8th Class Mathematics Comparing Quantities Comparing Queantities

COMPARING QUANTITIES

FUNDAMENTALS

Ratio and Proportion

• Ratio is method of comparing two quantities of the same kind by division.
• A ratio does not have any unit, it is only a numerical value.
• The symbol used to write a ratio is ':' and is read as 'is to
• A ratio is generally expressed in its simplest form.
• To express two terms in ratio, they should be in the same units of measurement.
• Multiplying or dividing the terms of a ratio by the same number gives equivalent ratio.
• When two ratios are equal, they are known to be in proportion. The symbol for proportion is ': :' and is read as 'as to'.

For e.g., 2 is to 3 as to 6 is to 9 is written as 2 : 3 : : 6 : 9 or, \[\frac{2}{3}=\frac{6}{9}\]

• If two ratios are equal or are in proportion, then the product of means is equal to the product of their extremes.

Example: If a:b::c:d then the statement ad = bc, holds good.

• If a : b and b : c are in proportion such that (symbol, |) \[{{b}^{2}}=ac\] then b is called the mean proportional of a : b and b : c.

1. Percentage

• Another way of comparing quantities is percentage. The word percent means per Thus 13% means 13 parts out of 100 parts.
• Fractions can be converted into percentages and vice versa.

e.g., (i)\[\frac{1}{5}=\frac{1}{5}\times 100%=20%\]

(ii) \[15%=\frac{15}{100}=\frac{3}{20}\]    

• Decimals can be converted into percentages and vice-versa.

e.g., (i) \[0.26\text{=}0.26\times 100%=26%\]              

(ii) \[33%=\frac{33}{100}=0.33\]

• If a number is increased by a%, and then decreased by a% or is decreased by a%, and then increased by a%, then the original number decreases by \[\frac{{{a}^{2}}}{100}%\]%.

Example: Express each of the following as a fraction:

(i) 18%                          (ii) 0.45%                       (iii) \[87\frac{1}{2}%\]

Solution: We have,

(i) \[18%=\frac{18}{100}=\frac{9}{50}\]

            (ii) \[0.45%=\frac{0.45}{100}=\frac{45}{10000}=\frac{9}{2000}\]

            (iii) \[87\frac{1}{2}%=\frac{87\frac{1}{2}%}{100}=\frac{7}{8}\]

            (Dividing numerator &denominator by \[12\frac{1}{2}\] as \[12\frac{1}{2}\]is factor of both \[87\frac{1}{2}\]and 100, we get \[\frac{7}{8}\]). [It would help if you remember table of\[12\frac{1}{2}:12\frac{1}{2}\times 1=12\frac{1}{2};\]\[12\frac{1}{2}\times 2=25;...............12\frac{1}{2}\times 8=100]\]

Example: Which is smallest amongst \[\mathbf{8}\frac{1}{3}\text{ }\!\!%\!\!\text{  },\frac{2}{9}\mathbf{and}\text{ }\mathbf{0}.\mathbf{21}\] ?

Solution:  We may write,

\[8\frac{1}{3}\text{ }\!\!%\!\!\text{  =}\frac{25}{3}%=\frac{25}{3}\times \frac{1}{100}=\frac{1}{12}=0.0833..\]

 \[\frac{2}{9}=0.222...\]       Clearly, \[8\frac{1}{3}%\]is the smallest