Percentage - Atreya Roy


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    Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.

    By the term percent we mean value for every 100. If it is said, A gets a profit of 20% of every sale in the company, we can say that for every Rs 100 profit obtained from sale, A will get Rs 20. So in the entire chapter we will consider all calculations based on 100.

    X percent means = X/100

    Dependency : K is X percent of Y. This means, K = (X/100)*Y = XY/100

    We might keep in mind some fractions and their corresponding Percentages for better use

    1 = 100%   1/2 = 50%   1/3 = 33.33%   1/4 = 25%   1/5 = 20% 

    1/6 = 16.67%   1/7 = 14.28%   1/8 = 12.5%  1/9 = 11.11%   1/10 = 10% 

    1/11= 9.09%   1/12= 8.33%   1/13 = 7.69%   1/14 = 7.14%   1/15 = 6.66% 

    1/16 = 6.25%   1/17 = 5.88%    1/18 = 5.55%   1/19 = 5.29%   1/20 = 5.00% 

    1/25 = 4.00%   1/50 = 2.00%    1/100= 1.00 %

    Addition/Subtraction to a Quantity:

    If A is X % more than B, we will write : A= ((100+X )/100)*B

    If A is X % less than B, we will write : A = ((100-X )/100)*B

    So, Percent Increase, increases the value and Percent Decrease, decreases the value of the item which is being considered.

    Example : The Salary of A is 20% more than B.  By what percent, the salary of B is less than B ?

    Solution:

    Let us take salary of B = 100

    Then Salary of A = 100 + (20/100)*100

    = 120

    Difference = 120-100= 20

    B is 20/120 = 1/6 = 16.67% less than A

    Successive Change in Percentage:

    When there are more than one percent change on any item/quantity, consecutively, then it is called Successive percentage change

    Suppose we come in a situation when the price of the tickets for a film are increased by 10% once, and again by 10% . What will be the final change in amount ?

    Let the Initial cost be= 100

    After 10% Increase , Cost = (100+10) /100 * 100 = 110

    After again an increase of 10%, Cost = 110/100 * 110 = 121

    Hence total increase = 121-100 =21

    Percent wise increase = 21/100 = 21%

    We will discuss some formulas which will easy our calculations and also the time.

    1. If, “a” and “b” are the two successive changes(increase), Then total change = a+b+(ab/100)
    2. If there is x% increase and then x% decrease, then the resultant change = -x^2/100 %
    3. If there is a x% increase and then y% decrease, then net change = x-y –xy/100 %

    Example :

    After two successive increase of 10% and 20% a number becomes 66. What was the number ?

    Solution :

    Net increase = 10+20 + 10*20/100 = 30+2 = 32%

    Let the number be X,  so, X * (100+32)/100 = 66

    132x/100 = 66

    2x= 100

    Or, x=50

    Practice Questions in Percentage :

    A company purchases a property for Rs 2.5lakhs and invested Rs 75k on repairs and modifications. After 2 years the company sells out the same property for Rs 2.86lakhs. 15 % is to be deducted on account of depreciation, find its approximate percentage of profit or loss.

    1. 2.76 %
    2. 3.25 %
    3. 3.35 %
    4. 3.53 %

    Answer:- option D

    Solution:-

    Amount of property = 2.5+.75 = 3.25lakhs

    Depreciation cut = 15 % of 3.25 lakhs = 0.4875lakhs

    Present value = (100-15) % of 3.25lakhs

    = 2.7625lakhs

    Profit = (2.86-2.7625) / 2.7625 lakhs

    = (9750 / 276250) * 100 %

    = 3.53 %

    A man can buy 3 1/3 kgs more rice for Rs 130 on a reduction of 15 % in the price. Find the reduced price of each kg of rice?

    1. 6.89
    2. 5.86
    3. 5.68
    4. 6.98

    Answer:- option B

    Solution:-

    Man can buy / kg more for Rs 130

    Let the price of rice be x

    Amount of brought in Rs 130 = 130/x

    New price = 85x/100 = 17x/20

    Thus, (130/17x/20) – 10/3 =130/x

    = > x = 6.89

    Reduced price = 85x/100 = 5.86

    In an interview, 70% of male and 30 % of female are selected. The number of females selected is 180, which is 3/5th of number of males not selected. What is the no of candidates appeared for the interview? Every women who appeared was selected.

    1. 300
    2. 600
    3. 900
    4. 1000

    Answer:- option C

    Solution :-

    Selected percentage of females = 30

    Selected percentage of males = 70

    Females selected = 180

    Males selected = x

    x* 3/5 = 180

    x = 60

    180 = 30%

    100% = 600

    Hence, 600 got seleced

    Males selected = 600 – 180

    Total candidate appeared = 420+180+300  = 900

    If 1/3 of a group of employers resigns from the company, by what percentage should the remaining men increases the working hours per day in order to compensate the loss of employees?

    1. 33.33 %
    2. 67.67 %
    3. 50 %
    4. None of these

    Answer: - option (C)

    Solution:-

    Let, x men works at y hours per day

    Number of employee after resignation = 2x/3

    If, these employers’ works for z hours per day

    Then, xy = (2x/3) * z

    = > z = 3y/2

    So, 1/2 hours per day increased

    So, in percentage it is increased by 100/2 = 50%

    A is 30% more efficient than B in doing a job. If the difference of their earning upon completing the work together is Rs. 30. How much did they receive in all?

    A. 160
    B. 165
    C. 170
    D. 175

    ANSWER :Option C (170)

    SOLUTION : 

    Let B do the job in 10 days;

    Hence, A does it in 7 days.

    Total part of A = 10x/17

    Total part of B = 7x/17

    Difference =  3x/17= 30

    = > x = 170

    A got  10*170/17 = 100 ; B got 7*170/17 = 70

    Total Income = Rs. 170

    Rohan takes a loan of 40,000 from XYZ bank. The bank offered the loan at an interest rate of 10%. Rohan started repaying the amounts from the 4th year as 3 annual instalments for the next 2 years. If the installment he pays is in arithmetic progression with a common difference of 5,000; find the amount left for repayment after the requisite time if the first installment was worth 10,000?

    A. 12320
    B. 14320
    C. 14820
    D. 13320

    Answer- Option (B )

    Solution-

    Loan amount= 40,000;     Rate of interest=10%

    After 3 years, interest incurred= 40,000*3*10/100=12,000

    Total amount=40,000+12,000=52,000

    Amount of first installment= 10,000

    Loan amount left= Total loan- loan repaid = 52,000-10,000=42,000

    Interest incurring that year= 42,000*1*10/100=4200

    New loan amount= 46200.

    Loan repaid in the next year= 15,000.

    Loan amount still to be paid=46,200-15,000=31,200

    Interest incurred=31200*1*10/100=3,120.

    Net loan amount= 31,200+3,120=34,320

    Loan paid in the third instalment = 20,000.

    Thus loan amount still to be paid= 34,320-20,000 = 14,320

    Petrol Prices rose up by 10% as a result of an extra import duty. Present petrol price is Rs 44. Atreya travels 300kms daily with his bike. His bike gives a fair mileage of 40km /lt. What is the rise in his expenditure if he gets only half of the extra money as incentives from his office ?

    1. 10
    2. 15
    3. 20
    4. 25

    Answer- Option (B )

    Solution:

    Present value of petrol= 44. So previous value = 44*100/110 = Rs 40.

    Mileage of bike= 40km/lt.

    Expenditure of Atreya per km= 40/40 = Re 1 /km

    Total expenditure=300km*1 = Rs 300.

    New expenditure after price rise = (44/40)*300 =330.

    Extra expenditure=30.

    Amount paid by office= ½ * 30 = 15.

    Extra money Atreya has to pay =Rs 15.

    After 3 successive rise in percentage, the total rise was measured to be 44.29 %. What could have been the value of each such rise ?

    A. 12%
    B. 14%

    C. 12.5%
    D. 13%

    Answer- Option (D)

    Solution-

    Solve through options

    Let us assume a value of 100. 13% rise in it makes it = 100+ 13% of 100 = 100+13 = 113

    Again increase in 13 % of this value will result in an increase of 13/100 * 113 = 14.69

    Again the third increase will result in an increase of 13/100 * 127.69 = 16.6

    Thus the final value will be 127.69 +16.6 = 144.29

    A’s salary is increased by 20% first, then decreased by 20% and again increased by 20% . While, B’s salary is increased by 25% first, then decreased by 25% and again increased by 25%

    1. 115:117
    2. 115:116
    3. 115:118
    4. 115:119

    Answer- Option (A)

    Solution:

    Let, A’s salary be 100 rs. After the first increase, the salary of A is Rs 120.

    Then 20% of this is decreased, so our new salary becomes 120*80 / 100 = 96

    Then again, 20% of this is increased, so A’s new salary becomes 96*120/100 =115.20

    Let, B’s salary be 100 rs. After the first increase, the salary of B is Rs 125.

    Then 25% of this is decreased, so B’s new salary becomes 125*75 / 100 = 93.75

    Then again, 25% of this is increased, so B’s new salary becomes 93.75*125/100 =117.20

    So the ratio is 115:117

    The ratio of the incomes of A,B,C are 5:6:8 . The ratio of their expenditure is 1:2:3. In this scenario, who has the maximum savings?

    1. A
    2. B
    3. C
    4. Cannot Be Determined

    Answer : Option (D)

    Solution:

    Let the incomes of A,B,C be Rs 50,60 and 80 respectively. And their expenditure is 25,50 and 75 respectively. So the savings are

    For A , savings = 50-25=25

    For B, savings = 60-50 = 10

    For C, savings= 80-75 = 5.

    So in this case, A has the highest Savings.

    But consider another scenario, where Incomes of A,B and C are 5,6 and 8. Expenditure are 2,4 and 6. Now the savings of A,B and C are respectively 3,2,2 and we can see that Savings of B and C are equal. So it cannot be determined. 

    The manager of a company increases the wages/ hour of his workers by 10% and decreases the working hours by 10%. Being a concerned brother of the manager, you will choose the state of work , i.e will this theory will come into work or not according to the way which will yield a maximum profit to the organization. Select the better case

    1. Revise the condition of work
    2. Not revise the condition
    3. Both will yield same
    4. Cannot Be Determined

    Answer- Option (A)

    Solution:

    Let the wage of a worker be 100.

    Number of working hours = 100

    Total cost = 100*100 = 10,000

    After increase, new wage = 100*110 / 100 = 110

    New working hours = 100*90/100 = 90

    Total Cost = 90*110 = 9,900

    So, the condition must be revised.

    Appreciation / Depreciation in Population

    We will face questions that are based on population increase or decrease that might be triggered due to various factors like migration / diseases etc.

    So the simple approach to tackle these sums are successive percentage change.

    1. If there is successive increase of a% and b%, then the net change will be  =  a + b + (a*b/100)
    2. If there is successive decrease of a% and b%, then the total discount will be = a + b – (a*b/100)
    3. If there is a% increase and then b% decrease, then the net change = a – b – (a*b/100)
    4. If there is a% increase and then a% decrease, then the net change = -(a*a/100)

    All are the increase or decrease in terms of percentage.

    Example 1 :

    The population of a village is 2000. In 2011, the population increased by 4 % and in 2012 it decreased by 4%. What is the population at the end of these 2 years.

    Solution :

    Net change = - (4*4/100)% = -.16%

    = 2000*.16/100 =3.2

    2000-3.2 = 1997 people ( rounded to nearest value)

    Example 2:

    A city has 10,000 residents. Its population grows at the rate of 10% per annum, what’ll be its total population after 5 years?

    Solution:

    Rate of increase = 10%

    Population = 10000

    This can be viewed as a problem of compound interest.

    So population after 5 years = 10000* (1+.1)^5 = 10000 * 1.61051 = 16105

    Example 3 :

    A city has 10,000 residents. Its population declines at the rate of 10% per annum, what’ll be its total population after 5 years?

    Solution:

    Decrease = 10%

    Proceeding like the last sum,

    10000* (1-10/100)^5

    = 10000* (0.9)^5

    = 10000* 0.59049

    = 5904.9 ~ 5905

     


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