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# Playing with Percentages and Net Discount!

Although many students study percentages in 4th grade, I still see most of them struggle with it during their SAT Math practice period. One of the reasons is the lack of sound basic understanding and the second is the lack of faster execution methods.

In this blog, I wanted to share a quick review of some of the basic yet typical problems. You might even use it in your daily life!

(a) Some commodity priced \$20 went up by say 20%. What is the new price?

Here, conventional method distates us to find the increment value as 0.20 x 20 and then adding it up to \$20. However, that eats away valuable time. What a SAT Math student must be well versed with is to solve it as (1 + 0.20) x 20 or simply 1.20 x 20 = 24

Here is a quick drill:

Original 30, increased by 10% => new value = 1.10 X 30 = 33 (here 1.10 = (100 + 10)/100)
Original 60, increased by 40% => new value = 1.40 X 60 = 84 (here 1.40 = (100 + 40)/100)
Original 160, deccreased by 30% => new value = 0.70 X 160 = 112 (here 0.70 = (100 - 30)/100)

(b) An extension of above concept is when successive increments / decrements or combination of increment and decrement are done.

Here is a quick drill:

Example-1 : Stock price 30, First increased by 10% followed by 15% => new value = 1.10 X 1.15 x 30 = 37.95
Example-2 : Stock price 50, First decreased by 20% followed by 15% => new value = 0.80 X 0.85 x 50 = 34.00
Example-3 : Stock price 70, First increased by 20% then decreased by 10% => new value = 1.20 X 0.90 x 70 = 75.60

Contrary to what most may believe, 10% increase followed by 15% is not equivalent to 25% increase!! As you can see from above, the net increment is actually [(37.95 - 30)/30] x 100 = 26.5%.

Incidentally, if you notice, the formula for two successive increment becomes (a + b + ab/100)%.
In the first example above, a=10 b=15, therefore net increment is 10 + 15 + 150/100 = 26.5 %
That is to say, final value is (1 + 0.265) times the initial value of 30 which becomes 37.95.

Likewise, with 2 successive decrements, we can see that net decrement (refer example 2 above) = [(50 - 34)/50] x 100 = 32%.
That is definitely not the same as 20 + 15 = 35% !!

Incidentally, if you notice, the formula for two successive decrement becomes (a + b - ab/100)%.
In the second example above, a=20 b=15, therefore net decrement is 20 + 15 - 300/100 = 32 %
That is to say, final value is (1- 0.32) times the initial value of 50 which becomes 34.

(c) An extension of above is the case when a radioactive material keeps reducing, say at the rate 2% per year. If original amount was 250 grams, what will be left after 5 years?

Here Initial Amount = 250 Grams. Final Amount = Initial Amount (1 - 0.02) to the power 5. Use calculator and you can solve.

Basically, what you are doing is:
Final Amount = 250((1 - 0.02)((1 - 0.02)((1 - 0.02)((1 - 0.02)((1 - 0.02) i.e. (1-0.02) repeated / multiplied 5 times => Final Amount = 250(0.9039) = 225.98 grams

So, net decrement is [(250 - 225.98)/250] x 100 = 9.6%
Alternatively, you could also write: Final Amount = 250(0.9039) = 250(1- 0.0961) = 225.98 => Net 9.6% decrement in 5 years!

(d) One more logical extension of (b) above is the case where say, a merchandise store is offering you two successive discounts of 10% and 20% (10% was already marked - red tag sale - but on Sat morning between 8 - 10 AM, it is offering additional 20% off). Assuming a commodity was worth \$500, what is the amount you will end up paying? Saving? Net Percentage Discount you are realizing?

Using earlier concepts, final value = (1-0.10)(1-0.20)500 = 360 => You will pay \$360 for \$500 commodity price => Saving = \$140
Now, what is the total or net discount did I get? If you are still thinking 10% + 20% = 30%, think again!
Your net discount was actually 28% !!! Why? Because [(500 - 360)/500]x100 = 28% = (Saving/Original price) x 100

Alternatively, a=10 b=20 and successive decrements => (a + b - ab/100) = 28%
Also, final value = (1-0.10)(1-0.20)500 = 0.90 x 0.80 x 500 = 0.72 x 500 = (1 - 0.28) x 500 => You realized 28% net discount

Note - The important thing here is that contrary to what you may think, you did not realize 30% discount - you realized 28% discount. So, if some other store offers a flat discount of 30% on that commodity of \$500, then you are better off purchasing there as opposed to the place that is offering you 2 successive discounts of 10% followed by 20%! Why? Because 10% followed by 20% is NOT equivalent to 30%. It may be worth your time to visit the other store!

Keep that in mind when you rush to JCP that next BIG SALE Saturday ;-)

Happy Learning!