7/28/2019 Percentage Theory

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Lecture 5 Percentage Theory

Prepared by Anup Sinha Page 1

The term percentage means parts per 100 or for every hundred. A fraction whose denominator is 100 iscalled 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 rupees he invested in the business

i.e., 20/100 rupees for each Rupee. The abbreviation of percent is p.c. and it is generally denoted by %.

Lets start with a number X (= 1 X)

X increased by 10% would become X + 0.1 X = 1.1 XX 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 XX 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 XX 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 differentanswers.

A Percentage can be expressed as a Fraction. 10% can be expressed as 10/100 or 1/10. To express apercentage as a fraction divide it by 100 a % = a/100.

To express a fraction as a percent multiply it by 100

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%.

Some Typical Formula To Remember

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

Successive change in percentage. If a number A is increased successively by X% followed by Y%, and thenby Z%, then the final value of A will be

7/28/2019 Percentage Theory

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Lecture 5 Percentage Theory

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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

Let the present population of a town be p and let there be an increase of X% per annum. Then:

(i) Population after n years = p [1 + (X/100) ]n

(ii) Population n years ago = p / [1 + (X/100) ]n

If the population of a town (or value of a machine) decreases at R% per annum, then :

(i) Population (or value of machine) after n years = p [1 (R/100) ]n

(ii) Population (or value of machine) n years ago = p / [1 (R/100) ]

n

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.

NOTE : In the problems on DISCOUNT remember the following:

1. Marked price is the price listed on the article (called list price).2. Discount is calculated on Marked price and NOT on Cost price.

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

Problems