Percentage:- The word ‘per cent’ means per hundred. Thus 9 percent means 9 parts out of 100 parts. This can also be written as \(\frac{9}{100}\).
Therefore percentage is a fraction whose denominator is 100, and the numerator of this fraction is called the Rate Percent. So \(\frac{9}{100}\) = 9 per cent. The sign of percent is %.
Some Important Percentages and their equivalent Fraction
● To convert any fraction \(\frac{a}{b}\) into rate percent, multiply it by 100 and put a ‘%’ sign
\(\frac{a}{b} = \frac{a}{b} \times 100\% \)
ex. \(\frac{3}{4} = \frac{3}{4} \times 100\% = 75\% \)
● To convert a rate percent to a fraction, divide it by 100 and delete the ‘%’ sign.
ex. 5% = \(\frac{5}{100}\)
A% of B = \(\left(\frac{A}{100}\right) \times B\)
ex. 25% of 100 = \(\frac{25}{100} \times 100\) = 25
● How to understand a Percentage.
● Two step change of percentage of a Number.
⟶ In the first step, a number is changed ( increased or decreased) by x%, and in the second step, this changed number is again changed (increased or decreased) by y%, then net percentage change on the original number can be found out by using the following formula:-
if x of y indicates decrease in percentage, then put a ‘-‘ sign before x or y, otherwise +ve sign will remains.
Percentage change & its effect on Product
If A is changed by x%, and also B is changed by y%, then the net percentage change of the product of A and B can be found out by the formula:-
if x or y indicates decrease in percentage, then put a -ve sign before x or y, otherwise +ve sign will remain.
⟶ % effect on expenditure, when rate and composition are changed, since rate × consumption = expenditure.
⟶ % effect on area of rectangle square/triangle/circle, when its side/radius are changed, since
side₁ × side₂ = area
or
radius × radius = area
⟶ % effect on distance converted, when time and speed are changed, since time × speed = distance
Example:- If length of a rectangle is increasesd by 20% and breadth is decreased by 30% then the % change in the area of the rectangle.
Solution:-
net % change in area = 20 – 30 + \(\frac{20 \times \left( – 30\right)}{100}\)
= -10 – 6
= -16%
So area will decrease by 16% Ans