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 Commercial Mathematics « PreviousNext » Profit & Loss 
Author: S.LAL
Created: 18 Aug, 2010; Last Modified: 17 May, 2026

Commercial Mathematics - 01

Percentage

Percent means “per hundred.” It is a special fraction where the denominator is always 100, while its numerator is called rate-percent. The symbol % is used to represent the word “percent.” Since the denominator is always 100, it is properly expressed as a decimal fraction. Thus, 4 % = 4 100 = 0.04

When a fractional part of a whole is expressed in hundredths or percent, it is referred to as a percentage of the total. Although this appears to be the same as the term percent, the usages differ. While percent is used with a number, percentage is used without a number. The following statement clarifies the distinction: "Although the exact percentage of people below poverty line is under dispute, the best estimate is 45 percent."

Any common fraction can be expressed as a decimal fraction, and any decimal fraction can be expressed as hundredths – thus both can be easily expressed as percentage. Since a ratio is just a common fraction, any ratio is also expressible as a percentage.

Rules for conversion

The following rules are helpful for converting to and from a percentage: (Thompson, JE, Arithmetic Practical Worker, p. 204-5)

  1. From a decimal fraction to a percentage: Shift the decimal places two places to the right.
    e.g. 0.65 = 65%
  2. From a common fraction or ratio to a percentage: Multiply the numerator by 100 (add two zeroes), and divide by the denominator.
    e.g. 53 79 = 5300 79 % = 67.1 %
    This essentially means that you are multiplying the fraction with 100 100 , i.e. unity, wherein the 100 of the denominator converts to the percent symbol. Thus,
    53 79 = 53 79 × 100 100 = 53 × 100 79 % = 67.1 %
  3. From a percentage to a decimal fraction: Shift the decimal places in the percentage number two places to the left.
    e.g. 43% = 0.43
  4. From a percentage to a common fraction: Write the percentage number as a numerator with 100 as denominator, and reduce the resulting fractions to lowest terms.
    e.g. 65 % = 65 100 = 13 20
Important equivalents

Some important equivalents to remember are:

2 1 2 % = 1 40 5 % = 1 20 10 % = 1 10 12 1 2 % = 1 8 20 % = 1 5
25 % = 1 4 33 1 3 % = 1 3 50 % = 1 2 66 2 3 % = 2 3 75 % = 3 4

 

Find the value of (i) 5% of `700 (ii) 7 1 2 % of `140. (Chakravarti 1992, p. 305 Examples 151: 6-7)

(i) 5 %    of    ` 700 = 5 1 00 × 7 00 =   ` 35 .

(ii) 7 1 2 %   of   ` 140 = 7 1 2 100 × 140 = 15 2 × 1 10 0 × 14 0 = ` 105 .

What percent of 60 is 15?

The fractional part (15) of a whole (60) has to be expressed in hundredths to get the percentage.

Required percentage = 15 1 60 4 × 100 % = 1 4 × 100 25   % = 25 % .

5% of the total population of a town are Englishmen; the rest are Hindus. If the population of the town is 37820, what is the number of Hindus? (Chakravarti 1992, p. 305 Examples 151: 13)

The total population of the town is 37820. Since 5% of it are Englishmen, while the rest Hindus, this means that 95% of the population are Hindus. Thus,

No. of Hindus = 95 % × 37820 = 95 10 0 × 3782 0 = 35929

50% of a man's salary is `3500. What is 75% of his salary?

Method 1:

50 %  of salary = ` 3500 ⇒ 1 %  of salary = 3500 50 = ` 70 ⇒ 75 %  of salary = 70 × 75 = ` 5250

Method 2:

50 %  of salary = ` 3500 ⇒ 75 %  of salary = 75 50 × 3500 = 3 2 × 3500 = ` 5250

Thus, 75% of salary corresponds to `5250.


Note: The method 2 in this example embodies an important shortcut technique – the "new-over-old" technique – which you should know about. It is explained below.

The "New-over-Old" technique

If the value of an item (old V A ) is equivalent to the value of another item (old V B ), or mathematically,

 old  V A   ⇔    old  V B

and there is a direct-relationship between the two items (i.e. when V A increases, V B increases, and when V A decreases, V B decreases), then if the value of the first item changes to new V A , then,

new  V B =  new  V A  old  V A ×  old  V B

A man's income in 1975 was `5000. In 1976 it was increased by 20%. What was his income in 1976?

Method 1:

An increase of 20% over the original income of `5000 means that the increase in income = 20 % × 5000 = 20 100 × 5000 = ` 1000 . Thus, the new income in 1976 is ` ( 5000 + 1000 ) = ` 6000

Method 2 ("new-over-old" technique):

The increase of 20% in income basically means that if the man's income was `100 in 1975, it became `120 in 1976. Thus,
1975's income of `100 ⇔ 1976's income of `120, so
1975's income of `5000 ⇔ 1976's income of 120   ×   5000 100 = ` 6000 .

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List of References

Chakravarti, JC, Arithmetic, 55th edn, Kolkata, India: MC Chakravarti, 1992.
Thompson, JE, Arithmetic for the Practical Man, 3rd edn, NJ: Van Nostrand's Reinhold Company Inc., 1962.

Bibliography

Chakravarti, JC, Arithmetic, 55th edn, Kolkata, India: MC Chakravarti, 1992.
Thompson, JE, Arithmetic for the Practical Man, 3rd edn, NJ: Van Nostrand's Reinhold Company Inc., 1962.

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