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## Quantitative Methods Session 2 – 11.07.13 Chapter 2 - PERCENTAGE

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**Quantitative MethodsSession 2 – 11.07.13Chapter 2 -**PERCENTAGE Pranjoy Arup Das**Percent …..Per-Cent …..Cent…..Century = 100**• Percent means per hundred. • Derived from the Latin word per centum. • The % symbol denotes percent or percentage. • 48% means 48 out of a total of 100. • 48% of X means 48/100 th. of X. = 48/100 * X**“ I spend 30% of my monthly income on eating out”.**• 30% means 30 out of a total of 100 • Spending 30% of monthly income means if my monthly income is Rs. 100, I spend Rs. 30 on eating out. • Out of every Rs. 100 of my income, I spend Rs 30 on eating out. • After spending Rs. 30 per Rs. 100, Rs. 70 is left per Rs. 100 • Which means, after spending 30% of my monthly income on eating out, 70% of my monthly income remains with me. “ My monthly income is Rs. 1000. I spend 30% of it on eating out”. • (30% of Rs. 1000)= 30/100 * 1000 = Rs. 300 • (70% of Rs. 1000)=70/100*1000 = Rs. 700 remains • Or (Rs.1000 – Rs. 300) = Rs. 700 remains**“ My monthly income is Rs. 2000. I spend 50% of it on**household expenses , 40% on eating out and keep the rest as savings”. • (50% of Rs. 2000) = 50/100 * 2000 = Rs. 1000 on household ex. • (40% of Rs. 2000) = 40/100 * 2000 = Rs. 800 on eating out • Remaining (Rs. 2000 – Rs. 1000 – Rs. 800) = Rs. 200 as savings. • Or we can say 10% of the monthly income remains as savings, i.e., 10% of Rs. 2000 = 10 / 100 * 2000 = Rs. 200. “ My monthly income is Rs. 3500. I spend 40% of it on household expenses and 30% of the remaining amount on eating out and whatever remains I keep it as savings”. • (40% of Rs. 3500 = 40/100 * 3500)= Rs. 1400 on household ex. • Remaining amount = Rs. 3500 – Rs. 1400 = Rs. 2100 • (30% of Rs. 2100 = 30/100 * 2100 = Rs. 630 on eating out • Remaining amount( Rs. 2100 – Rs. 630) = Rs. 1470as savings.**“ My monthly income is Rs. 3500. I spend 40% of it on**household expenses and 30% of the remaining amount on eating out and whatever remains I keep it as savings”. What percent of my monthly income is being saved? • From previous solution, Savings = Rs.1470 • Out of total monthly income of Rs. 3500, savings is Rs. 1470 • To find percentage of savings, means that we have to find how much will be the savings out of a total monthly income of Rs. 100. • So if out of Rs. 3500, Rs. 1470 is saved, • Then out of Re. 1 , Rs. (1470/3500) is being saved • Therefore out of Rs. 100, how much is being saved? = Rs. (1470/3500 * 100) = Rs. 42 • So out of Rs. 100, Rs. 42 is being saved. • Which means out of every Rs. 100 of the monthly income of Rs. 3500, Rs. 42 is being saved. • In other words, Rs. 42 is the savings per Rs. 100 of the monthly income • Or Rs. 42 is the Savings Per Cent of the monthly income. • Or 42% of the monthly income is being saved.**“ My monthly income is Rs. 3500. I spend 40% of it on**household expenses, 30% on eating out and whatever remains I keep it as savings”. What percent of my monthly income is being saved?**“ My salary has been increased by 30% but my allowance has**been reduced by 25%”. • This means that if my salary earlier was Rs. 100, now it is Rs. 130 and if my allowance earlier was Rs. 100, it is now Rs. 75. • New salary = Old salary + 30% of Old Salary = Old Salary + (30/100 * Old salary) = 130/ 100 * Old Salary • In other words, New salary = 130 % of Old Salary and Old Salary = 100/130 * New Salary • New allowance = Old Allowance – 25% of Old Allowance = Old Allowance – (25 /100 * Old Allowance) = 75 / 100 * Old Allowance • In other words, New Allowance = 75% of Old Allowance and Old Allowance = 100/ 75 * New Allowance**“ My salary has been increased from Rs. 3500 to Rs. 5000,**but my allowance has been reduced from Rs. 2000 to Rs. 1500”. What is the increase and decrease percent? • Increase in Salary = Rs. 5000 – Rs. 3500 = Rs. 1500 • Salary of Rs. 3500 has increased by Rs. 1500 • So a salary of Re 1 has increased by Rs. (1500 / 3500) • Therefore a salary of Rs. 100 will increase by Rs. (1500/3500* 100) In short , Percentage increase in salary = 1500 / 3500 * 100 i.e., Percentage increase = Percentage decrease =**“My monthly income is Rs. 5000. My neighbour earns exactly**20% more than me”. What is my neighbours monthly income? • Neighbour earns 20% more than what I earn. • That means my neighbours income = My income + 20% of My income Neighbours income = Rs. 5000 + 20% of Rs 5000 = 5000 + 20/100 * 5000 = 5000 + 1000 = Rs. 6000 ALTERNATIVE: “My monthly income is Rs. 5000. My neighbour earns exactly 20% more than me”. What is my neighbours monthly income? • Neighbour earns 20% more than what I earn. • So If I earn Rs. 100, my neighbour earns Rs. 120 • If I earn Re. 1, my neighbour earns Rs. 120 / 100 • Therefore my earning is Rs. 5000, my neighbours earning is Rs.(120 / 100 * 5000) = Rs. 6000 NOTE: Neighbours income is 120% of my income**Solved Examples – Arithmetic Dr. R.S. AggarwalPage 180 to**183**Ex. 7 If A earns 15 % more than B, then how much percent**less does B earn than A? • Point to note: If A earns Rs. X more than B it implies that B earns Rs. X less than A. But if A is x% more than B, it does not imply that B is x% less thanA. Solution 1 • Let B’s earning be Rs. 100 • A’s earning = = Rs. 115 Difference between A’s & B’s earnings = 115 -100= Rs. 15 When A’s earning is Rs. 115, B earns Rs.15 less than A If A’s earning was Rs. 100, B’s earning will be less than A by 15/115 x 100 =___________% B’s earning + 15% of B’s earning = 100 + 15 /100 x 100**Ex. 7 – Solution 2**• Let A’s earning be Rs. 100 • Let B’s earning be Rs. X • A’s earning = B’s earning + 15% of B’s earning => 100 = X + (15/100 * X) => 100 = 115X/100 => X = 100 * 100/115 = Rs. 86.96 • So when A’s earning is Rs. 100, B earns Rs. 86.96 • % that B earns less than A = 100 – 86.96=__%**= X + 15 % of X**Ex. 7 – Solution 3 • Let B’s earning be Rs. X • A’s earning When A’s earning is Rs. 115X/100, B’s earning is Rs. X If A’s earning is Rs. 100, B’s earning will be= X / 115X/100 * 100 = 100X/115 X * 100 = 10000/115 = Rs. 86.96 So Difference in earnings = 100 – 86.96 = ______% = X + 15/100 * X = Rs. 115X/100**Points to note :**If A is x% more than B, then A = B + x % of B or A = (100+x)% of B If A is x % less than B, then A = B – x% of B or A = (100-x) % of B Ex. 8 If A earns 10 % less than B, then how much percent more does B earn more than A? • Let B’s earning be Rs. 100 • A’s earning = = 100 – 10% of 100 = Rs. 90 Difference between B’s & A’s earnings = 100 -90= Rs. 10 If A’s earning is Rs. 90 B’s earning is more than A by Rs. 10 If A’s earning was Rs. 100, B’s earning will be more than A by 10/90 x 100 =_____% B’s earning - 10% of B’s earning**Ex. 9 If the price of tea is increased by 8%, by how much**percent must a family reduce their consumption to avoid spending more money? Solution 1: • Let us assume that the family consumes x Kg of tea • Let the old cost of x kg of tea was Rs. 100. So the new cost of x Kg is Rs. 108 • Increase in expenditure on tea= (108 – 100) = Rs. 8 - So the expenditure on tea needs to be brought down from Rs.108 to Rs. 100 by Rs. 8 - Which means, tea worth Rs. 8 will have to be reduced from x Kg. • If Rs. 108 is the worth of x kg of tea, Then Rs. 8 is the worth of x/108 * 8 = 8x/108 Kg. • So If x Kg needs to be reduced by 8x/108 Kg, That means 100 kg will have to be reduced by 8x/108/x * 100 = 8/108 * 100 = ___%**Ex. 9 Solution 2**• If we assume that the old price of tea was Rs. 100 per kg, then the new price is Rs. 108 per kg. • Let us also assume that the family consumes 1 Kg of tea. With the price rise the family has to spend Rs. 8 extra on 1 Kg of tea Which means the tea worth Rs. 8 will have to be reduced from 1 Kg. • If Rs. 108 is the cost of 1 kg of tea Then Rs. 8 is the cost of = 8/108 Kg of tea. = 0.074 Kg That means consumption has to be reduced by 0.074 Kg • So If 1 Kg needs to be reduced by 0.074 Kg That means 100 Kg will have to be reduced by 0.074 /1 x 100 =_%**Ex. 13 (i) : Population of a town is 176400. It increases by**5% per annum. What will be the population after 2 years? Solution : • Present population = 176400 • It is given that popn increases by 5% every year • Population after 1 year = Present population + 5% of Present population = 176400 + 5/100 * 176400 = 185220 • Population after 2 years = Popn after 1 year + 5% of population after 1 year = 185220 + 5/100 * 185220 =________**Ex 13 (ii) : What was the population 2 years ago?**Solution: Since pop. increases by 5% every year • Present population = => 176400 = 105 / 100 * Previous years pop. => 176400 * 100/105 = Previous years population => 168000 = Previous years population Again, previous years pop. = 105% of pop. 2 years ago => 168000 = 105/100 * pop. 2 years ago => Pop. 2 years ago = 168000 * 100 / 105 = ____________ 105 % of Previous years pop.**Ex. 11. When tax on a commodity is reduced by 15%, its**consumption increases by 20%. What is the effect on tax revenue? • Point to remember: Sales Revenue = Price per unit of the commodity * No. of units consumed Tax Revenue = Tax amount per unit * No. of units consumed Solution: • Let us assume that the original tax was Rs. 100 per unit and the no. of units originally consumed was 100 units. • The original tax revenue = Rs. 100 x 100 units = Rs. 10000 Now after the reduction in tax: • The new tax becomes Rs. 85 per unit and consumption becomes 120 units. • The new tax revenue = Rs. 85 x 120 units= Rs. 10,200 • There is an increase in the revenue (The effect) • So the increase in revenue = 10200- 10000 = Rs. 200 • And Percentage increase in revenue = 200/10000 x 100 = ___%**Ex.14 : The value of a machine depreciates at the rate of**10% per annum. Its present value is Rs. 10,00,000. What will be its value after 3 years? Solution: • Present value of the car = Rs. 10,00,000 • Since the value of the car depreciates (is reduced) by 10% every year, - The value of the car after 1 year = (Present value – 10% of present value) OR 90% of p.v. = 90/100 x 1000000 = Rs. 9,00,000 • The value of the car after 2 years = 90% of value of the car after 1 year = 90/100 x 900000 = Rs. 8,10,000 • The value of the car after 3 years = 90% of the value after 2 yrs = 90/100 x 8,10,000 = Rs.____________**80% of Previous years value**Ex. 15 : The value of a machine depreciates @ 20% p.a. Its present value is Rs. 64000. What was its value 2 years ago? Solution: • Present value = • 64000 = 80/100 x Previous years value • Previous years value = 64000 x 100 / 80 = Rs.80000 • Previous years value = 80% of value 2 years ago • 80000 = 80/100 x Value 2 years ago • Value 2 years ago = 80000 x 100 /80 = Rs. ___________**Class assignment**• Ref book – Arithmetic, R.S.Aggarwal • Page no. 205 to 208, Exercise 10B • Please solve problem nos. 1, 6, 9,22,32 & 50 HOME ASSIGNMENT- From Exercise 10B Problem nos. 2, 3, 4, 10, 19, 26, 29, 35, 36, 40, 41, 50 & 51.