# Percentage: Basic Concept and Shortcut Tricks

We all use percentage in our daily routine. The term percent mans for every hundred. In its simplest sense percent i.e. per hundred. Thus the expression 20% means 20/100

To express n% as a fraction: n%/100

To express a/b as percentage: (a/b x 100)%

Thus 3/5= (3/5 x100)%

=60%

**For example**: India won 50% of the matches she played means India played 100 matches and won 50 out of them. Pay close attention to “50% of the matches she played” Suppose India played 30 matches then number of matches won by India= 50% of 30

i.e. 30 x 50 /100=15

As seen above n% is nothing but a fraction with numerator n and denominator 100. By remembering certain percentage in reduced form of fraction can be very useful during exams.

½=50% 1/3=33.33%, ¼=25%

1/5=20% 1/6=16.66% 1/8=12.5%

1/9=11.11% 1/11=9.09% 1/12=8.33%

1/15=6.66% 1/16=6.25% 1/20=5%

2/3=66.66% ¾=75% 5/6=83.33%

3/8=37.5% 5/8=62.5% 7/8=87.5%

We will see in further chapter of Ratio that ratio is nothing but fraction. Thus there is strong relation between fraction, ratio and percentage.

For example : In a mixture of 40 liter of milk and 30 liter of water , the ratio of milk an water is 2:3. So this can be converted to fraction of milk as 2/5^{th}. As seen 2/5 = 2/5X100=40%

**Concept of Multiplying Factor:**

Consider the following calculation when 60 has to be increased by 20%.

Increased value= 60 + 20% of 60

=60+ 20×60/100= 72

One can be rearrange this in following manner

60(1+20%)

=60(1+0.2)

=60 x 1.2=> 72

Here 1.2 can be called multiplying factor. It can be simplifying lots of calculation.

**Tips and Tricks with Example**:

**Shortcut tricks 1**: If two values are respectively a% and b% more than third value then the first is (100+a/100+b)/100 % of the second**.**

For example: If two numbers are respectively 30% and 50 % more than a third then first is the

(100+30/100+50)x 100= (130/150 )x 100= 86.66% of the second.

**Tricks 2:**

If A is p% of C and B is q% of C, then A is p/q x 100 % of B.

For Example: If two numbers are respectively 30% and 40% of a third number. Then first of second in percentage = 30/40 X 100= 75%

**Tricks 3**:

If price of commodity is increased by r% , then the reduction in consumption so as not to increase the expenditure , is

[r/(100+r) x100] %

**Tricks 4**:

If price of commodity is decreased by r% , then the increase in consumption so as not to decrease the expenditure , is

[r/(100-r) x100] %

**Trick 5**:

If the original population of town is P, and the annual increase is r% the population after n year =

P (1+r/100) ^{n}

For Example: If annual increase in population of town is 4% and present number of people is 15,625, then population after 3 year

15625(1+ 4/100)^{3}

=17576

**Trick 6**:

If the original population of town is P, and the annual decrease is r% the population after n year =

P (1-r/100) ^{n}

For Example: If annual decrease in population of town is 4% and population two year ago was is 62,500, then present population

62,500(1- 4/100)^{2}

=57,600

**Trick 7**:

If first value is r% more than second value, then the second is [r/(100+r) x 100] % less than the first value.

For Example Ram’s salary is 30% more than that Of Rahul, then Rahul salary is (30/130 x 100 )= 23.07% less than Ram.

**Trick 8**:

If first value is r% less than second value, then the second is [r/(100-r) x 100] % more than the first value.

For Example Ram’s salary is 30% less than that Of Rahul, then Rahul salary is (30/70 x 100 )= 42.85% more than Ram.

**Successive Percentage Change**

If the value is increased and decreased successively by a% and b% then final increase is given by

**[a+b+ab/100]%**

The value is increased or decreased according to positive and negative sign obtained.

**For Example**

If the price of petrol increased successively by 20% and then by 10% the net change in percentage term=

20+10+20×10/100= 32%

The same concept is apply to any relation of this type. Few relations of this types are

Area= Length x Breath

Total sales in rupees= Sales in volumes x Rupees per units

Total sales= Market shares x Total market sales

If the order of increased and decreased is changed, the result remain unaffected so in general

Effect= % increase-% decrease- (% increase x %decrease)/100

The value is increased or decreased according to +ve or –Ve sign obtained.

**If the value is first increased by a% and then decreased by a% then net change is a ^{2}/100**

For Example

The salary of worker is first increased by 10% and then decreased by 10 % then change in their salary is 10^{2}/100=1%