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1 Sriramanujan1729.weebly.com Ratio Ratios are used to compare quantities. To compare two quantities, the units of the quantities must be the same. Or A Ratio is an ordered comparison of two quantities. Ratios help us to compare quantities and determine the relation between them. We write ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, then the ratios are said to be equivalent. Ex ; In a class no.of boys = 27 and no of girls = 21 Ratio of boys and girls = 27 : 21 = 9 : 7 Proportion When two ratios are equivalent, the four quantities are said to be in proportion. Ex: If a:b = c:d then a,b,c and are in proportion Ratio and proportion problems can be solved by using two methods, the unitary method and equating the ratios to make proportions, and then solving the equation. 1. Unitary method Unitary method is the method of finding the value of one unit (unit rate) at first and then the value of required number of units. Ex: e.g. Cost of 6 pens is Rs 90. What would be the cost of 10 such pens? Solution: Cost of 6 pens = Rs 90 Cost of 1 pen = 90 6 = Rs 15 cost of 10 pens = 10 15 = Rs 150. Compound ratio: Two simple ratios are expressed like a single ratio as the ratio of product of antecedents to product of consequents and we call it Compound ratio of the given two simple ratios i.e. ratios are compounded by multiplying together the fractions which denote them. Ex: a : b and c : d are any two ratios, then their compound ratio is a X c = ac = ac: bd. b d Ex: Find the compound ratios of 3 : 4 and 2 : 3 Sol: The compound ratios of 3 : 4 and 2 : 3 = ( 3:4 ) X ( 2:3 ) = 3 4 X 2 3 = 3X2 4X3 = 6 12 = 1 2 = 1:2 Find the compound ratios of the following. (i) 4 : 5 and 4 : 5 (ii) 5 : 7 and 2 : 9 Ex: If the compound ratio of 7:5 and 8:x is 84:60. Find x. bd

2 Sol: compound ratio of 7:5 and 8:x = 7 5 X 8 x = 56 5x = 56:5x But The compound ratio of 7:5 and 8:x is 84:60 56 : 5x=84:60 5xX84 = 56X60 ( product of extremes = product of means ) x = 56X60 5X84 = 8 For Practice 1. Smita works in office for 6 hours and Kajal works for 8 hours in her office. Find the ratio of their working hours. 2. If the compound ratio of 5:8 and 3:7 is 45:x. Find the value of x. 3. speed of a cycle is 15km/h and speed of the scooter is 30km/h then find the their ratio? Percentage: Percent (or per cent) means one hundredth. The symbol for percent is %. 1% means 1/ or one hundredth, 7% means 7/ or seven hundredths. Since percentages are just hundredth parts, we can very easily write them as fractions and as decimals. i.e., 75% = 75 2 = 0.75 and 2% = = 0.02 Examples 1. A test has 20 questions. If yuvanth gets 80% correct, how many questions did yuvanth missed? Sol: The number of correct answers is 80% of 20 or 80/ 20 = 80/ 20 = 0.80 20 = 16 Since the test has 20 questions and he got 16 correct answers, the number of questions he missed is 20 16 = 4 2. In a school, 25 % of the teachers teach basic math. If there are 50 basic math teachers, how many teachers are there in the school? Sol: Let Total teachers in a school = x 25% of x = 50 25 X x = 50 25x = 50 25x = 50X X = 5000 25 = 200 total teachers in the school = 200 3. 24 students in a class took an algebra test. If 18 students passed the test, what percent do not pass? Sol: Number of students who did not pass is 24 18 = 6 x% of 24 = 6 or x% times 24 = 6 5%means = 5

3 x X 24 = 6 Percentage Increases and Decreases X = 6X 24 = 25 % 25% of students did not pass Often prices are increased or decreased by a percentage. Example: Ramya earns Rs4000 per week for her part-time job. She is to be given a 5% pay rise. How much will she earn per week after the pay rise(new pay)? Sol: 5% of 4000 = 5 X 4000 = 200 New pay = old pay + rising pay (OR) = Rs.4200 = Rs 4000 + Rs 200 Example : In a sale the cost of a computer is reduced by 30%. The normal price of the computer was Rs9000. Calculate the sale price of the computer. Sol: 30% of Rs.9000 = 30 X 9000 Sale price = (-30)% of 9000 For Practice: Sale price = Rs.2700 (OR) = normal price reduced price = Rs.9000 Rs.2700 = Rs. 6300 1. In a primary school there shall be 3 teachers to 60 students. If there are 400 students Enrolled in the school, how many teachers should be there in the school in the same ratio? 2. There were 2075 members enrolled in the club during last year. This year enrolment is Decreased by 4%.(a) Find the decrease in enrolment. (b) How many members are enrolled during this year? 3. A farmer obtained a yielding of 1720 bags of cotton last year. This year she expects her crop to be 20% more. How many bags of cotton does she expect this year? Discounts: Prices of the items are marked according to the price list quoted by the factory which is called List price or catalogue price. Or marked price. A discount is an amount that is subtracted from the regular price of an item. Sale price: The reduced cost of an item Sale price = List Price Discount New pay = ( + 5 )% 0f 4000 = 105 = 70 X 4000 X 9000 = Rs.6300

4 Discount rate: Percent that the price is reduced. Discount = (List Price) (Discount Rate) Ex : Find the sale price of an item that has a list price is Rs.24 and a discount rate of 50% Sol: Discount = 24 50% = 24 50 = 12 Sale Price = List Price - Discount = 24 12 = Rs.12 Ex: Find the sale price of an item that has a list price is Rs.27 and a discount rate of 33.33333% Sol: Discount = 27 33.33333% = 27 0.3333333 = 8.9999999 Sale Price = List Price - Discount = 27 8.9999999 = Rs.18.0000001 Ex: A cycle is marked at ` 3600 and sold for ` 3312. What is the discount and discount percentage? discount percentage? Sol: Discount = marked price sale price = 3600 3312 = 288 Discount percent = Discount M.P X Profit and loss = 288 3600 = 8% X Profit or Gain: If the selling price is more than the cost price, the difference between them is the profit incurred. Profit or Gain = S.P. C.P. Loss: If the selling price is less than the cost price, the difference between them is the loss incurred. Loss = Cost price (C.P.) Selling Price (S.P.). Profit or Loss is always calculated on the cost price. Llist of some basic formulas used in solving questions on profit and loss: Gain % = Gain * CP Loss % = Loss * CP ( + Gain%) SP = * CP SP = ( Loss %) * CP

5 If an article is sold at a gain of 10%, then SP = 110% of CP. If an article is sold at a loss of 10%, then SP = 90% of CP. CP = * SP ( + Gain%) CP = * SP ( Loss%) Ex: A man sold a fan for Rs. 465. Find the cost price if he incurred a loss of 7%. Sol: CP = [ / ( Loss %)] * SP cost price of the fan = (/93)*465 = Rs. 500 Ex: An article is purchased for Rs. 450 and sold for Rs. 500. Find the gain percent. Sol: Gain = SP CP = 500 450 = 50. Gain% = (50/450)* = /9 % Ex: In a transaction, the profit percentage is 80% of the cost. If the cost further increases by 20% but the selling price remains the same, how much is the decrease in profit percentage? Sol: Let CP = Rs.. Then Profit = Rs. 80 and selling price = Rs. 180. The cost increases by 20% New CP = Rs. 120, SP = Rs. 180. 1.615 : 1 This ratio is called Golden ratio. ratio. Profit % = 60/120 * = 50%. Profit decreases by 30%. Ex: Vinay bought a flat for ` 4,50,000. He spent 10,000 on its paintings and repair. Then he sold it for 4,25,500. Find his gain or loss and also its percent. Sol: Total cost price = purchasing price+ repair charges. = (4,50,000 + 10,000) = 4,60,000.

6 But, we can observe Selling price < cost price. So there is a loss. Loss = cost price selling price = 4,60,000 4,25,500 = Rs. 34,500. Loss percent = LOSS C.P X = 34,500 X 4,60,000 = 7.5% Complete the following table with appropriate entries Pro. no C.P In RS. Expenses In RS. S.P In RS. Profit In RS. Loss In RS. % of Profit % of Loss 1 750 50? 80 XXXXXXXX? XXXX 2 4500 500? XXX 0 XXXXXXXX? 3 46,000 400 60,000???? 4 300 50?? XXXXXXXX 12% XXXXX 5 330 20? XXX?? 10% Solutions : 1. Actual C.P = C.P + Expenses = 750+50 = Rs.800 S.P = A.C.P + Profit = 800+80= Rs.880 % 0f Profit = P 80 X = X = 10% ACP 800 4. A.C.P = 300+50 = 350 But, SP = ( + Gain%) * CP S.P = (+12) X350 = Rs.392 Now try yourself remaining problems Sales Tax (or) Value Added Tax (VAT) A value-added tax (VAT) is a consumption tax levied on products at every point of sale where value has been added, starting from raw materials and going all the way to final retail purchase by a consumer. Ultimately, the consumer pays VAT; buyers earlier in the chain of production receive reimbursements for previous VAT taxes paid. Simply, VAT is charged on the Selling Price of an item and will be included in the bill. VAT is an increase percent of selling price. Example:

7 Item Seling Price(Rs) VAT % VAT 1 0 23 % 230 2 0 23 % 230 3 0 9 % 90 4 1500 10% 150 Bill Amount = (0+230)+(0+230)+(0+90)+(1500+150) =Rs.5190 Compound interest Simple interest (1) = Amount PTR, where P = principal T = Time in years R = Rate of interest. = Principal + Interest = P + PTR Compound interest allows you to earn interest on interest. (Or) Compound interest is the interest earned on the principal amount and on its accumulated interest. Compound Interest = Total amount of Principal and Interest in future (or Future Value)less Principal amount at present C.I = [P (1 + I ) n ] P = P [(1 + i) n 1] (Where P = Principal, i = nominal annual interest rate in percentage terms, and n = number of compounding periods.) If the number of compounding periods is more than once a year, "i" and "n" must be adjusted accordingly. The "i" must be divided by the number of compounding periods per year, and "n" is the number of compounding periods per year times the loan or deposit s maturity period in years. For example: The compound interest on $10,000 compounded annually at 10% (i = 10%) for 10 years (n = 10) would be = Rs.25,937.42 Rs.10,000 = Rs.15,937.42 The amount of compound interest on Rs.10,000 compounded semi-annually at 10% (i = 10 =5%) for 10 years (n = 20) would be = Rs.26,532.98 Rs.10,000 = 2 Rs.16,532.98 The amount of compound interest on Rs.10,000 compounded monthly at 10% (i = 10 12 =0.833%) for 10 years (n = 120) would be = Rs.27,070.41 Rs.10,000 = Rs.17,070.41

8 Note: A credit-card balance of Rs.20,000 carried at an interest rate of 20% (compounded monthly) would result in total compound interest of Rs.4,388 over one year or about Rs.365 per month. Look at the table that shows the monthly interest for February through June. Simple interest( 5% ) would have the money growing fifty dollars per month. Month Previous Balance Interest New Balance Feb Rs.1,000 Rs.1,000 * 0.05 = Rs.50 Rs.1,050 Mar Rs.1,050 Rs.1,000 * 0.05 = Rs.50 Rs.1 Apri Rs.1, Rs.1,000 * 0.05 = Rs.50 Rs.1,150 May Rs.1,150 Rs.1,000 * 0.05 = Rs.50 Rs.1,200 June Rs.1,200 Rs.1,000 * 0.05 = Rs.50 Rs.1,250 Look at the same table but with compound interest instead of simple interest: Month Previous Balance Interest New Balance Feb Rs.1,000 Rs.1,000 * 0.05 = Rs.50 Rs.1,050 Mar Rs.1,050 Rs.1,050 * 0.05 = Rs.52.50 Rs.1,102.50 Apri Rs.1,102.50 Rs.1,102.50 * 0.05 = Rs.55.13 Rs.1,157.63 May Rs.1,157.63 Rs.1,157.63 * 0.05 = Rs.57.88 Rs.1,215.51 June Rs.1215.51 Rs.1215.51*0.05= Rs.60.78 Rs.1276.29 Examples: 1. What will be the amount and compound interest, if 5000 is invested at 8% per Annum 2 years? S0l: P = ` 5000; R = 8% and n = 2 years A = P( 1+ R )n or P [(1 + i) n 1] ( Here i = R/ )

9 = 5000( 1+ 8 )2 = 5832 compound interest = A-P = 5832 5000 = Rs.832 The time period after which interest is added to principal is called conversion Period. When interest is compounded Half yearly, there are two conversions Periods in a year,each after 6 months. In such a case, half year rate will be Half of the annual rate. 2. Calculate Compound interest onrs. 0 over a period of 1 year at 10% per annum if interest is compounded half yearly. Sol: As interest is compounded half yearly. 2 conversion periods in a year. n=2 Rate of interest( R ) = 1 X 10 = 5% ( for 6 months ) 2 A = P(1+ R )n = 0(1+ 5 )2 = 0X 21 20 X21 20 = Rs.1102.50 C.I = A-P = 1102.50 0 = Rs.102.50 3. The population of a village is 6250. It is found that the rate of increase in population is 8% per annum. Find the population after 2 years. Sol: Here, let P = 6250 ; R = 8% ; n = 2 Population after 2 years A = P(1+ R )n = 6250( 1+ 8 ) 2 = 6250( 1+ 2 25 )2 = 6250( 27 25 )2 = 7290 4. Kathleen deposited Rs.6500 into a new superannuation account. This amount decreased by2% each year for 3 consecutive years. What was the value of her superannuation after 3 years? Sol: Deposited amount (P) = Rs.6500 The more common form of depreciation is reducing-balance depreciation. The rules for calculating this kind of depreciation are similar to calculating Compound interest. The formula is just slightly different. Deposited amount = Rs.6500 n = 3 R = 2% But, The amount decreased by2% each year

10 For Practice R = -2% A = P( 1+ R )n = 6500 ( 1+ 2 )3 = 6500 ( 49 50 )3 = Rs.6117.75 1. Find the amount and the compound interest on ` 8000 at 5% per annum, for 2 years compounded annually. 2. Calculate compound interest on _ 0 over a period of 1 year at 10% per annum, if interest is compounded quarterly? 3. Machinery worth _ 00 depreciated by 5%. Find its value after 1 year. 4. I borrowed _ 12000 from Prasad at 6% per annum simple interest for 2 years. Had I borrowed this sum at 6% per annum compounded annually, what extra amount would I have to pay?

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