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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)X increased by 10% would become X + 0.1 X = 1.1 X
X increased by 1% would become X + 0.01 X = 1.01 X
X increased by 0.1% would become X + 0.001 X = 1.001 X
X decreased by 10% would become X – 0.1 X = 0.9 X
X decreased by 1% would become X – 0.01 X = 0.99 X
X decreased by 0.1% would become X – 0.001 X = 0.999 X
X increased by 200% would become X + 2X = 3X
X decreased by 300% would become X – 3X = –2X
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 DECREASEIncrease % \(= \frac{Increase}{Original Value} \times 100\)
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 that2 = 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.
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