The term percentage means parts per 100 or “for every hundred”. A fraction whose denominator is 100 is called percentage and the numerator of the fraction is called the rate percent. Thus, when we say a man made a profit of 20 percent we mean to say that he gained Rs.20 for every hundred dollars he invested in the business, i.e., 20/100 dollars for each dollar. The abbreviation of percent is p.c. and it is generally denoted by %.

Let’s start with a number X (= 1 X)



Similarly, you can work mentally with any specifically chosen number (say 500) and work out different answers.





A Percentage can be expressed as a Fraction. 10% can be expressed as \(\frac{10}{100}\) or \(\frac{1}{10}\). To express a percentage as a fraction divide it by 100, \(a\)% \(= \frac{a}{100}\).

To express a fraction as a percent multiply it by 100 ⇒ \(\frac{a}{b}= \left[ \begin{array}{cc|r} (\frac{a}{b}) \times 100 \end{array} \right]\) %


To express percentage as a decimal we remove the symbol % and shift the decimal point by two places to the left. For example, 10% can be expressed as 0.1. 6.5% = 0.065 etc.

To express decimal as a percentage we shift the decimal point by two places to the right and write the number obtained with the symbol % or simply we multiply the decimal with 100. Similarly 0.7 = 70%.


Percent INCREASE and DECREASE


Increase % \(= \frac{Increase}{Original Value} \times 100\)

Decrease % \(= \frac{Decrease }{ Original Value} \times 100
\)

In increase %, the denominator is smaller, whereas, in decrease %, the denominator is larger.

Change % \(= \frac{Change }{ Original Value} \times 100\)


Successive change in percentage. If a number A is increased successively by X% followed by Y%, and then by Z%, then the final value of A will be \(A(1+\frac{X}{100})(1+\frac{Y}{100})(1+\frac{Z}{100})\)

In a similar way, at any point or stage, if the value is decreased by any percentage, then we can replace the same by a negative sign. The same formula can be used for two or more successive changes. The final value of A, in this case, will be \(A(1-\frac{X}{100})(1-\frac{Y}{100})(1-\frac{Z}{100})\)


Also, let us remember that

2 = 200% (or 100% increase), 3 = 300% (or 200% increase), 3.26 = 326% (means 226% increase), fourfold (4 times) = 400 % of original = 300% increase, 10 times means 1000% means 900% increase, 0.6 means 60% of the original means 40% decrease, 0.31 times means 31% of the original means 69% decrease etc.

1/2 = 50%, 3/2 = 1 + 1/2 = 100 + 50 = 150%, 5/2 = 2 + 1/2 = 200 + 50 = 250% etc.,

2/3 = 1 - 1/3 = 100 - 33.33 = 66.66%, 4/3 = 1 + 1/3 = 100 + 33.33 = 133.33%,

5/3 = 1 + 2/3 = 100 + 66.66 % = 166.66%, 7/3 = 2 + 1/3 = 200 + 33.33 = 233.33%,

8/3 = 2 + 2/3 = 200 + 66.66 = 3 - 1/3 = 300 - 33.33 = 266.66% etc.

1/4 = 25%, 3/4 = 75%, 5/4 (1 + 1/4) = 125% (= 25 increase), 7/4 (1 + 3/4 = 2 - 1/4) = 175% (= 75% increase), 9/4 (2 + 1/4) = 225% (= 125% increase), 11/4 = 275% = (175% increase).

1/5 = 20%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%, 6/5 = 120%, 7/5 (1 + 2/5) = 140% etc.

1/6 = 16.66%, 5/6 = 83.33%, 7/6 (1 + 1/6) = 116.66%, 11/6 = 183.33%

1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%, 9/8 = (1 + 1/8) = 112.5%, 11/8 = (1 + 3/8) = 137.5%, 13/8 = 162.5%, 15/8 = 187.5% etc.

1/9 = 11.11%, 2/9 = 22.22%, 4/9 = 44.44%, 5/9 = 55.55%, 7/9 = 77.77%, 8/9 = 88.88%, 10/9 = 111.11%, 11/9 = (1 + 2/9) = 122.22% etc.


If we have a problem that deals with a discount, it is useful to remember:

Marked Price - Discount = Sale Price. Also Cost Price + Profit = Sale Price.