Percentage Concept of BY and TO - problems

Percentage Concept of BY and TO - problems

Due to fall in manpower, the production in the factory decreases by 60% by what per cent should the working hour be increased to restore the original production in the factory? 

Solution:

Manpower x working hours = production

  (-60%) = 3/5 -------> 3/2 = (+150%)                        (3/5 -->  3/5-3  = 3/2)

hence by 150% working hours will be increased. It means the new working hours will be 2.5 times (not 1.5 times) of the original time.


Two numbers are respectively 25% and 40% less than a third number. What per cent is the second of the first?

Solution:

consider A,B,C three numbers and assume C = 100 ( as a base)

A                 B                    C
75                60 <-----------100
                             (-40%)

<------------(-25%)------------

Now  60/75 x100 = 80%


A person gives 10% to his wife 10% of the remaining to a hospital (as a donation) again 10% of the remaining to prime minister's relief fund. Then he has only 7290rs. with him. What was the initial sum of money with that person?

Solution:

Since he gives 10% so he is left with 90% of the original sum and since he does the same with the remaining (or left) amount. so it forms a chain.

therefore remaining amount = (x) x 0.9 x 0.9 x 0.9
                                             = 0.729x = 7290

=> x = 10,000, where x is supposed to be initial amount.

Initially a shopkeeper had n chocolates. A customer bought 10% chocolate from n then another customer bought 20% of the remaining chocolates, after that one more customer purchased 25% of the remaining chocolates. Finally shopkeeper is left with 270 chocolates in his shop. How many chocolates were there initially in his shop?

Solution:

n x 0.9 x 0.8 x 0.75 = 270

n = 270 x 10,000 / 9 x 8 x 75

n =500

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.