Lesson Introduction to Ratios

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Introduction 
A <a href="ratio.wikipedia">ratio</a> is a comparison or a relationship between two numbers. It is represented using a colon ':', the word 'to' or using a fraction '/'. For example the <a href="ratio.wikipedia">ratio</a> of a and b is represented as:
<center>a:b &nbsp;&nbsp;&nbsp;&nbsp; OR &nbsp;&nbsp;&nbsp;&nbsp; a to b &nbsp;&nbsp;&nbsp;&nbsp;OR &nbsp;&nbsp;&nbsp;&nbsp; a/b</center>
We say that the <a href="ratio.wikipedia">ratio</a> is 'a to be' or 'a is to b'. For example if in a class there are 30 boys and 15 girls, we say that the <a href="ratio.wikipedia">ratio</a> of boys to girls in the class is 2 to 1 or 2:1. Thus given the actual numbers we can calculate the <a href="ratio.wikipedia">ratios</a>. However, if we are given only the <a href="ratio.wikipedia">ratio</a>, it is not possible to calculate the actual numbers. For example if we know that the ratio of pens to pencils in a box is 1 to 4, we can only say that there are 4 times as many pens as pencils. There can be 4 pens and 1 pencil or 8 pens and 2 pencils or 12 pens and 3 pencils. We cannot comment on that.

Expressing ratios in simplest form 
Ratios must always be expressed in their simplest form. This means that the numerator and denominator must be divided by the highest common factor to obtain a simple form. For example the ratio 15/10 should be expressed as 3/2 by dividing the numerator and denominator by 5 which is the HCF. Another example is 48/40, which should be expressed as 6/5.

Comparing ratios 
Ratios are used frequently for comparison. For example we may want to know that which class has the highest <a href="ratio.wikipedia">ratio</a> of boys to girl. In this case we would need to first calculate the ratios of boys to girl in each class and then compare them. To compare the ratios, we express them as fractions and then cross multiply. The general steps to compare 2 ratios (a:b and c:d) would be:

1. Express the ratios as fractions: a/b and c/d
2. Cross multiply the fractions to get: a*d and b*c
3. If a*d > b*c, then a/b > c/d. If a*d < b*c, then a/b < c/d. And if a*d = b*c, then a/b = c/d.

Lets consider an example now. 
Q. Compare 2 ratios 3:4 and 5:7.
Solution.
First we express the ratios as fractions. The ratios become 3/4 and 5/7. Then we cross multiply to get 3*7 = 21 and 4*5 = 20. Since 21>20, 3/4 is greater than 5/7. Thus 3:4 is greater than 5:7.