Percentage is not new to us. We deal with it every day; while giving tips, while calculating discounts etc. In CAT and other exams, application of percentage can be seen in various topics like TSD, Profit/Loss, Ratio/Proportion, Number theory etc. and not to forget DI.

**Concept 1: X% of Y = Y% of X**

What is 27% of 90?

Our thought process mostly would be 10% of 90 is 9, 5% is 4.5 and 2% is 1.8

hence 27 % of 90 = 18 + 4.5 + 1.8 = 24.3 (5 seconds max)

There would be faster methods but we are not going to any math Olympiad. Practice this method religiously and you will be quick enough for CAT and similar exams. :)

Now what is 90% of 27?

No need to calculate again, we already did it and got the answer as 24.3!

27% of 90 = 27 x 90 / 100 = 90 x 27 / 100 = 90% of 27

That means when we solve one percentage sum we are actually solving two. And this comes very handy if one problem is much easier to solve than the other one.

For example, what is 36% of 25?

We know 36% of 25 = 25% of 36

as second one is much easier to calculate we can say 36% of 25 = 9 in no time.

Let’s recap how to convert percentage to fractions and vice versa.

60% = 60/100 = 3/5

so to calculate 65% of 60, we can easily calculate as 65% of 60 = 60% of 65 = 3 x 65 / 5 = 39

Now to convert 3/5 to percentage, (3/5) x 100 = 60%

Before going further, make sure you know values like 1/3 = 0.33, 1/7 = 0.1428, 1/9 = 0.111, 1/11 = 0.0909 etc. If we get a question like a 140 is increased by 28.56% we can immediately know it has increase 2/7 times which is 140 (1 + 2/7) = 140 x 9/7 = 180. Saved time, isn’t it? Mug up!

**Concept 2: Successive Percentage Increase/decrease**

Let’s start with a simple problem, what would be the final value if 60 is increased by 30%

Final value = 60 x (1 + 30/100) = 78

(Else we know 10% of 60 = 6, so 30% increase should add 18 yielding 78 )

Now what would be the final value if 60 is increased by 30% and then by 50%

Final Value = 60 x (1 + 30/100) (1 + 50/100) = 117

Let the successive increase in percentages be a% and b%. In that case, the total increase will be (a + b + ab/100) % (for decrease use a negative sign)

Here an increase of 30% and 50% can be considered as a net increase of 95%

Final value = 60 (1 + 95/100) = 60 x 195/100 = 117

(Smart way would be 95% increase means 5% ( = 3 ) less than double the original value, so 60 x 2 – 3 = 117)

Now what would be the final value if 60 is increased by 30% first and then decreased by 50%?

Net change = (30 – 50 – 15) = -35% => 35% decrease

Final value = 60 – 21 = 39 (because 35% of 60 is 6 x 3 + 3 = 21)

What if a 50% off + 50% off is given on a brand? Will we get it for free? :)

Net change = (-50) + (-50) + (-50) x (-50)/100 = -100 + 25 = -75% = 75% discount

Now an interesting one, what if 60 is increased by 50% first and then decreased by 50%

Net change = 50 – 50 – 25 = -25 => 25% discount.

What if 60 is decreased by 50% and then increased by 50%?

Net change = -50 + 50 – 25 = -25% discount

Got the idea right, cool!

**Concept 3: Product Constancy**

Petrol/Diesel price again hiked today. But we are not affected as we always fill for 100 Rupees. ;)

Formula:

If Z = XY and if there is p% increase in X (or Y). Then Y (or X ) should decrease by P/ (100+P) x 100 to keep Z constant.

Also, and if there is p% decrease in X (or Y). Then Y (or X ) should increase by P/ (100 - P) x 100 to keep Z constant.

If Milk price is hiked by 25%, how much % should we reduce the consumption to keep the expenditure constant (say at 100)?

Expenditure = Price x Consumption

Price went up by 5%

so Consumption should come down by 25/ (100+25) x 100 = 2500/125 = 20%

Also, if price is decreased by 10%, then consumption should increase by 10/ (90) x 100 = 11.11% (we knew 1/9 = 0.1111)

A man spends Rs.54 on apples every month. Price of Apple is decreased by 20% and this man can buy 10 apples more. What is the reduced price per dozen?

If price is 20% decreased => consumption should increase by 25% (as seen in previous case)

25% increase = 10 apples => He used to buy 40 apples before.

54 Rupees for 40 Apples.

Now price is reduced by 20% (Multiplying factor = 1-20/100 = 80/100 = 0.8 )

so new price per dozen = 0.8 x 54/40 x 12 = 13 (approx)

Ramesh takes 90 minutes to reach his office from home. If his speed becomes 5/4 of original speed. How much less time will he take to reach his office?

Here distance is same and we can see Speed increased by 25%. So Time should reduce by 20% = 18 minutes.

Once you master this concept, many problems in TSD, Profit/Loss, and Geometry will be a cake walk.

**Some extra gyan:**

If A’s income is 16.66% than B. Does that mean B’s income is 16.66% less than that of A. Of course not.

If A is X% higher than B then B is 100 * X/ (100+X) % lower than A

So In the given question, B’s income is 100 x 16.66 / (100 + 16.66) = 14.28% lesser than A

We don’t need this formula here if you have mastered the fractional values. We know 1/6 = 0.1666

So A = (1 + (1/6) B ) = 7/6 B

=> B = (6/7) A => (1 – 1/7) A => 14.28% less.

Also, If A is X% lower than B then B is 100 * X/ (100-X) % higher than APercentage change (increase/decrease) = (Change/original) x 100

A was earning 60 rupees and he got a hike and started earning 70. What is the percentage change?

Percentage change = [(70 – 10)/60] x 100 = 16.66% increase (we know 1/6 = 0.1666).

Now, how much percentage decrease should happen for A to earn 60 rupees again?

= [(60 – 70)/70] x 100 = 14.28% (we know 1/7 = 0.1428 )If current value, P increases at a rate of R% per annum then

Value after n years = P*(1 + R/100)^n

Value n years ago = P/(1 + R/100)^n

If value decreases at a rate of R%, use –R instead of R in the above formula.

Percentage is not new to us. We deal with it every day; while giving tips, while calculating discounts etc. In CAT and other exams, application of percentage can be seen in various topics like TSD, Profit/Loss, Ratio/Proportion, Number theory etc. and not to forget DI.

**Concept 1: X% of Y = Y% of X**

What is 27% of 90?

Our thought process mostly would be 10% of 90 is 9, 5% is 4.5 and 2% is 1.8

hence 27 % of 90 = 18 + 4.5 + 1.8 = 24.3 (5 seconds max)

There would be faster methods but we are not going to any math Olympiad. Practice this method religiously and you will be quick enough for CAT and similar exams. :)

Now what is 90% of 27?

No need to calculate again, we already did it and got the answer as 24.3!

27% of 90 = 27 x 90 / 100 = 90 x 27 / 100 = 90% of 27

That means when we solve one percentage sum we are actually solving two. And this comes very handy if one problem is much easier to solve than the other one.

For example, what is 36% of 25?

We know 36% of 25 = 25% of 36

as second one is much easier to calculate we can say 36% of 25 = 9 in no time.

Let’s recap how to convert percentage to fractions and vice versa.

60% = 60/100 = 3/5

so to calculate 65% of 60, we can easily calculate as 65% of 60 = 60% of 65 = 3 x 65 / 5 = 39

Now to convert 3/5 to percentage, (3/5) x 100 = 60%

Before going further, make sure you know values like 1/3 = 0.33, 1/7 = 0.1428, 1/9 = 0.111, 1/11 = 0.0909 etc. If we get a question like a 140 is increased by 28.56% we can immediately know it has increase 2/7 times which is 140 (1 + 2/7) = 140 x 9/7 = 180. Saved time, isn’t it? Mug up!

**Concept 2: Successive Percentage Increase/decrease**

Let’s start with a simple problem, what would be the final value if 60 is increased by 30%

Final value = 60 x (1 + 30/100) = 78

(Else we know 10% of 60 = 6, so 30% increase should add 18 yielding 78 )

Now what would be the final value if 60 is increased by 30% and then by 50%

Final Value = 60 x (1 + 30/100) (1 + 50/100) = 117

Let the successive increase in percentages be a% and b%. In that case, the total increase will be (a + b + ab/100) % (for decrease use a negative sign)

Here an increase of 30% and 50% can be considered as a net increase of 95%

Final value = 60 (1 + 95/100) = 60 x 195/100 = 117

(Smart way would be 95% increase means 5% ( = 3 ) less than double the original value, so 60 x 2 – 3 = 117)

Now what would be the final value if 60 is increased by 30% first and then decreased by 50%?

Net change = (30 – 50 – 15) = -35% => 35% decrease

Final value = 60 – 21 = 39 (because 35% of 60 is 6 x 3 + 3 = 21)

What if a 50% off + 50% off is given on a brand? Will we get it for free? :)

Net change = (-50) + (-50) + (-50) x (-50)/100 = -100 + 25 = -75% = 75% discount

Now an interesting one, what if 60 is increased by 50% first and then decreased by 50%

Net change = 50 – 50 – 25 = -25 => 25% discount.

What if 60 is decreased by 50% and then increased by 50%?

Net change = -50 + 50 – 25 = -25% discount

Got the idea right, cool!

**Concept 3: Product Constancy**

Petrol/Diesel price again hiked today. But we are not affected as we always fill for 100 Rupees. ;)

Formula:

If Z = XY and if there is p% increase in X (or Y). Then Y (or X ) should decrease by P/ (100+P) x 100 to keep Z constant.

Also, and if there is p% decrease in X (or Y). Then Y (or X ) should increase by P/ (100 - P) x 100 to keep Z constant.

If Milk price is hiked by 25%, how much % should we reduce the consumption to keep the expenditure constant (say at 100)?

Expenditure = Price x Consumption

Price went up by 5%

so Consumption should come down by 25/ (100+25) x 100 = 2500/125 = 20%

Also, if price is decreased by 10%, then consumption should increase by 10/ (90) x 100 = 11.11% (we knew 1/9 = 0.1111)

A man spends Rs.54 on apples every month. Price of Apple is decreased by 20% and this man can buy 10 apples more. What is the reduced price per dozen?

If price is 20% decreased => consumption should increase by 25% (as seen in previous case)

25% increase = 10 apples => He used to buy 40 apples before.

54 Rupees for 40 Apples.

Now price is reduced by 20% (Multiplying factor = 1-20/100 = 80/100 = 0.8 )

so new price per dozen = 0.8 x 54/40 x 12 = 13 (approx)

Ramesh takes 90 minutes to reach his office from home. If his speed becomes 5/4 of original speed. How much less time will he take to reach his office?

Here distance is same and we can see Speed increased by 25%. So Time should reduce by 20% = 18 minutes.

Once you master this concept, many problems in TSD, Profit/Loss, and Geometry will be a cake walk.

**Some extra gyan:**

If A’s income is 16.66% than B. Does that mean B’s income is 16.66% less than that of A. Of course not.

If A is X% higher than B then B is 100 * X/ (100+X) % lower than A

So In the given question, B’s income is 100 x 16.66 / (100 + 16.66) = 14.28% lesser than A

We don’t need this formula here if you have mastered the fractional values. We know 1/6 = 0.1666

So A = (1 + (1/6) B ) = 7/6 B

=> B = (6/7) A => (1 – 1/7) A => 14.28% less.

Also, If A is X% lower than B then B is 100 * X/ (100-X) % higher than APercentage change (increase/decrease) = (Change/original) x 100

A was earning 60 rupees and he got a hike and started earning 70. What is the percentage change?

Percentage change = [(70 – 10)/60] x 100 = 16.66% increase (we know 1/6 = 0.1666).

Now, how much percentage decrease should happen for A to earn 60 rupees again?

= [(60 – 70)/70] x 100 = 14.28% (we know 1/7 = 0.1428 )If current value, P increases at a rate of R% per annum then

Value after n years = P*(1 + R/100)^n

Value n years ago = P/(1 + R/100)^n

If value decreases at a rate of R%, use –R instead of R in the above formula.