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 Salaryand 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 Allowanceand 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. 6000ALTERNATIVE:“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 isRs.(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. 115Difference between A’s & B’s earnings = 115 -100= Rs. 15When A’s earning is Rs. 115, B earns Rs.15 less than AIf A’s earning was Rs. 100, B’s earning will be less than A by15/115 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=__%

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. XIf A’s earning is Rs. 100, B’s earning will be= X / 115X/100 * 100 = 100X/115 X * 100 = 10000/115 = Rs. 86.96So Difference in earnings = 100 – 86.96 = ______%

= X + 15 % of X= 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 BIf A is x % less than B, then A = B – x% of B or A = (100-x) % of BEx. 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. 10If A’s earning is Rs. 90 B’s earning is more than A by Rs. 10If 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 teaThen Rs. 8 is the cost of = 8/108 Kg of tea. = 0.074 KgThat 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 populationAgain, 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 consumedSolution: • 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. 10000Now 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.____________

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 = Þ 64000 = 80/100 x Previous years valueÞ Previous years value = 64000 x 100 / 80 = Rs.80000• Previous years value = 80% of value 2 years agoÞ 80000 = 80/100 x Value 2 years agoÞ Value 2 years ago = 80000 x 100 /80 = Rs. ___________

80% of Previous years value

Class assignment• Ref book – Arithmetic, R.S.Aggarwal• Page no. 205 to 208, Exercise 10B• Please solve problem nos. 1, 6, 9,22,32 & 50HOME ASSIGNMENT- From Exercise 10B Problem nos. 2, 3, 4, 10, 19, 26, 29, 35, 36, 40, 41, 50 & 51.