# Percentages

Tag:
The term percent comes from Latin and means "for every hundred". So when you hear a statistic such as "10% of all people are left-handed" that means, for every hundred people, 10 are left-handed.
Percent of change (percent of increase or percent of decrease) can be calculated using the following formula:

Example: Richard's health insurance premium for last year was \$1440. If he paid \$1512 this year, what is the percent of increase on his health insurance premium?
Answer:(\$1512-\$1440)/\$1440 — 100% = 72/1440 — 100% = 5% Hence, the percent of increase on the health insurance premium is 5%.

Example:
A T-shirt is sold at 20% off the original price of \$32. What is the sales price?
Answer: Let the sales price be x dollars. (\$32-x)/\$32 — 100% = 20% (\$32-x)/\$32 = 0.2 \$32 - x = \$6.4 x = \$25.6 Hence, the sales price of the T-shirt is \$25.6.

The percent of a number can be found based on the type of the question asked.

Type 1: x is y percent of n
Translation 1: x = y% — n
Example: 24 is 15% of what number?
Answer: 24 = 15% — n; n = 24 · 15% = 160

Type 2: x percent of y is n
Translation 2: x% — y = n
Example: 11% of 50 is what number?
Answer: 11% — 50 = n; n = 5.5

CATrick
What is 320% of 25?
Many of the students will start doing like this:(320/100)*25 , little realizing that 320% of 25 is same as 25% of 320. so the better and faster method is to deal with fractions and this question is best solved like this: (25/100)*320 i.e 1/4th of 320 and hence answer is 80.

A common error when using percentages is to imagine that a percentage increase is cancelled out when followed by the same percentage decrease.
A 50% increase from 100 is 100 + 50, or 150. A 50% reduction from 150 is 150 - 75, or 75. The end result is smaller than the 100 we started out with. This phenomenon is due to the change in the "initial" value after the first calculation. In this example, the first initial value is 100, but the second is 150.In general, the net effect is:
(1 + x) (1 - x) = 1 - x2,
that is a net decrease proportional to the square of the percentage change

Example: If a quantity increases by 20% and then decreases by 10%, then what is the net increase/decrease?
Answer: If we take the quantity to be x, then the answer will be found as: x(1.2)(0.9) = 1.08x and hence 8% increase. This calculation is more tedious than when quantity chosen is 100. When quantity is 100, after the 20% increase, the quantity becomes 120 and a decrease of 10% on this will be a decrease of 12 units. Therefore, the quantity becomes 108 and hence 8% increase.

Example: If Rama's income is 20% more than Shyama's income, by what percentage is Shyama's income less than Rama's?
Answer: Instead of using x or y, take Shyama's income as Rs 100. Then, Ram's income = Rs 120.
Therefore, Shyama's income is Rs 20 less or 1/6th less or 16.66% less than Rama's income.
Similarly, if Ram's income were less by 20%, then Rama's income would be Rs 80 and hence Shyam's income would be Rs 20 more or 25% more than Rama's income.

Note that the percentage is changed because of base (though the quantity of increase or decrease of Shyam's income is same).

Successive Changes

Whenever there is a successive changes of a particular vale , the net change can be expressed as a single percentage

[a + b + (ab/100)]%.

Here a and b are the first change & the second percentage changes in that order The above formula is applicable when the successive changes are on the same parameter .

Ex. Two successive changes in the price of an item.
The length & breadth of a rectangle changes by a% and b%.

Calculating Percent Change when the Base is a Negative Number percent change is a meaningless statistic when the underlying quantity can be positive or negative (or zero). The actual change means something, but dividing it by a number that may be zero or of the opposite sign does not convey any meaningful information, because the amount by which a profit changes is not proportional to its previous value. Yet, such a percentage is often requested, and in reasonable cases seems useful. So what do we do? Lets discuss about it in our forum @ http://www.cat4mba.com/forum/

Undoing Percentage Changes
If original amount is A, and the percent increase is p, then the new amount is A' = A(1+p)
You want to decrease it by some percentage q, to get back to A.
That is, you want to find q such that A = A(1+p)(1-q) 1 = (1+p)(1-q) 1/(1+p) = 1 - q q = 1 - 1/(1+p) = (1 + p - 1)/(1+p) =

p/(1+p)

Let's check this with a simple example.
If we increase something by 100%, we should have to decrease it by 50% to get back to where we started: q = 1.0 / 2.0 = 0.5 If we increase something by 1/3, we should have to decrease it by 1/4: q = (1/3) / (4/3) = 1/4 So this seems to work okay. So if p is 2 percent, q would be q = 0.02 / 1.02

Question: A substance is 99% water. Some water evaporates, leaving a substance that is 98% water. How much of the water evaporated
Answer:  Let's say we start with W units of water, and S units of other stuff. We originally have 99% water,
so W/ (W+S) = 99/100
Now we want to reduce the water to some fraction, F, of the original amount. And we want to end up with 98% water: FW/( FW+S) =98/100
We can solve each of these equations for S:
W/ (W+S)= 99/100 100W = 99(W+S) 100W = 99W + 99S W = 99S W/99 = S and
FW/FW+S =98/100 100FW = 98(FW+S) 100FW = 98FW + 98S 2FW = 98S 2FW/98 = S

Two things equal to the same thing are equal to each other, so W/99 = 2FW/98 1/99 = 2F/98 98/(99*2) = F 0.495 = F So 49.5% of the water remains, which means that 50.5% evaporated.

population formula:
Pn = P0 (1 + r/100)n ,
where, r = rate of growth;
n = number of time periods (generally in years);
P0 is the population at the start of the first time period and
Pn is the population at the end of the last time period.

Example:
If the population today is 10,000 and increases at the rate of 5% per annum, what was the population 4 years ago. Answer: Note that, in this example, Pn = 10,000, r = 5%, n = 4 years and Po is required to be calculated. Therefore, 10,000 = P0 (1+5/100)4 =>P0=8227

Naturally, if the population is decreasing, rate of growth will be taken as negative and Pn = P0 (1 - r/100)n ,

Further, if every year, the population increases at a different rate, then

Pn = P0 (1 + r1/100)n(1 + r2/100)n (1 + r3/100)n .

In case of a decrease in a particular year replace + rn with -rn

Example: If a bacteria population increases at the rate of 6% in the first 10 minutes, and then at the rate of 10% in the next 10 minutes, then what is the overall percentage increase in the population?

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