Objective

In our daily life, there are many occasions when we compare two quantities.

This topic gives an overview of;

 

• Ratios and Proportions
• Equivalent Ratios
• Proportion
• Solving Ratio and Proportion

Ratios and Proportions

Ratios are used to compare quantities. Ratios help us to compare quantities and determine the relation between them. A ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. Since a ratio is only a comparison or relation between quantities, it is an abstract number. For instance, the ratio of 6 miles to 3 miles is only 2, not 2 miles. Ratios are written with the” : “symbol.

If two quantities cannot be expressed in terms of the same unit, there cannot be a ratio between them. Hence to compare two quantities, the units must be the same.

Consider an example to find the ratio of 3 km to 300 m.First convert both the distances to the same unit.

 

So, 3 km = 3 × 1000 m = 3000 m.

Thus, the required ratio, 3 km : 300 m is 3000 : 300 = 10 : 1

Equivalent Ratios

Different ratios can also be compared with each other to know whether they are equivalent or not. To do this, we need to write the ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, we say the given ratios are equivalent. We can find equivalent ratios by multiplying or dividing the numerator and denominator by the same number. Consider an example to check whether the ratios 1 : 2 and 2 : 3 equivalent.

To check this, we need to know whether 

We have, 

 

We find that  which means that 

Therefore, the ratio 1 :2 is not equivalent to the ratio 2 : 3.

Proportion

 

The ratio of two quantities in the same unit is a fraction that shows how many times one quantity is greater or smaller than the other. Four quantities are said to be in proportion, if the ratio of first and second quantities is equal to the ratio of third and fourth quantities. If two ratios are equal, then we say that they are in proportion and use the symbol ‘:: ’ or ‘=’ to equate the two ratios.

 

Solving Ratio and Proportion

 

Ratio and proportion problems can be solved by using two methods, the unitary method and equating the ratios to make proportions, and then solving the equation.

 

For example,

To check whether 8, 22, 12, and 33 are in proportion or not, we have to find the ratio of 8 to 22 and the ratio of 12 to 33.

 

 

Therefore, 8, 22, 12, and 33 are in proportion as 8 : 22 and 12 : 33 are equal. When four terms are in proportion, the first and fourth terms are known as extreme terms and the second and third terms are known as middle terms. In the above example, 8, 22, 12, and 33 were in proportion. Therefore, 8 and 33 are known as extreme terms while 22 and 12 are known as middle terms.

The method in which we first find the value of one unit and then the value of the required number of units is known as unitary method.

 

Consider an example to find the cost of 9 bananas if the cost of a dozen bananas is Rs 20.

 

1 dozen = 12 units

 

Cost of 12 bananas = Rs 20

∴ Cost of 1 bananas = Rs 

∴ Cost of 9 bananas = Rs 

This method is known as unitary method.

Summary

 

• We have learnt, Ratios are used to compare quantities.

• Since a ratio is only a comparison or relation between quantities, it is an abstract number.

• Ratios can be written as fractions. They also have all the properties of fractions.

• The ratio of 6 to 3 should be stated as 2 to 1, but common usage has shortened the expression of ratios to be called simply 2.

• If two quantities cannot be expressed in terms of the same unit, there cannot be a ratio between them.

• If any three terms in a proportion are given, the fourth may be found. The product of the means is equal to the product of the extremes.

• It is important to remember that to use the proportion; the ratios must be equal to each other and must remain constant.