Chapter PERCENTAGE What is Percentage Percentage is a way to express a number or quantity as a fraction of 100 (per cent meaning "per hundred"). It is denoted using the sign "%". For example, 45% (read as "forty-five percent") is equal to 45 100 = 0.45. Percentage can be also seen as a common platform It can be further understood with the help of the following table which gives us the marks obtained by different students in their class-10 exams: Student in country Marks obtained Total Marks Marks obtained/ 100 marks America 100 1000 10% India 25 25 100% China 45 300 15% France 50 100 50% Student in America gets 100 marks out of 1000 marks. If we convert the marks obtained on a base of 100, then it becomes 10. According to the definition of percentage, this student has obtained 10% marks. In fact we use percentage as a common platform of 100 to compare the given values. Using only Marks Obtained we cannot say student from which country has performed best. We need to know Total Marks too. Needless to say, if we know the percentage marks obtained only (as given in rightmost column), we can find out the best performer. So, this way, percentage provides a common platform for comparing similar quantities. Before we move ahead, it is important to understand the Basic statements used in percentage.
Basic Statement 1: What is x % of y? x y 100 It can also be seen that x % of y = y % of x Example - 4.5% of 20,000 = 20,000 % of 4.5 This one simple fact can be used to divide or multiply any number by 50 or 25 or so. Let us see this with the help of an example We are trying to find out the value of 25 32, which is nothing but 32 100/4 = 800. Similarly if we have to divide any number by 50, we should be multiplying the number by 100 and dividing it by 2 finally. Using this, we can see that if we have to calculate 24% of 25(or any other calculation of similar nature), it is better to find out 25% of 24(=24 100/4) = 600. Q1. What is 20% of 50% of 40% of 20? Solution %age means per hundred. So, 20% of 50% of 40% of 20 = (20/100) (50/100) (40/100) 20 = 0.8 What we can observe here that even if we change the order of values here, the final result will be same. Basic Statement 2 : What %age of x is y? = y 100 x Basic Statement 3 : Change a. Percentage change = Change Initial..value 100 b. Percentage point change It is the numerical difference between the values for which we have to calculate change. Let us assume some values to understand the above written concept:
Market share 2008-09 2009-10 Maruti 40% 48% Honda 30% 26% Percentage change in the market share of Maruti over the years = 48 40 100 = 20% 40 Percentage point change in the market share of Maruti over the years = 48 % 40% = 8% S Similarly, if we have to increase any quantity N by S%, then it is equal to N (1 + ) and when the 100 S same quantity N is to be decreased by S%, then final quantity = N (1 - ) 100 Observations: i. An increase of 100% is equal to the final amount becoming 200% of initial value or twice the initial value. ii. An increase of 500% is equal to the final amount becoming 600% of initial value or six times the initial value. iii. A decrease of 100% is equal to the final amount becoming zero. Hence 0% of initial value. iv. Concept of Multiplier Multiplier is the factor which provides the final value. 100 20% 100 1.2 = 120 100 20% 100 0.8 = 80 In the above examples, 1.2 and 0.8 are the multipliers obtained as a result of increasing by 20% and decreasing by 20% respectively. 150 30% 150 1.3 = 195 150 30% 150 0.7 = 105 In the above examples, 1.3 and 0.7 are the multipliers obtained as a result of increasing by 30% and decreasing by 30% respectively.
210 27% 210 1.27 = 266.7 210 27% 210 0.73 = 153.3 In the above examples, 1.27 and 0.73 are the multipliers obtained as a result of increasing by 27% and decreasing by 27% respectively. So, if Final Value and %age increase or %age decrease is given and we have to find out the Initial Value, then it can be done in the similar way. Using S 30% S 1.3 = 195 So, if Final value 195 and 30% is given, then initial value S = 195 = 150 1.3 Q6. If 120 is 20% of a number, then 120% of that number will be: a) 20 b) 120 c) 360 d) 720 Solution: Let the number be x. Then, 20% of x = 120 (20/100 x) = 120 x = (120 100/20) = 600. Hence 120% of x = (120/100 600) = 720. Hence option (D) is the answer. Alternatively, if 20% = 120, so 120 % will be six times of 120 = 720 Q7. 30% of 28% of 480 is the same as a) 15% of 56% of 240 b) 60% of 28% of 240 c) 60% of 56% of 240 d) None of these Solution - Clearly, 60% of 28% of 240 = (60/100 28/100 240) = (30/100 28/100 2 240) = (30/100 28/100 480) = 30% of 28% of 480. Hence option (B) is the answer. Q8. When 35 is subtracted from a number, it reduces to its 80 %. What is four-fifth of that number? Solution: a) 70 b) 90 c) 120 d) 140 Let the number be x. Analyze the statement and look at the preposition to it reduces to its 80% - it means a loss of 20% = 35 subtracted from the number. Hence 100% = 35 5 = 175.
Four fifth of number = 175 4 5 = 140. Hence option (D) is the answer. Q9. 45% of 150 + 35% of 170 =? % of 317.5 a) 30 b) 35 c) 45 d) None of these Solution: Let N% of 317.5 = 45% of 150 + 35% of 170. Then, N 317.5/100 = (45 150/100 + 35 170/100) = 67.5 +59.5 = 127 So, N = (127 100/317.5) = 40. Hence option (D) is the answer. Alternatively, this question can be done with the help of options too. Type 2 Questions based on Percentage Change Q10. Tatto s working hours per day were increased by 25% and her wages per hour were increased by 20%. By how much percent were her daily earnings increased? Solution: 1.20% 2.25% 3.50% 4.45% Assume that she works for 10 hours daily at the rate of Rs. 10 / hour. Hence her Old earning = 10 10 = 100 New earning = 12.5 12 = 150. Hence net percentage increase = 50%. Hence option (3) is the answer. Q11. If the price of an article rose by 25% every odd year and fell by 20% every even year, what would be the percentage change after 180 years? 1. 10% increase 2. 25% increase 3. No change 4. 20% decrease Solution: There are 90 odd years and 90 even years. Or in other words, 90 pairs of (25% rise and 20% fall) are there. Net result of rise of 25% and fall of 20% = 0% change. Hence percentage change after 180 years = 0%. Hence option (3) is the answer. Q14. The entry fee in an exhibition was Rs.10. Later this was reduced by 25%, which increased sales revenue by 20%. Find the percentage increase in the number of visitors.
1. 54 2. 57 3. 60 4. 66 Solution: Assume that earlier there were N visitors. Hence initial revenue = Number of tickets sold Entry fee (price / ticket) = Rs. 10 N = 10 N. New entry fee = 0.75 Rs. 10 = Rs. 7.5 New sales revenue = 1.2 10N = 12 N 12 N So, new number of visitors = = 1.6 N 7.5 Increase in number of visitors = 1.6N N = 0.6 N Hence percentage increase in the number of visitors = 60%. Hence option (3) is the answer. Type 3 Questions based on Sets Q15. In an examination a total of 6,00,000 students appeared. 40% of them were females while the rest were males. Pass percentage among males is 75% and the overall pass percentage is 70%. What is the pass percentage for females? 1. 37.5 % 2. 50% 3. 62.5% 4.70% Solution: Look at the following tree: It is given that a total of 70% students pass the examination, So total number of students passed = 4,20,000. So total number of females passed = 4,20,000 2,70,000 = 1,50,000. Hence pass percentage for females = 150000 100 = 62.5% Hence option (3) is the answer. 240000 Q16. In an examination, 80% students passed in Philosophy and 70% students passed in Maths. Solution At the same time 15% failed in both the subjects. If 390 students passed in both the subjects, then how many students appeared in the examination? 1.500 2.400 3.800 4. 600
1 st of all, understand all the possibilities and statements: Philosophy Pass Fail Pass Fail Maths Pass Pass Fail Fail 15% failed in both the subjects does not mean that 85% passed in both the subjects. It means Summation of following three possibilities: Philosophy Pass Fail Pass Maths Pass Pass Fail We have the information regarding the Philosophy pass percentage (and obviously fail percentage too can be calculated from this data), and Maths pass percentage (and obviously fail percentage too can be calculated from this data). Using Set Theory will provide a better view of the whole scenario: According to the question: 85% = (70% x) + x + (80% x) x = 65% = Students who passed in both the subjects = 390 100% = 390 65 100 = 600. Hence option (4) is the answer. Practice Exercise 1 Q1. The difference between a number and its two-fifth is 510. What is 10% of that number? a) 12.75% b) 85 c) 204 d) None of these Q2. If 35% of a number is 12 less than 50% of that number, then the number is: a) 40 b) 50 c) 60 d) 80 Q3. Three candidates contested an election and received 1136, 7636 and 11628 votes respectively. If these three were the only candidates contesting the election, what percentage of the total valid votes did the winning candidate get?
a) 57% b) 60% c) 65% d) 90% Q4. In an election, only two candidates are contesting. Winner of the election gets 84% of the valid votes and is elected by a majority of 476 votes. What is the total number of valid votes? a) 672 b) 700 c) 749 d) 848 Q5. Two labors A and B are paid a total of Rs. 550 per week. If A is paid 120 percentage of the sum paid to B, how much is B paid per week? a) Rs. 200 b) Rs. 250 c) Rs. 300 d) None of these Q6. 1100 boys and 700 girls are examined in a test. 42% of the boys and 30% of the girls pass. The percentage of the total who failed is: a) 58% b) 62 ⅔% c) 64% d) 78% Q7. If x% of x is 49, then x is equal to: a) 7 b) 70 c) 700 d) 4900 Q8. Subtracting 16% of x from x is equivalent to multiplying x by how much? a) 0.094 b) 0.94 c) 9.4 d) 9.4 Q9. Dilip spends 30% of his monthly income on food articles, 40% of the remaining on conveyance and clothes and saves 50% of the remaining. If his monthly salary is Rs. 1,840, how much money does he save every month? a) Rs. 362.4 b) Rs.386.4 c) Rs. 426.4 d) Rs. 588.8 Q10. If 8% of x = 4% of y, then 20% of x is: a) 10% of y b) 16% of y c) 80% of y d) None of these Q11. If A = x% of y and B = y% of x, then which of the following is true? a) A is smaller than B. b) A is greater than B. c) Relationship between A and B cannot be determined. d) A = B. Q12. In an examination, there are three papers and a candidate has to get 35% of the total to pass. In one paper, he gets 62 out of 150 and in the second 35 out of 150. How much must he get, out of 180, in the third paper to pass the examination? a) 60.5 b) 68 c) 70 d) 71
Q13. A number is decreased by 10% and then increased by 10%. The number so obtained is 10 less than the original number. What was the original number? a) 1000 b) 1050 c) 1500 d) 2000 Q14. Entry fee in an exhibition was Re. 10. Later, the entry fee was reduced by 25% which increased the sales by 20%. The percentage increase in the number of visitors is a) 54 b) 57 c) 60 d) 66 Q15. Price of an article increases at the rate of 8% p.a. What will be the new price of a Rs. 20 article two years later? a) Between Rs. 24 and Rs. 25 b) Between Rs. 21 and Rs. 22 c) Between Rs. 22 and Rs. 23 d) Between Rs. 23 and Rs. 24 Q16. The present population of a country estimated to be 10 crores is expected to increase to 13.31 crores during the next three years. The uniform rate of growth is: a) 8% b) 10% c) 12.7% d) 15% Q17. Amar s salary is 50% more than Barun s. How much percent is Barun s salary less than Amar s? a) 33% b) 33¼% c) 33⅓% d) 33½% Q18. 5% of income of A is equal to 15% of income of B and 10% of income of B is equal to 20% of income of C. If C s income is Rs. 2000, then the total income of A, B and C is: a) Rs. 6000 b) Rs. 14, 000 c) Rs. 18, 000 d) Rs. 20, 000 Q19. A Milk mixture contains 5% water. What quantity of pure milk should be added to 10 litres of this mixture to reduce the water concentration to 2%? a) 5 litres b) 7 litres c) 15 litres d) Cannot be determined Q20. 405 chocolates were distributed equally among students in such a way that the number of chocolates received by each student is 20% of the total number of student. How many chocolates did each student receive? a) 9 b) 15 c) 18 d) 45 Answer Grid
Practice Exercise 1 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 B D A B B B B B B A Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 D D A C D B C C C A Disclaimer Nishit Sinha This chapter is taken from Numerical Aptitude and Data Interpretation (Author Nishit Sinha), published by Pearson publication. You may freely distribute this material as long as (a) it is not for profit, and (b) this disclaimer is not removed. TO see the full list http://www.pearsoned.co.in/web/authors/2726/nishit-k_sinha.aspx
Chapter - Profit Loss and Discount Basic Terminology (1) Cost Price (CP) This is the price which a person pays to purchase something or cost incurred while manufacturing something. Types of Cost:- 1.Fixed cost As obvious from the name, it is that kind of cost which is fixed in all the cases. 2. Variable cost Variable costs are those costs which vary according to the no. of units produced. 3. Semi-variable cost Semi variable costs are those costs which are fixed in one particular strata, but the costs varies among the different layers. One good example of Fixed Cost, Variable Cost and Semi-variable cost is the bill we receive for the telephone connections at our home. A part of that bill, rental, is fixed cost; and the rest part of the bill is calculated on the basis of the no. of calls made. (2) Selling Price (SP) - This is the price at which something is sold. Now, there are three situations possible: Case 1: Selling Price > Cost Price, then Profit Occurs. Profit = SP - CP Profit Percentage = Profit CP 100 Case 2: Cost Price > Selling Price, then Loss Occurs. Loss = CP SP Loss Percentage = Loss CP 100 Case 3: Selling Price = Cost Price, then there is no profit no loss. We call it Break-Even case. (3) Marked price Or Mark-up price (MP) This is the price which the shopkeeper fixes in anticipation of some discount being asked by customer. List price Or Tag price As obvious from the name, this is the price which is printed on the tag of the article. For our calculations related to the concept of PLD, till the moment nothing is stated in the questions we won t see much difference between Marked Price and List price.
Types of Questions: Type 1: CP and SP are given, and profit percentage or loss percentage is to be calculated: Q1. An article is bought for Rs. 600 and sold for Rs. 750. What is the profit percentage? 1. 20% 2. 25% 3. 30% 4. None of these Solution: CP = Rs. 600 and SP = Rs. 750. Since SP is more than CP, there will be profit. Profit = SP CP = Rs 150 Profit Percentage = Profit CP 100 = 150 100 = 25% 600 Type 2: CP and profit percentage / loss percentage are given, and SP is to be calculated: If one of CP or SP is given alongwith Profit percentage or Loss percentage, using the concept of multiplier makes the whole calculation simple. [To know about the concept of multiplier, read Percentage Chapter] In general, (a) CP Multiplier = SP (b) CP = SP Multiplier (c) If there is a profit, multiplier will be more than 1, and if there is a loss, multiplier will be less than 1. [Irrespective of the fact that we have to find out CP or SP]. Alternatively If there is a profit of R%, and CP = C, then CP = SP 100 100 R If there is a loss of R%, CP = SP 100 100 R Q2. Nitika buys a kinetic for Rs. 16,000. If she wants to gain 40%, how much should she charge for the kinetic? Solution: CP = Rs. 16000. Profit percentage = 40%. SP = CP + 40% of CP = 1.4 CP [Here the multiplier = 1.4] = 1.4 16000 = Rs. 22400.
Q3. By selling a VCD player for Rs. 1,950, I got a profit of 30%. At what price should I have sold it in order to get a profit of 40%? 1. Rs. 2,000 2. Rs. 2,100 3.Rs. 2,500 4. None of these Solution: SP = Rs 1950 and Profit percentage = 30%. Understand that if profit percentage = 30%, multiplier has to be equal to 1.3. Now only thing you have to think is : Shall I multiply SP by 1.3 or divide SP by 1.3? Understand this further that if there is profit, then SP > CP. So, multiplying SP by 1.3 will make it larger than SP, hence we conclude that SP should be divided by 1.3. CP = SP = 1950 Multiplier 1.3 = Rs. 1500. Now, I want to earn a profit of 40%. Hence multiplier = 1.4 So, SP = CP 1.4 = 1500 1.4 = Rs. 2100 Q4. By selling an article for Rs. 360, loss incurred is 10%. At what minimum price should the article be solve to avoid loss? 1. Rs. 320 2. Rs. 324 3. Rs. 396 4. Rs. 400 Solution : SP = Rs. 360 Type 3: Loss percentage = 10%, hence multiplier = 0.9 Now CP = SP = 360 = Rs. 400. Multiplier 0.9 Hence to avoid any loss, SP should be atleast equal to the CP =Rs. 400. Questions involving Marked Price / Tag Price Q5. A shopkeeper wants to earn a profit of 20% and at the same time, the minimum discount which he wants to offer is 25%. What should be the minimum percentage mark-up over CP? 1. 50% 2. 42.5% 3. 62.5% 4. 35% Solution: Assume that Cost Price = Rs. 100. To earn a profit of 20%, Multiplier = 1.2 Hence SP = CP 1.2 = 100 1.2 = Rs. 120..(1) Now discount offered = 25%. Discount is always provided on Tag Price or MarkUp price. Multiplier related to 25% discount = 0.75. Assume that MarkUp Price = M
Hence Selling Price = 25% discount on MarkUp Price = 0.75 M Using (1), 0.75 M = Rs. 120 M = 120 = Rs. 150 0.75 Hence percentage mark up = MarkUp CP 100 = (150 100) 100 = 50% 100 Type 4: Questions involving No. of articles sold and No. of articles bought Q6. Cost price of 40 apples is equal to Selling price of 30 apples. What is the profit percentage? Solution: Best way to solve these questions is by assuming a value (ideally LCM of 30 and 40). CP of 40 apples = SP of 30 apples = Rs. 120 (LCM of 30 and 40) CP of one apples = Rs 3 and SP of one apple = Rs. 4 Hence profit = SP CP = Re. 1 Hence profit percentage = 1 100 = 33.33% 3 In general, Profit percentage / Loss Percentage = Goods Left /GoodsAdded Number of Articles Sold 100 In this case, there are 10 apples left out (after selling 30 apples out of 40 apples, 10 are left out). Hence Profit Percentage = 10 100 = 33.33% Profit 30 Q7. Cost price of 40 apples is equal to Selling price of 50 apples. What is the profit / loss percentage? Solution: Type 5: Since number of articles sold is more than number of articles bought, hence there is loss. [Shopkeeper is selling more number of articles than he has got]. Loss percentage = 10 100 = 20% Loss 50 Questions based upon Faulty balance In these questions, shopkeeper cheats his customers by selling less quantity than what he is professing. If shopkeeper sells x grams instead of 1000 grams (where x < 1000),
Profit percentage = 1000 x 100 % x Q8. A shopkeeper professes to sell his goods at Cost Price. However he sells only 800 grams at the place of 1000 grams. What is his profit percentage? Solution: Profit will be obtained for 200 grams (1000 800). Profit percentage = 200 100 % = 25% 800 Q9. A shopkeeper professes to sell his goods at Cost Price. However he sells x grams at the place of 1000 grams, and thus earns a profit of 20%. What is the value of x? Solution: 1000 x x 100 = 20 1000 x x Some important results: = 1/5 x = 833.33 grams When SPs of two articles are same: i. First article is sold at a profit of x% and second article is sold at a loss of x%. In this case, there will be always a loss. Net Loss = x 2 % of CP Practice Exercise 1 100 Q1. A sells an article which costs him Rs. 400 to B at a profit of 20%. B then sells it to C, making a profit of 10% on the price he paid to A. How much does C pay B? a) Rs. 472 b) Rs. 476 c) Rs. 528 d) Rs. 532 Q2. A fruit seller sells mangoes at the rate of Rs. 9 per kg and thereby loses 20%. At what price per kg, he should have sold them to make a profit of 5%? a) Rs. 11.81 b) Rs.12 c) Rs. 12.25 d) Rs. 12.31 Q3. A man gains 20% by selling an article for a certain price. If he sells it at double the price, the percentage of profit will be: a) 40 b) 100 c) 120 d) 140
Q4. Profit earned by selling an article for Rs. 1060 is 20% more than the loss incurred by selling the article for Rs. 950. At what price should the article be sold to earn 20% profit? a) Rs. 980 b) Rs.1080 c) Rs. 1800 d) None of these Q5. If the selling price of 50 articles is equal to the cost price of 40 articles, then the loss or gain percent is: a) 20% loss b) 20% gain c) 25% loss d) 25% gain Q6. On an order of 5 dozen boxes of a consumer product, a retailer receives an extra dozen free. This is equivalent to allowing him a discount of: a) 15% b) 161/6% c) 16⅔% d) 20% Q7. A man bought some fruits at the rate of 16 for Rs. 24 and sold them at the rate of 8 for Rs. 18. What is the profit percent? a) 25% b) 40% c) 50% d) 60% e) None of these Q8. By selling 12 toffees for a rupee, a man loses 20%. How many for a repee should he sell to get a gain of 20%? a) 5 b) 8 c) 10 d) 15 Q9. A dairyman pays Rs. 6.40 per litre of milk. He adds water and sells the mixture at Rs. 8 per litre, thereby making 37.5% profit. The proportion of water to milk received by the customers is: a) 1: 10 b) 1: 12 c) 1: 15 d) 1: 20 Q10. A dishonest dealer uses a scale of 90 cm instead of a metre scale and claims to sell at cost price. His profit is: a) 9% b) 10% c) 12% d) None of these Q11. A shopkeeper cheats to the extent of 10% while buying as well as selling, by using false weights. His total gain is: a) 10% b) 11% c) 20% d) 21% e) 222/9% Q12. A man buys an article for 10% less than its List Price and sells it for 10% more than its List Price. His gain or loss percent is:
a) no profit, no loss b) 20% loss c) less than 20% profit d) more than 20% profit Q13. If a man reduces the selling price of a fan from Rs. 400 to Rs. 380, his loss increases by 2%. The cost price of the fan is: a) Rs. 480 b) Rs.500 c) Rs. 600 d) None of these Q14. The difference between the cost price and sale price of an article is Rs. 240. If the profit is 20%, the selling price is: a) Rs. 1240 b) Rs.1400 c) Rs. 1600 d) None of these Q15. A house and a shop were sold for Rs. 1 lac each. In this transaction, the house sale resulted into 20% loss whereas the shop sale resulted into 20% profit. The entire transaction resulted in: a) no loss, no gain b) loss of Rs. 1/12 lakh c) loss of Rs. 1/18 lakh d) gain of Rs. 1/24 lakh Q16. Piyush purchased 20 dozen notebooks at Rs. 48 per dozen. He sold 8 dozen at 10% profit and the remaining 12 dozen with 20% profit. What is his profit percentage is the transaction? a) 6.68 b) 15 c) 16 d) 19.2 Q17. A vegetable vendor sold half of his stock at 20% profit, half of the remaining at 20% loss and the rest was sold at the cost price. In the total transaction, his gain or loss will be: a) Neither loss, nor gain b) 5% loss c) 5% gain d) 10% gain Q18. Cost Price of two cameras taken together is Rs. 840. By selling one at a profit of 16% and the other at a loss of 12%, there is no loss or gain in the whole transaction. Find the C.P. of the two watches: a) Rs. 360, Rs. 480 b) Rs.480, Rs. 360 c) Rs. 380, Rs. 460 d) Rs. 400, Rs. 440 Q19. List price of an article at a showroom is Rs. 2000. It is sold at successive discounts of 20% and 10%. Its net selling price will be: a) Rs. 1400 b) Rs. 1440 c) Rs 15020 d) Rs. 1700 Q20. A trader marked the price of his commodity so as to include a profit of 25%. He allowed discount of 16% on the marked price. His actual profit was: a) 5% b) 9% c) 16% d) 25%
Answer Grid Practice Exercise 1 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 C D D A C C B A Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 D D B C A Disclaimer Nishit Sinha This chapter is taken from Numerical Aptitude and Data Interpretation (Author Nishit Sinha), published by Pearson publication. You may freely distribute this material as long as (a) it is not for profit, and (b) this disclaimer is not removed. TO see the full list http://www.pearsoned.co.in/web/authors/2726/nishit-k_sinha.aspx
SIMPLE INTEREST and COMPOUND INTEREST Interest is the cost of borrowing money. In other words, Interest is also defined as Time Value of Money. There are two types of Interest: (1) Simple Interest In case of Simple Interest, Interest as well as Principal remains fixed for every compounding period. Simple interest is calculated for original principal only. Accumulated interest from previous periods is not used in calculations for the next periods. Example - If the rate of interest = 10%, and the principal = Rs. 1000, Then: Interest for 1 st year = 10% of Rs 1000 = Rs. 100 Interest for 2 nd year = 10% of Rs 1000 = Rs. 100 Interest for 3 rd year = 10% of Rs 1000 = Rs. 100 It can be seen that Interest generated every year = Rs. 100 Principal Rate Interest 1st Year 1000 10% 100 2nd Year 1000 10% 100 3rd Year 1000 10% 100 (2) Compound Interest In case of Compound Interest, Interest as well as Principal keeps on changing for every compounding period. Interest keeps on increasing every compounding period because principal increases every year. Understand this in the following way: Principal of 1 st year (Initially) = P Principal of 2 nd year = P + Interest of 1 st year Principal of 3 rd year = P + Interest of 1 st year + Interest of 2 nd year In case of Compound Interest, interest gets added to the principal and for next years, interest is accrued over (Principal + Interest accrued so far). So in that way, Compound interest is interest that is paid on both the principal and also on any interest from past years. Example - If the rate of interest = 10%, and the principal = Rs. 1000, Then: Interest for 1 st year = 10% of Rs 1000 = Rs. 100 Principal Rate Interest 1st Year 1000 10% 100 2nd Year 1000 + 100 = 1100 10% 110 3rd Year 1000 + 100 + 110 = 1210 10% 121 Expression for Simple Interest (SI) / Compound Interest (CI):
Principal Rate of Interest Time SI = 100 CI = Principal 1 + R 100 N - Principal Principal = Sum invested or lent R = Rate of Interest per annum N = Number of years It should be noted that the unit of Rate of interest and Time should be same. So, if rate of interest is per year, then time should also be in Year. In case of Compound Interest, if the compounding is not done annually, then formula changes like the following: (a) Half Yearly compounding It means interest is given after every six months. In this case, after every six months, interest will be added to the principal. Rate of Interest Compounding Period Interest in 6 months (half Year) Number of compounding period in a year R% per year Half Yearly R%/2 2 (12 months / 6 months) CI = Principal 1 + R/2 100 2N - Principal (b) Quarterly compounding It means interest is given after every three months. In this case, after every three months, interest will be added to the principal. Rate of Interest Compounding Period Interest in 3 months Number of compounding period in a year R% per year Quarterly R%/4 4 (12 months / 3 months) CI = Principal 1 + R/4 100 4N - Principal Understand that the expression {Principal 1 + R Interest. To calculate interest, we need to subtract Principal from this. Q1. Find Simple interest for the following data: Principal = Rs. 400, Rate of interest = 20% per annum, Time = 4 months Solution: 100 N } in Compound Interest provides the amount = Principal + In this case, we can see that units of rate and time are not same. We can convert any one of the two to be in one single unit either in months / year. Time = 4 months = 4 12 year = 1 3 year
Principal Rate of Interest Time SI = = 400 20 1 3 400 20 1 = = Rs. 26.66 100 100 100 3 Q2. Find Compound interest for the following data: Principal = Rs. 400, Rate of interest = 20% per annum, Time = 12 months. Interest is compounded half yearly. Solution: Since interest is compounded half yearly, in 12 months, interest will be added (or compounded ) twice. Rate of interest for six months = 20 2 = 10% CI = 400 1 + 10 2 100-400= 400 (1.1) 2 400 = Rs 484 Rs 400 = Rs 84 Alternatively, it can be calculated through simple addition too: Interest for 1 st six months = 10% of Rs 400 = Rs 40 Interest for next six months = 10% of Rs 40 (interest for the interest for 1 st six months) + 10% of Rs 400 = Rs. 44 Hence Total Interest = Rs 30 + Rs 44 = Rs 84. Q3. A sum of money becomes 3 times in 5 years. Find in how many years will the same sum become 6 times at the same rate of SI? Solution Sum of money gets 3 times, it means 200% is being added up to the original sum (Principal) in 5 years. 1 12 So, 500% will be added up in 2 years. Q4. Difference between two years of Compound Interest and Simple Interest at 10% over Rs. X is Rs. 10. What is the value of X? Solution - SI CI At the end of 1 st year 10% 10% At the end of 2 nd year 10% 10% + 10% of 10%= 11% =20% = 21% So, 1% = Rs. 10 100% = Rs. 1000 Important Points:
1. If the rate of Interest = R% per annum for both CI and SI, then the difference between CI and SI for 2 years will be equal to (R% of R)% of Principal = R2 % of Principal In the above case, R = 10%, so the difference between CI and SI for 2 years = 1% 2. If a sum doubles itself in n years at Simple Interest, then rate of interest = 100 Q5. A sum of money doubles itself in 5 years at SI. What is the rate of interest? Solution Rate of interest = 100 5 = 20% 100 Q6. A sum of money amounts to Rs. 2600 in 3 years, and to Rs. 3000 in next two years at Simple Interest. What is the rate of interest? Solution: Increase in interest in two years = Rs. 400 Increase in interest in ONE year = Rs. 200 Principal = 2600 (3 200) = Rs. 2000 Interest Rate = 200 100 = 10% 2000 Comparison between CI and SI Assume two different sums are getting double at their respective rates of S.I. and CI in 5 years. Following table gives us the mechanism of getting money n times in above situation: After 5 years After 10 years After 15 years After 20 years At SI 2 times 3 times 4 times 5 times At CI 2 times 4 times 8 times 16 times n Exercise 1 Q1. A certain amount earns simple interest of Rs. 1750 after 7 years. Had the interest been 2% more, how much more interest would it have earned? a) Rs. 35 b) Rs. 245 c) Rs. 350 d) Cannot be determined Q2. A sum of money amounts to twice the original sum in 5 years at SI. What is the rate of interest? a) 10% b) 12.5% c) 20% d) 25%
Q3. The simple interest on a certain sum of money at the rate of 5% p.a. for 8 years is Rs. 840. at what rate of interest the same amount of interest can be received on the same sum after 5 years? a) 6% b) 8% c) 9% d) 10% Q4. A sum of money amounts to three times in 5 years at CI. In how many years will the same sum amount to nine times the original sum? a) 10 b) 15 c) 20 d) 9 Q5 What will be the ratio of simple interest earned by certain amount at the same rate of interest for 6 years and that for 9 years? a) 1: 3 b) 1: 4 c) 2: 3 d) Data inadequate Q6. Ravi Shankar wishes to buy an AC with the money in the bank, which is currently earning interest at the rate of 15 pcpa compounded annually. But Tanzar, his friend forecasts that the inflation rate applicable to AC is going to be 14%; 15% and 16% respectively for the next 3 consecutive years and advises Ravi Shankar to postpone the purchase by 3 years. Does Ravi Shankar gain monetarily, if he takes Tanzar s tip? a.yes c.he neither gains nor losses year. b.no d.he gains only if the purchase is made in the second Q7. The simple interest on Rs. 10 for 4 months at the rate of 3 paise per rupee per month is: a) Rs. 1.20 b) Rs. 1.60 c) Rs. 2.40 d) Rs. 3.60 Q8. Likhit earns x% on the first Rs. 2000 and y% on the rest of his income. If he earns Rs. 700 from Rs 4000 income and Rs 900 from Rs 5000 income, find y%. a.20% b.15% c.25% d.none of these Q9. If a sum of money at simple interest doubles in 6 years, it will become 4 times in: a) 12 years b) 14 years c) 16 years d) 18 years Q10. Likhit earns x% on the first Rs. 2000 and y% on the rest of his income. If he earns Rs. 700 from Rs 4000 income and Rs 900 from Rs 5000 income, find x%. a.20% b.15% c.25% d.none of these Answer Grid Practice Exercise 1 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 D C B A C A A A D B
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