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 ?
Let us take salary of B = 100
Then Salary of A = 100 + (20/100)*100
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.
- If, “a” and “b” are the two successive changes(increase), Then total change = a+b+(ab/100)
- If there is x% increase and then x% decrease, then the resultant change = -x^2/100 %
- If there is a x% increase and then y% decrease, then net change = x-y –xy/100 %
After two successive increase of 10% and 20% a number becomes 66. What was the number ?
Net increase = 10+20 + 10*20/100 = 30+2 = 32%
Let the number be X, so, X * (100+32)/100 = 66
132x/100 = 66
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.
- 2.76 %
- 3.25 %
- 3.35 %
- 3.53 %
Answer:- option D
Amount of property = 2.5+.75 = 3.25lakhs
Depreciation cut = 15 % of 3.25 lakhs = 0.4875lakhs
Present value = (100-15) % of 3.25lakhs
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?
Answer:- option B
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.
Answer:- option C
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?
- 33.33 %
- 67.67 %
- 50 %
- None of these
Answer: - option (C)
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?
ANSWER :Option C (170)
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?
Answer- Option (B )
Loan amount= 40,000; Rate of interest=10%
After 3 years, interest incurred= 40,000*3*10/100=12,000
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
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 ?
Answer- Option (B )
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.
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 ?
Answer- Option (D)
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%
Answer- Option (A)
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?
- Cannot Be Determined
Answer : Option (D)
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
- Revise the condition of work
- Not revise the condition
- Both will yield same
- Cannot Be Determined
Answer- Option (A)
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.
- If there is successive increase of a% and b%, then the net change will be = a + b + (a*b/100)
- If there is successive decrease of a% and b%, then the total discount will be = a + b – (a*b/100)
- If there is a% increase and then b% decrease, then the net change = a – b – (a*b/100)
- 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.
Net change = - (4*4/100)% = -.16%
= 2000*.16/100 =3.2
2000-3.2 = 1997 people ( rounded to nearest value)
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?
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?
Decrease = 10%
Proceeding like the last sum,
= 10000* (0.9)^5
= 10000* 0.59049
= 5904.9 ~ 5905