin{aligned} \frac{2}{5}:4 &= \left(\frac{2}{5}\div\frac{2}{5}\right):\left(4\div\frac{2}{5}\right) \\ &= 1:\left(4\times\frac{5}{2}\right) \\ &= 1:10 \end{aligned}\]

Remember that dividing by a fraction is the same as multiplying by its reciprocal.

E) If both parts are fractions, the same idea still works. Divide the second fraction by the first fraction carefully. For example, with \(\frac{3}{4}:\frac{9}{8}\):

\[\begin{aligned} \frac{3}{4}:\frac{9}{8} &= \left(\frac{3}{4}\div\frac{3}{4}\right):\left(\frac{9}{8}\div\frac{3}{4}\right) \\ &= 1:\left(\frac{9}{8}\times\frac{4}{3}\right) \\ &= 1:\frac{3}{2} \end{aligned}\]

F) A useful fact to remember is:

\[ a:b = 1:\frac{b}{a} \]

This works because dividing both parts of \(a:b\) by \(a\) gives:

\[\begin{aligned} a:b &= (a\div a):(b\div a) \\ &= 1:\frac{b}{a} \end{aligned}\]

This shortcut is especially helpful when checking your answer.

G) Always check that your final ratio really starts with \(1\). If it does not, it is not yet in the required form. Also make sure you have divided both parts by the same number or fraction, otherwise the ratio changes and becomes incorrect.