Chapter 6: PERCENTAGE


Introduction


If the price of a commodity increases by R%, then the reduction in consumption so  as not to increase the expenditure is

New expenditure = [R / ( 100 + R )) * 100] %

If the price of the commodity decreases by R%,then the increase in consumption so as to decrease the expenditure is 

New expenditure = [ ( R / ( 100 - R ) * 100] %

Results on Population : Let the population of the town be P now and suppose it increases at the rate of    
       R% per annum, then : 

Population after n years = P * [1+(R/100)]^n
Population n years ago = P / [1+(R/100)]^n

Results on Depreciation :  Let the present value of a machine be P. Suppose  it depreciates  at the rate 
       R% per annum.

Value of the machine after n years = P * [1-(R/100)]^n
Value of the machine n years ago = P / [1-(R/100)]^n

Questions & Answers


Q. If the sales tax  reduced from  3 1/2 % to  3 1/3%, then what difference does it make to a person who purchases an article with market price of Rs. 8400 ?

A. difference in final rate  = (3 1/2% rate on Rs 8400 ) - (3 1/3 % rate on Rs. 8400)

= (3 1/2 - 3 1/3)% of Rs 8400

= (1/6)% of Rs 8400


Q. In an election between two candidates, 75% of the voters cast their votes, out of which 2% of the votes were declared invalid. A candidate got 9261 votes which were 75% of the total valid votes. Find the total number of votes enrolled in that election.

A. total votes be 'x'. votes cast = 3x/4. invalid votes = (2/100)*(3x/4)

so valid votes are (98/100)*(3x/4)

we know 9261 = (3/4) * (98/100)*(3x/4) = 16800


Q. Reema's test had 75 problems i.e. 10 arithmetic, 30 algebra and 35 geometry problems. Although she answered 70% of the arithmetic , 40% of the algebra, and 60% of the geometry problems correctly.  she did not pass the test because she got less than 60% of the problems right. How many more questions she would have to answer correctly to earn 60% of the passing grade?

A. Reema's correct answers = 7 arithmetic, 12 algebra, 21 geometry.

Total correct = 7+12+21 = 40

total = 75, passing = 45. so she needs =5.


Q. Mr.J gave 40% of the money he had to his wife. he also gave 20% of the
remaining amount to his 3 sons. half of the amount now left was spent on
miscellaneous items and the remaining amount of Rs.12000 was deposited in the bank. how much money did Mr.J have initially? 

A. let 'x' be the initial amount. wife gets = 2x/5, remaining amount = 3x/5; sons get = (1/5)*(3x/5) = 3x/25; amount now left = 12x/25;

amount spent on miscellaneous = 6x/25.

remaining amount = 6x/25 which is 12000 so x = 12000*25 / 6 = 50000


Q. Raman`s  salary  was  decreased  by  50%  and  subsequently  increased  by 50%. How much percent does he lose?

A. Raman's salary be Rs. 100 [assume]. Then it becomes Rs. 50 as its reduced by 50%. Then its increased by 50% so becomes Rs. 75 so overall decrease is Rs. 25 so 25%.


Q. Paul  spends 75% of his  income. His  income  is  increased by 20% and he increased his expenditure by 10%. Find the percentage increase in his savings

A. Assume Paul earn's Rs. 100. he spends Rs. 75. so he saves Rs. 25 Now he earns Rs. 120 and he spends Rs. 82.5 so new savings is Rs. 37.5.

increasing in savings = 12.5 which is 50%.


Q. How many kg of pure salt must be added to 30kg of 2% solution of salt and water to increase it to 10% solution ?

A. salt in 2% solution of 30 kg is 0.6 kg now we want to find 'x' which will satisfy equation (0.6 + x) / (30 + x) = 1/10 [new ratio].


Q. In  the  new  budget  ,  the  price  of  kerosene  oil  rose  by  25%.  By  how much percent must a person reduce his consumption so that his expenditure  on it does not increase ?

A. Assume old price is 100 and new price becomes 125 so we must find how much is reduction in percentage is needed to get 100 from 125.

so we calculate how much % is 25 i.e. (125-100) of 125 it is 20% so a 20% reduction is needed.


OR


Directly apply short cut formula (see above).


Q. The population  of  a  town  is  1,76,400  .  If  it  increases  at  the  rate  of  5% per annum , what will be its population 2 years hence ? What was it 2 years ago ?

A. Formula: Population n years after= P * [1+(R/100)]^n

Population n years ago   = P / [1+(R/100)]^n

CAT Problems for practice


Q.In a stockpile of products produced by three machines M1, M2 and M3, 40% and 30% were manufactured by M1 and M2 respectively. 3% of the products of M1 are defective, 1% of products of M2 defective, while 95% of the products of M3 are not defective. What is the percentage of defective in the stockpile?


  1. 3
  2. 5
  3. 2.5
  4. 4

Ans.a


Q.Gopal went to a fruit market with certain amount of money. With this money he can buy either 50 oranges or 40 mangoes. He retains 10% of the money for taxi fare. If he buys 20 mangoes, then the number of oranges he can buy is


  1. 25
  2. 18
  3. 20
  4. none

Ans.c



Q.Gopal went to a fruit market with certain amount of money. With this money he can buy either 50 oranges or 40 mangoes. He retains 10% of the money for taxi fare. If he buys 20 mangoes, then the number of oranges he can buy is


  1. 25
  2. 20
  3. 18
  4. 6

Ans.b


Q.Ghosh Babu has a certain amount of property consisting of cash, gold coins and silver bars. The cost of a gold coin is Rs. 4000 and the cost of a silver bar is Rs. 1000. Ghosh Babu distributed his property among his daughters equally. He gave to his eldest daughter gold coins worth 20% of the total property and Rs. 25000 in cash. The second daughter was given silver bars worth 20% of the remaining property and Rs. 50000 cash. He then gave each of the third and fourth daughters equal number of gold coins and silver bars both together accounting each for 20% of the property remaining after the previous distribution and Rs. 25000 more than what the second daughter had received in cash.

Q.The amount of property in gold and silver possessed by Ghosh Babu is


  1. 2,25,000
  2. 2,75,000
  3. 4,25,000
  4. None of these

Ans.b


Q.Total property of Ghosh Babu (in Rs.lakh) is


  1. 5
  2. 7.5
  3. 10
  4. 12.5

Ans.a


Q.If Ghosh Babu had equal number of gold and silver bars, the number of silver bars he has is


  1. 90
  2. 60
  3. 75
  4. 55

Ans.d



Q.The rate of increase of the price of sugar is observed to be two percent more than the inflation rate expressed in percentage. The price of sugar, on January 1, 1994, is Rs. 20 per kg. The inflation rate for the years 1994 and 1995 are expected to be 8% each. The expected price of sugar on January 1, 1996 would be


  1. 23.6
  2. 24
  3. 24.2
  4. 24.6

Ans.c


Q.The number of votes not cast for the Praja Party increased by 25% in the National General Election over those not cast for it in the previous Assembly Polls, and the Praja Party lost by a majority twice as large as that by which it had won the Assembly Polls. If a total 2,60,000 people voted each time. How many voted for the Praja Party in the Assembly Elections


  1. 1,10,000
  2. 1,50,000
  3. 1,40,000
  4. 1,20,000

Ans.c


Read and answer


Alphonso, on his death bed, keeps half his property for his wife and divide the rest equally among his three sons Ben, Carl and Dave. Some years later Ben dies leaving half his property to his widow and half to his brothers Carl and Dave together, shared equally. When Carl makes his will he keeps half his property for his widow and the rest he bequeaths to his younger brother Dave. When Dave dies some years later, he keeps half his property for his widow and the remaining for his mother. The mother now has Rs. 1,575,000.


Q.What was the worth of the total property?


  1. Rs. 30 lakh
  2. Rs. 8 lakh
  3. Rs. 18 lakh
  4. Rs.24 lakh

Ans.d


Q.What was Carl’s original share?


  1. Rs. 4 lakh
  2. Rs. 12 lakh
  3. Rs. 6 lakh
  4. Rs. 5 lakh

Ans.a


Q.What was the ratio of the property owned by the widows of the three sons, in the end?


  1. 7 : 9 : 13
  2. 8 : 10 : 15
  3. 5 : 7 : 9
  4. 9 : 12 : 13

Ans.b


Q.Ram purchased a flat at Rs.1 lakh and Prem purchased a plot of land worth Rs.1.1 lakh. The respective annual rates at which the prices of the flat and the plot increased were 10% and 5%. After two years they exchanged their belongings and one paid the other the difference. Then


  1. Ram paid Rs.275 to Prem
  2. Ram paid Rs.475 to Prem
  3. Ram paid Rs.375 to Prem
  4. Prem paid Rs.475 to Ram

Ans.a


Q.A person who has a certain amount with him goes to market. He can buy 50 oranges or 40 mangoes. He retains 10% of the amount for taxi fares and buys 20 mangoes and of the balance he purchases oranges. Number of oranges he can purchase is


  1. 36
  2. 40
  3. 15
  4. 20

Ans.d


Q.2/5 of the voters promise to vote for P and the rest promised to vote for Q. Of these, on the last day 15% of the voters went back of their promise to vote for P and 25% of voters went back of their promise to vote for Q, and P lost by 2 votes. Then the total number of voters is


  1. 100
  2. 110
  3. 90
  4. 95

Ans.a


Q.The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which is equal in area to the triangle, and has a width of 10 units. Then the perimeter of the rectangle is


  1. 30
  2. 36
  3. 13
  4. none

Ans.d


Q.The length of a ladder is exactly equal to the height of the wall it is learning against. If lower end of the ladder is kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then the height of the wall is


  1. 12
  2. 15
  3. 18
  4. 11

Ans.b


Q.A stockist wants to make some profit by selling sugar. He contemplates about various methods. Which of the following would maximise his profit?
I. Sell sugar at 10% profit.
II. Use 900 g of weight instead of 1 kg.
III. Mix 10% impurities in sugar and selling sugar at cost price.
IV. Increase the price by 5% and reduce weights by 5%.


  1. I or III
  2. II
  3. II, III and IV
  4. Profits are same

Ans.b


Percentage problems - CAT


The data given in the table shows the investment details in country 'Fortune Land' of companies A, B, C, D, E and F. Figures in the table are in US dollars in billions.

percentage problems

Q.What is the percentage increase in investment of B, C, D and E from year 1 to year 3?


  1. 121%

  2. 321%

  3. 221%

  4. 300%


Ans.c


Q.What is the ratio of investments of E to F for the years 1 to 3?


  1. 31 : 19

  2. 19 : 31

  3. 20 : 29

  4. 41 : 53


Ans.b


Q.What is D’s contribution as a percentage of total investments in year 2?


  1. 8.2%

  2. 4.5%

  3. 7.4%

  4. 9.2%


Ans.c


Q.For which company is investment not increased from year 1 to year 3?


  1. c

  2. d

  3. f

  4. none


Ans.d


Q.What is the percentage difference in investments of companies A, B, C and companies D, E, F in year 2?


  1. 75

  2. 81

  3. 67.5

  4. 42.3


Ans.b


Q.I bought 5 pens, 7 pencils and 4 erasers. Rajan bought 6 pens, 8 erasers and 14 pencils for an amount which was half more what I had pai(d) What per cent of the total amount paid by me was paid for the pens?


  1. 37.5

  2. 62.5

  3. 50

  4. none


Ans.b


Q.The price of a Maruti car rises by 30% while the sales of the car come down by 20%. What is the percentage change in the total revenue?


  1. -4

  2. -2

  3. 4

  4. 2


Ans.c


Q.A student gets an aggregate of 60% marks in five subjects in the ratio 10 : 9 : 8 : 7 : 6. If the passing marks are 50% of the maximum marks and each subject has the same maximum marks, in how many subjects did he pass the examination?


  1. 2

  2. 3

  3. 4

  4. 5


Ans.c


Q.Fresh grapes contain 90% water while dry grapes contain 20% water. What is the weight of dry grapes obtained from 20 kg fresh grapes?


  1. 2

  2. 2.5

  3. 2.4

  4. none


Ans.b


Read and answer


A survey of 200 people in a community who watched at least one of the three channels — BBC, CNN and DD — showed that 80% of the people watched DD, 22% watched BBC and 15% watched CNN.


Q.What is the maximum percentage of people who can watch all the three channels?


  1. 12.5%

  2. 8.5%

  3. 15%

  4. Data insufficient


Ans.c


Q.If 5% of people watched DD and CNN, 10% watched DD and BBC, then what percentage of people watched BBC and CNN only?


  1. 2%

  2. 5%

  3. 8.5%

  4. cannot be determined


Ans.a


Q.Referring to the previous question, what percentage of people watched all the three channels?


  1. 3.5%

  2. 0%

  3. 8.5%

  4. Cannot be determined


Ans.d


Q.A man earns x% on the first Rs. 2,000 and y% on the rest of his income. If he earns Rs. 700 from income of Rs. 4,000 and Rs. 900 from if his income is Rs. 5,000, find x%


  1. 20

  2. 15

  3. 25

  4. none


Ans.b


Q. One bacterium splits into eight bacteria of the next generation. But due to environmental condition only 50% survives and remaining 50% dies after producing next generation. If the seventh generation number is 4,096 million, what is the number in first generation?


  1. 1 million

  2. 2 million

  3. 4 million

  4. 8 million


Ans . A


Read and answers


A company purchases components A and B from Germany and USA respectively. A and B form 30% and 50% of the total production cost. Current gain is 20%. Due to change in the international scenario, cost of the German mark increased by 30% and that of USA dollar increased by 22%. Due to market conditions, the selling price cannot be increased beyond 10%.


Q. What is the maximum current gain possible?


  1. 10%

  2. 12.5%

  3. 0%

  4. 7.5%


Ans . A


Q. If the USA dollar becomes cheap by 12% over its original cost and the cost of German mark increased by 20%, what will be the gain? (The selling price is not altered.)


  1. 10%

  2. 20%

  3. 15%

  4. 7.5%


Ans . B


Q. Forty per cent of the employees of a certain company are men, and 75% of the men earn more than Rs. 25,000 per year. If 45% of the company’s employees earn more than Rs. 25,000 per year, what fraction of the women employed by the company earn less than or equal to Rs. 25,000 per year?


  1. 2/11

  2. 1/4

  3. 1/3

  4. 3/4


Ans . D


Q. A student took five papers in an examination, where the full marks were the same for each paper. His marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all papers together, the candidate obtained 60% of the total marks. Then the number of papers in which he got more than 50% marks is


  1. 2

  2. 3

  3. 4

  4. 5


Ans . C


Q. The owner of an art shop conducts his business in the following manner: every once in a while he raises his prices by X%, then a while later he reduces all the new prices by X%. After one such updown cycle, the price of a painting decreased by Rs. 441. After a second up-down cycle the painting was sold for Rs. 1,944.81. What was the original price of the painting?


  1. Rs. 2,756.25

  2. Rs. 2,256.25

  3. Rs. 2,500

  4. Rs. 2,000


Ans . A


Q. Fresh grapes contain 90% water by weight while dried grapes contain 20% water by weight. What is the weight of dry grapes available from 20 kg of fresh grapes?


  1. 2

  2. 2.4

  3. 2.5

  4. NONE


Ans . C


Q. At a certain fast food restaurant, Brian can buy 3 burgers, 7 shakes, and one order of fries for Rs. 120 exactly. At the same place it would cost Rs. 164.50 for 4 burgers, 10 shakes, and one order of fries. How much would it cost for an ordinary meal of one burger, one shake, and one order of fries?


  1. 31

  2. 41

  3. 21

  4. cant say


Ans . A


Q. A student took five papers in an examination, where the full marks were the same for each paper. His marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all papers together, the candidate obtained 60% of the total marks. Then the number of papers in which he got more than 50% marks is


  1. 2

  2. 3

  3. 4

  4. 5


Ans . C


Quiz

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