# Dealing with Percentage – Ravi Handa

• A percentage, for the lack of a better definition, is a number or ratio as a fraction of 100. Probably the oldest application of it, like so many other things in today’s world, was at the time of the Roman empire. Augustus, founder of the Roman Empire and its first Emperor, levied a tax of 1/100 on goods sold at auction known as centesima rerum venalium. Computation with these fractions were similar to computing percentages. The word itself is derived from the Latin per centum meaning “by the hundred”. The percent sign evolved by gradual contraction of the phrase per cento. The "per" was often abbreviated as "p." and eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line from which the modern "%" is derived. But enough about history and let us come back to the present and some practical uses of the same.

Percentage as Fractions:

One of the most useful tips, when it comes to percentage, is the use of fractions in calculating them. Given below is a table which can help you get started. There is no better way to remember these values than to start using them in your daily calculations.

Fraction%ageFraction%age
1/250.00%1/911.11%
1/333.33%1/1010.00%
1/425.00%1/119.09%
1/520.00%1/128.33%
1/616.66%1/137.69%
1/714.28%1/147.14%
1/812.50%1/156.66%

Example: Find out 28.5% of 476
a) 135.66 b) 136 c) 136.28 d)136.43

As you can see here the option are really close and you might think that you would have to calculate the actual value. But if you are well versed with the table given above you can save some effort and time in doing so. These seconds you save might be vital in the exam.

To calculate 28.5%, I urge you to have a closer look at 1/7.

1/7 represents 14.28%, so 2/7 would represent 28.56% which is freakishly close to the percentage that we are looking for. Also, calculating 2/7 of 476 is not that difficult as 476/7 is divisible by 7 and it is 68. So, 2/7 of 476 will be 136. Now, we know that 2/7 is 28.56% and we also know that that it is 136. Our answer will be little less than 136 as we are trying to find out 28.5% which is a little less than 28.56%. So, our answer would be 135.66. Option A

Percentage Change:

To find out the percentage change, you can use the formula
% change = (Final value - Initial value) * 100 / Initial value

A lot of people make a mistakes in calculating values related to the given formula because they do not calculate % change based upon the initial value.

Example: My current salary is Rs. 1000 a day, which is 25% higher than what it was last year. What was my salary last year?

If you think that the answer to the above question is Rs. 750 per day then you are making the same mistake that lot of students make. The raise that I got (25%) was not on my current salary but the initial value, which was my salary previous year.
To calculate it correctly, let us assume my last year’s salary to be ‘x’ Rs. per day.
A 25% increase means an increase of 1/4
My current salary is 5/4 of my previous year’s salary.
This means that my salary last year was 4/5 of my current salary.
My last year’s salary = 4/5 * 1000 = 800

Changing Quantities by Percents:

The simplest way to change a number by a given percent is to simply multiply it by the ‘final’ percentage.
That is, to increase a number by x%, simply multiply it by (100 + x)% or (100 + x)/100 , and to decrease it by x%, simply multiply it by (100 – x)% or (100 - x)/100

Example: A piece of paper is in the shape of a right angled triangle and is cut along a line that is parallel to the hypotenuse, leaving a smaller triangle. There was a 35% reduction in the length of the hypotenuse of the triangle. If the area of the original triangle was 34 square inches before the cut, what is the area (in square inches) of the smaller triangle?
a.16.665 b. 16.565 c. 15.465 d. 14.365

If the hypotenuse reduced by 35%, it became 100 – 35 = 65% of the original or in other words it became 0.65 times the original. The other two sides also become 0.65 times the original. Area will become (0.65)^2 = 0.4225 times the original.
New area = 0.4225*34 = 14.365. Option D

Successive Percentage Change:

I am sure you would have seen retail stores with offers like 50% + 40% off. If my memory serves me right, these type of offers were made extremely popular by Koutons. I always thought of it as a 90% off offer but clearly I was wrong. I guess some of you might have made the same mistake as well. A 50% + 40% off does not mean a 90% off. It actually means that you will be given a discount of 50% first and then on the reduced price you will given another 40% discount. Let us calculate to figure out how much this actually means.

Let us assume that the T-shirt you are trying to buy costs 100 Rs. A 50% discount would bring down the price of the T-shirt to 50 Rs. Another 40% discount on the reduced price of 50 Rs. would further bring down the price by 20 Rs., which is 40% of 50, to 30Rs. So, the price has effectively gone down from 100 Rs. to 30 Rs. which means that the effective discount has been 70% and not 90%.

Example 1: Shop A is selling a T-shirt at a discount of 50% + 40% on the MRP whereas shop B is selling the same T-shirt at a discount of 40% + 50%. You should buy the T-shirt from Shop A or Shop B?

Now, there might be a few things going on in your head like ‘A’ is better because it is offering higher percentage first or ‘B’ is better because it is offering higher percentage later or this cannot be determined until and unless we know the MRP of the T-shirt. Well, stop your internal monologue.

The answer is that it would not make a difference whether you buy it from Shop A or from Shop B. Don’t believe me – do the math.
Let us assume that the MRP of the T-shirt is ‘x’
From Shop A, a 50% discount would bring down the price to 0.5x and another 40% discount would bring down the price to 0.3x
From Shop B, a 40% discount would bring down the price to 0.6x and another 50% discount would bring down the price to 0.3x
As you can see the sale price is coming out to be the same in both cases.

In case you are wondering as to why it is the same in both cases, it is because percentages are multiplicative in nature and p * q is always the same as q * p.

For future reference, you can use the formula for effective % change in case of successive % changes of a% and b% is (a + b + ab/100)

Note: Please keep in mind the +/- sign while using the formula.

If you put a = 50% and b = 40% (from the above example), you would get 110% which would be the incorrect answer. You should use -50% and -40% to get the correct answer of -70%. You will have to use –ive values for the above example because the % changes considered are discounts.

Example 2: A company’s revenue grew by 10% and 20% in the 2009 and 2010 but fell by 25% in 2011. What was the net % change?
Use the formula for the first two years = 10 + 20 + 10 * 20/100 = 32
Combine this with the last year = 32 – 25 + (32)(-25)/100 = 7 – 8 = -1%

Compensating Percentage Change:

I am sure you must have encountered questions like these:

a) The price of beer has gone up by 25%, by how much should you reduce your consumption so that the your expenses do not change.

b) Ravi got a salary hike of 10%. The new boss thought this shouldn’t have happened because Ravi doesn’t deserve it, so he slashes Ravi’s salary by 10%. Is Ravi back to the original? If no, then by what % should Ravi’s boss deduct his salary.

Now there are couple of ways of doing these questions. One is using the concepts of fractions, proportionality, etc. The other one is by using the formula. Let us discuss both of them

a) Price of beer has gone up by 25%
Price of beer has increased by 1/4th
Price of beer has become 5/4th
Consumption should become 4/5th
Consumption should reduce by 1/5th
Consumption should reduce by 20%

b) A hike of 10% and then a reduction of 10% won’t be fair as Ravi would end up getting less than the original. Using the successive % change formula {10 – 10 + 10* (-10)/100 = -1}, we can say that he would end up getting 1% lesser than the original. To get the correct deduction value:
Ravi’s salary was hiked by 10%
Ravi’s salary has increased by 1/10th
Ravi’s salary has become 11/10th
To go back to the original, it should become 10/11th of the current value
It should reduce by 1/11
It should reduce by 9.09%

We could have also used the formula for compensating a change of r% : - 100r/(100 + r) %

a) r = 25%. Compensating % change = – (100 * 25)/(100+25)
–20% = reduction of 20%

b) r = 10%. Compensating % change = – (100 * 10)/(100+10)
–9.09% = reduction of 9.09%

I hope you found this post useful and it would help you get a higher percentage and percentile in your exam.

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