Concept of 'BY' and 'TO' Percentage

Concept of 'BY' and 'TO' Percentage

Please note that there is a clear difference between "by" and "to" e.g., the income is reduced by 40% it means the new income is 60% of the original and the income is reduced to 40% means the new income is 40% of the original value.  Thus "by" represents difference and "to" represents final value.

e.g., The income of sarika is increased by 20% means its new income is 100+20 = 120% of the original income. The income of sarika is increased to 120% means the new income of sarika is 120% of the original income.

Example 1: 

In an election between two candidates, the candidate who got 57% valid notes won by a majority of 420 votes. Find the total no. of valid votes.

Solution:

winner                          loser

0.57x                             0.43x                                   (100-57=43)

       -----------------------                                            (57-43 = 14)

                 0.14x

0.14x = 420

x = 3000

Hence total valid votes = 3000

Product Constancy Conditions

Product Constancy Conditions

i) When one factor of a product is increased by p% then the other factor will be decreased by (p/100+p X 100)%

 It means when one factor of a product is increased by n/d then the other factor is decreased by n/(d+n)

ii) When one factor of a product is decreased by p% then the other factor will be increased by
(p/100-p x 100)%

 It means when one factor of a product is decreased by n/d then the other factor will must be increased by n/(d-n)

Example 1:

If the price of a commodity be raised by 20% then by how much per cent a house holder reduce his consumption of the same commodity so that his expenditure does not increase.

Solution:

since here product (i.e., expenditure) is constant rate x consumption = expenditure

initially ------> 1 x 1 = 1

after change   1.2 * X = 1

X = 0.833 . therefore decrease in value = 16.66%


Example 2:

If the price of petrol falls down by 20% by how much per cent must a person increase its consumption, so as not to decrease the expenditure on this items?

Solution:

since product is constant
decrease by                              increase by

20% = 1/5 ------------------>       1/4 = 25%                ( n/d -----> n/(d-n))


Example 3:

Due to 50% increase in the price of rice. We purchased 5 kg less rice with the same amount of rs.60. What is the new price of rice?

Solution:

Increase in price                           Decrease in amount
50% = 1/2           ---------------->       1/3 = 33.33%

since the new quantity of rice decreased by 33.33% which is equal to 5kg it means initially there was 15kg rice to be used.

so, the initial price = rs.4           (60/15 = 4)

and final price = rs.6                   (60/10=6)


Example 4:

The length of a plot is decreased by 33.33%. By how much % the breadth of the plot will be increased so that the area remains constant?

Solution:

(Decrease)                             (Increase)

33.33=1/3 --------------->    1/2 = 50%

Concept of Product Constancy

Concept of Product Constancy

It is the same as we know the inverse proportion in the chapter of ratio, proportion and variation.

eg., when the rate of pencil is rs.1.25 then we can purchase 16 pencils by paying rs.20. If the rate of a pencil is decreased  by rs.0.25 then we can purchase 20 pencils by paying rs.20.

Explanation: 

Rate x No. of Pencils = Price

1.25 x 16 = 20
1.00 x 20 = 20

so you can see that here product 20 is constant in both the cases. Thus it is clear that if we reduce the price of a pencil to rs.0.50, then we can purchase 40 pencils in rs.20.

some more examples of product constancy:

speed x time = distance
rate x time = cost
efficiency x time = work
length x breath = area
average x no of elements = total value
rate x quantity = price

eg., The price of sugar is increased by 25% then by how much per cent should a customer reduce the consumption (ie., quantity used) of sugar so that he has not to increase his expenses on sugar.

price x quantity = expenditure

100 x 100 = 10,000
125 x X = 10,000

x = 10,000/125 = 80. Therefore, % reduction = 20%

or

1 x1 = 1
 1.25 x k =1 = > k = 0.8

thus there will be 20% decrease in the consumption of sugar in order to maintain to same expenditure on sugar.

Problems on Percentage

Example 1:

Height of arun is 50% greater than the height of amir khan. Height of amir khan is how much per cent less than that of arun?

Solution:

                             +50%  = 1/2

Amir khan --------------------------->  Arun

                 <----------------------------

                     -33.33% = 1/3 = 1/2+1

thus the height of amir is 33.33% less than arun

Example 2:

Due to irregular working habits of ramachandran his salary was reduced by 20% but after same month his salary was increased to the original salary. What is the percentage increase in salary of ramachandran? 

Solution:

Let initial salary be rs.100  

                -20%

100 -------------------> 80

      <-------------------

               +25%

Example 3:

Kajolu usually wear saree, which is 16.66% less than the actual length of the saree. By how much per cent the actual length of the saree is greater than the length of saree which kajol usually wears?

Solution:

Actual length                                                              Usual length

                                    -16.66% = 1/6

     -------------------------à A---------------------àUß-------------------       

                                          +20% = 1/5 = (1/6-1)

So the actual length of the saree is 20% greater than the usually used saree   

Example 4:

Initially mr. rakhi sawant has rs. 200 in her wallet then she increased it by 20%. Once again she increased her amount by 25%. The final value of money in her wallet will be how much percent greater than the initial amount.

Solution:

            +20%                                       +25%

200 --------------------à 240 -----------------------à 300

So the required % increase = 300-200/200 x 100 = 50%

Example 5:

The age of B is 50% greater than the age of A. The age of C is 20% less than the age of B. By how much percentage the age of C is greater than the age of A.

Solution:

            +50%                                       -20%

A -----------------------à B-----------------------------àC

100                              150                                          120

So the required percentage change = 20/100 x 100

                                                       = 20%

       

Advanced Concept of Percentage Change

Advanced Concept of Percentage Change

(A) If a value p is increased by x%, then we have to decrease the resultant value by
 (x/x+100 x 100)% to get back to the original value p.

original value p ------------->  p * x /100 --------> ( p + px/100) = p (100 + x/100)

Now the percentage decrease

= (p(100 + x /100) - p) / p (100 + x/100 ) x 100 = ( x/100+ x  * 100) %

In other words (i.e., in terms of fraction) if a value is increased by n/d then to get back the same number p from the resultant value, we have to decrease the increased value by ( n/ d+n).

Example 1:

Salary of arun in 2001 was 100rs per day and his salary in 2002 was 125rs per day. Again in 2003 his salary was rs 100 per day.

i) what is the percentage increase in his salary in 2002?
ii) what is the percentage decrease in his salary in 2003 over 2002?

Solution:

i) 125 - 100 /100  x 100 = 25%

ii) 125 - 100/125 x 100 = 20%

In fact is same absolute change but percentage change is different due to different denominator (i.e., base change). So the base change is as much important as the numerator or absolute change in quantity.


Example 2: 

Age of arun is 20 years. If the udhay age is 25% greater than that of arun then how much per cent arun age is less than udhay age?

Solution:

Age of arun                                    age of udhay

20 -------------+25%------------------->     25
     <-------------(-20%)-----------------

Solution using percentage change formula
here x = 25, then

percentage decrease (or less)  =  25/125 x 100 =20%

Solution using fractional change formula

since increase = 1/4                        ( 25% = 1/4)

Therefore decrease = 1/4+1 = 1/5 = 20%


Example 3

Arun has 33.33% more pencils than udhay has. By how much per cennt less pencils udhay has than that of arun?

Solution:

arun --------33% = 1/3-------------------> udhay
        <------(-25% = 1/4 = 1/3+1 )--------------

so udhay has 25% less pencils than arun has.

Percentage Change and Percentage Point change

Percentage Change and Percentage Point change

            Last year arun slaray was rs.10,000 and aruna huberta salary was rs.8,000. This year arun salary is rs. 12,000 while huberta salary is rs.10,000.

1.      I) What is the percentage increase of arun salary?

Ii) What is the percentage increase of huberta salary?

iii) Percentage increase in huberta salary is how much percent greater than the percentage increase in arun salary?

iv) What is the percentage point change in the salary of huberta and arun?

Solution:

i)                    12,000-10,000/10,000 x 100  = 20%

Or 2/10 --à 1/5 = 20%

ii)                  2/8 = 1/4 = 25%

iii)                Percentage increase of huberta salary = 25

Percentage increase of arun salary = 20

So the required percentage = 25-20/20 x 100 = 25%

It means percentage increase of huberta salary is 25% greater than the percentage increase of arun salary.

iv)                Percentage point change = (Percentage increase in huberta salary – Percentage increase in arun salary)

= 25-20 = 5 percentage point

Infact percentage point change is the difference between two percentage values.

Percentages (increase / decrease in quantity)

Some Most Important Values

1/2 =50%                    2/5=40%

1/3 = 33.33%              3/5=60%

2/3 = 66.66%              1/6=16.66%

1/4=25%                     5/6 =83.33%

3/4 =75%                    1/5 = 20%

 Example 1:

The height of arun some times ago was 110cm. Now his height is 120cm. Find the percentage change in his height.

Solution:

120-110/110 x 100 = 9.09%

Shortcut 

10/110 = 1/11 = 9.09%

Alternatively : 12/11 = 109.09%, so increase = 9.09%

Example 2

The total expenses of a hostel were rs.8000 per month. Some students left the hostel due to which the new expenses come down by rs.1000. Find the percentage decreases in expenses of the hostel.

Solution:

1000/8000 = 1/8 = 12.5%

Example 3

Salary of raja is rs.9000 per month and salary of rani is rs. 10,000 per month.

             i) What per cent is the salary of rani to that of raja?

            ii) What per cent is the salary of raja to that of rani?

Solution:

i) 10,000/9,000 x 100 = 111.11%

Alternatively:

10/9 = 9/9 + 1/9 = 100% + 11.11% = 111.11%

ii) 9/10 = 90%                                ( 1/10 = 10%)

Hence, salary of raja is 90% to the salary of rani.

Percentage Problems

Percentage Problems:

A tin contains 24 litres of milk. Due to leakage, 720 ml is lost. What per cent of milk is still present in the tin?

Solution:

Ans: 97%

Price of an item increased from 16.50 to rs. 41.25. Find the percentage increase in price.

Solution:

Ans: 150%

The excise duty on a certain item has been reduced to rs.3480 form rs.5220. Find the percentage reduction in the excise duty for that item.

Solution:

Ans: 33 1/3%

Out of total production of 6450 tonnes of  a coalmine a quantity of 645 tonnes was lost during extraction. What per cent of the total production  was the net coal extracted?

Solution:

Ans: 90%

A cricket team played 24 matches. The team won 9 matches and lost 3 matches. 12 matches ended in draw. What per cent of the total matches did the team lose?

Solution:

Ans: 12.5%

In a particular month, Rs.10,000 were allocated for the food items in a hostel out of total rs.50,000 budget. Further rs.2000 is allocated for the fruits out of rs.10,000 ( allocated for food items) what percent of total budget is spend on fruits only?

Solution:

Ans: 4%

Percentage ( Expressing One Quantity as a Percentage of Another Quantity)

Percentage ( Expressing One Quantity as a Percentage of Another Quantity)

Example 1 
What per cent is number 3 of number 20 ?

Solution:

As per cent means out of 100. Then by unitary method

out of 20 ----> 3

out of 1 -----> 3/20

out of 100 ----> 3/20 x 100 = 15%

Hence to find what percent the first number is of 2nd number, we divide the first method by the second number and multiply the result by 100.

Example 2:
 Ravi obtained 325 marks out of a maximum of 400 marks. Find the percentage of marks obtained by him.

Solution: 

required percentage of marks = 325/400 x 100 = 81.25%

Example 3:
In a factory of 150 workers, 18 were absent in a day what percentage were present?

Solution:

Present = 150-18 = 132

Percentage presence = 132/150 x100 = 88%

Example 4
Kurla obtained 480 marks out of 600 and birla obtained 560 marks out of 800. Whose performance is better.

Solution:

% marks of kurla = 480/600 x 100 = 80%

%marks of birla = 560/800 x 100 = 70%

so, obviously krula's performance is better than that of birla even though getting less absolute marks.

Percentage of a Quantity

Percentage of a Quantity

Find the no of male students (i.e boys), if there are 42% male students in the school and the total no. of students in the school is 1000.

Solution:

Required number of male students

                                = 42% of 1000

                               = 42/100 x 1000 = 420

A student scored 85% marks. Total marks were 400. How much did he score?

Solution:

Mark scored = 85% of 400

                    = 85/100 x 400

                    = 340


In an orchard 16 2/3% of the trees are mango trees. If the total number of trees in the orchard is 360, find the number of other types of trees in the orchard.

Solution:

Total number of trees = 360

Number of mango trees = 16 2/3% of 360

                                      = 50/3x100 x 360 = 60

Therefore, the number of other trees = 360-60 = 300

Alternatively: 

Number of mango trees = 16 2/3%

It means no. of other types of trees = (100 - 16 2/3)%

                                                       = 83 1/3%

Thus number of other types of trees = 83 1/3% of 360 = 300


Arun purchased 80 meters of cloth, out of which 35% was used for making trousers. How much cloth was used by her for making trousers?

Solution:

               = 80 x 35/100 = 28 meter

Aruna's monthly salary was rs.1140. Her salary is increased by 16.66%. How much increase has she gotten? Also find the salary if her salary increased by 33.33%.

Solution:

                      = 1140 x 16.66/100 = 190

                         = 1140 x 33.33/100 = 379.962  

                         = 1140 + 379.962 = 1520

A Shopkeeper announces a reduction of 8.33% on all its prices after new year. If a wrist watch was earlier for rs.2400. How much would it costs now?

Solution:

= 2400 x 8.33% = 199.92

= 2400-200 = 2200rs

Conversion of Fraction into a Percentage

Conversion of Fraction into a Percentage

NUMERATORS

D E N O M I N A T O R S		1	2	3	4	5	6	7	8	9	10	11	12
	1	100	200	300	400	500	600	700	800	900	1000	1100	1200
	2	50	100	150	200	250	300	350	400	450	500	550	600
	3	33.33	66.66	100	133.33	166.66	200	233.33	266.66	300	333.33	366.60	400
	4	25	50	75	100	125	150	175	200	225	250	275	300
	5	20	40	60	80	100	120	140	160	180	200	220	240
	6	16.66	33.33	50	66.66	83.33	100	116.66	133.33	150	166.66	183.33	200
	7	14.28	28.56	42.85	57.13	71.42	85.71	100	114.28	128.56	142.85	157.13	171.42
	8	12.5	25	37.5	50	62.5	75	87.5	100	112.5	125	137.5	150
	9	11.11	22.22	33.33	44.44	55.55	66.66	77.77	88.88	100	111.11	122.22	133.33
	10	10	20	30	40	50	60	70	80	90	100	110	120
	11	9.09	18.18	27.27	36.36	45.45	54.54	63.63	72.72	81.81	90.9	100	109.09
	12	8.33	16.66	25	33.33	41.66	50	58.33	66.66	75	83.33	91.66	100
	15	6.66	13.33	20	26.66	33.33	40	46.66	53.33	60	66.66	73.33	80

(Percentage fraction conversion table)

Remember: 

1/7 = 14.28%  and  1/14 = 7.14%

1/6 = 16.66% and  1/15 = 6.66%

1/9 = 11.11% and  1/11 = 9.09%

1/15 = 6.66% and 1/16 = 6.25%

1/3 = 33.33% and 3/10 = 30%

99.99% is equivalent to 100% ( in calculation)

Some  other typical values

1/13 = 7.69% = 7.7%,     1/17 = 5.88%,  1/19 = 5.26%,

1/21 = 4.76%,   1/23 = 4.35%,  1/24 = 41.66%

Learning from the table :

i) This table is a first hand support as a percentage values of some frequently used fractions.

ii) All the percentage values whose decimal part in 0.33, 0.66, 0.00 contain the denominator 3 in the fraction.

iii) Similarly if there is 0.16, 0.33, 0.50, 0.66, 0.83, 0.00 it means there is a 6 in the denominator of the fraction.

iv) Similarly if there is 0.28, 0.56, 0.85, 0.13, 0.42, 0.71 it means there is a 7 as denominator.

v) If there is 0.11, 0.22, 0.33, 0.44.....0.99 etc. It means there is 9 as the denominator of the fraction.

vi) If there is 0.09, 0.18, 0.27... it means there is 11 as the denominator of the fraction.

vii) If two percentage values have different decimal values ( there must be different denominators) then their addition or subtraction results always in decimal i.e., never as an integer.