Introduction to Everyday Math Now In this section we are going to discuss along with the application of some other operations of arithmetic like Ratio, Proportion, Rates, Percentage, which are very helpful to solve everyday problems and business related problems. Concept of Ratio Ratio: Sometimes, we have to deal with the quantities whose comparison is required. Suppose in a class of 45-students, 15 of the students are girls, 30 of the students are boys, and then we can compare the number of boys with number of girls in two different ways. 1. There are 15 more boys than girls in the class. In this case we are comparing the number of boys and number of girls by finding their difference. 2. The number of boys is twice the number of girls in the class. Here, we are comparing the number of boys and number of girls by finding a fraction consisting of the number of boys over the number of girls. i.e., fraction is “Ratio is fraction of two quantities of the same kind” In other words “Ratio is the comparison of two homogeneous quantities” or ” Ratio is the division of two quantities and having the same units” It is denoted by or (read as “a ratio b”) or The ratio has no unit. It is a numerical quantity which indicates how many times one quantity is greater than the other. e.g. The boy : girl, i.e. 30 : 15 or 2 : 1 indicates that the number of boys is twice the number of girls. e.g. The girl : boy, i.e. 1 : 2 or indicates that there are half girls than boys. Example: Find the ratio of (1) 50g to 200g (2) 700g to 1kg Solution: 1. The ratio of 50g to 200g can be found in two ways. 50 : 200 = or or 50 : 200 (divided by 50) = or 1:4 =1 : 4 Hence, ratio of 50g to 200g is 1 to 4. 2. 700g and 1kg have different units. First, we express them in the same units we know that 700g : 1kg =700g : 1000g (since 1kg=1000g) =700 : 1000 (divided by 100) =7 : 10 Hence, ratio of 700g to 1kg is 7 to 10. Example: Suppose three men receive their profits as 4000, 3000 and 1000. Find the ratio between their profits. Solution: The ratio of their share (profit) is 4000 : 3000 : 1000 =4:3:1 (divided by 1000) Example: Simplify the ratio of 3.5 : 2.5 Solution: We are given that 3.5 : 2.5 = = 35 : 25 =7:5 (multiply by 10) (Dividing by 5) Example: Express in the form of (1) 25, 45 (2) 30min, 1hour Solution: and fraction 1. 25, 45 = 25 : 45 =5 : 9 (Divided by 5) And in fraction form we have 2. 30min, 1hour 30min : 1hour (since 60min=1hour) = h : 1h (1min= = :1 (30min= hour) hour= h) =1:2 (Multiply by 2) And in fraction form we have Uses of Ratio and Continued Ratio Uses of Ratio: Ratio is used for calculating continued ratio, proportion, rates, percentage as well as continued proportion. Ratio can also be used for dividing the profit or amount between two or more people. For this we have the following process: 1. First, we find sum of given ratios. 2. Share can be divided by the following formula. Share of a person= (given amount) (Ratio/Sum of ratios) Continued Ratio: So far, we have learnt the method of comparing two quantities of the same kind. But there may be the situation when we have to compare more than two quantities, which are in a continued ratio. e.g. Suppose that Rs. 74000 are to be divided among three friends A, B, C such that A : B = 4 : 5 and B : C = 3 : 2 Sum of ratio = 12 + 15 + 10 = 37 Share of A = x 74000 = 12 x 2000 = 24000 Share of B = x 74000 = 15 x 2000 = 30000 Share of C = x 74000 = 10 x 2000 = 20000 Example: Divide 60000 in ratio 5 : 7 Solution: Let A and B be two persons having ratio of share 5 : 7 Sum of ratios = 5 + 7 = 12 Share of A = x 60000 = 25000 Share of B = x 60000 = 35000 Example: Three partners invested Rs. 12500, 9000 and 7500 respectively. If the total profit earned Rs. 5800, how much profit each partner will receive. Solution: Let A, B and C be three partners A invested = 12500 B invested = 9000 C invested = 7500 Profit = 5800 Ratio among their investments is Sum of ratios = 25 + 18 + 15 = 58 Share of A = x 5800 = 2500 Share of B = x 5800 = 1800 Share of C = x 5800 = 1500 Example: A profit of Rs. 8920 is to be divided among four persons in the ratios 5 : 9 : 11 : 15 respectively. How much does partner get? Solution: Given Profit = 8920 Ratio between Share = 5 : 9 : 11 : 15 Sum of ratios = 5 + 9 + 11 + 15 = 40 1st Partner's Share = x 8920 = 1115 2nd Partner's Share = x 8920 = 2007 3rd Partner's Share = x 8920 = 2453 4th Partner's Share = x 8920 = 3345 Examples Uses of Ratio and Continued Ratio Example: A, B and C starts business from 5000 for 1 year, 6000 for 9 months and 3000 for 7 months. They earned profit 3375. Find share of each. Solution: Let total profit = 3375 Ratio of Investments Ratio between their periods Ratio of investments with respect to time Sum of ratios = 20 + 18 + 7 = 45 Share of A = x 3375 = 1500 Share of B = x 3375 = 1350 Share of C = x 3375 = 525 Example: Three partners invested Rs. 16000, 14500 and 10500 respective. The profit was distributed among the partners. If the third partner got Rs. 273 as profit, what was (a) total profit (b) profit of 1st and 2nd partner? Solution: Let total profit = P Let A, B, C be three partners. Ratio between their partner investments Sum of ratio = 32 + 29 + 21 = 82 Profit receive by third partner = 273 = xP xP Total Profit = 1066 Share of 1st partner = x 1066 = 416 Share of 2nd partner = x 1066 = 377 Example: A servant receives house rent allowance of his basic pay. Medical allowance of his basic pay. If he receives Rs. 450 as medical allowance, find his total basic pay an his house rent allowance. Solution: Let basic pay = Let amount received as medical allowance = 450 But medical allowance = (basic pay) = = 9000 Rs House rent allowance = x 9000 = 900 Rs Increase and Decrease in Ratio Decrease and Increase in Ratio: If the no. of teachers in a college is increased from 50 to 60. Then the ratio of new staff and old staff is i.e. We say that, no. of teachers has been increased in ratio 6 : 5. In other words, no. of new staff is no. of old staff. Hence Rule: To increase a no. “ ” we multiply by an improper fraction. Similarly to decrease a no. “ ” we multiply e.g. (1) Increase Rs. 20 in ratio 6 : 5 New value = e.g. (2) Decrease 56 in ratio 7 : 8 New value = Concept of Rate by an proper fraction. times the Rate: Before defining rate, first we consider some examples. 1. If one dozen eggs cost 24 Rs., what is the cost of 1 egg? (Ans). One dozen eggs cost = 24 Rs Cost of 1 egg = 24/12=2Rs/egg 1. A car travels 540 km consumed 60 liters petrol. Find distance travelled in liter. (Ans). In 60 liters distance travelled= 540km In 1 liter distance travelled = 540/60=9km/liter 1. A boy works 5-hour and paid 120 Rs. Find his pay for one hour. (Ans) In 5 hours boy gets = 120 Rs. In 1 hour boy gets = 120/5=24Rs/hour In each of the above examples, the two quantities that is the quantity in numerator and the quantity in denominator have different units, whereas in ratio both quantities have same units. “The division of two quantities having different unit, is called RATE” In the above examples, we have 1. Rate = 24/12 = 2Rs/egg 2. Rate = 540/60 = 9km/liter 3. Rate = 120/5 = 24Rs/hour Example: A car uses 40 liters of petrol to travel 320 km. How far can is travel for 32 liters? Solution: In 40 liters car traveled = 320km In 1 liter car traveled = 320/40 = 6km/liter In 32 liter car traveled = (8km/liter)(32liter) = 256km Concept of Proportion Proportion: “A statement of equality of two ratios is called proportion” Four numbers a, b, c, d are said to be in proportion when the ratio first two a and b is equal to ratio of last two c and d. i.e. or a:b=c:d e.g. or 2:3=6:9 Some authors used the notation for proportion as a : b :: c : d, but this notation is not preferred now. Here or a:b=c:d If four numbers are in proportion, then we can also derived some other proportion from it. Let be the given proportion, then (1) (2) (3) (4) (5) are called the Derived Proportions. Here or a : b = c : d, then The numbers a and d are called extremes of proportion, and the numbers b and c are called means of proportion. Hence Product of extremes = Product of Means To solve proportion, we use above principal, A single term in the proportion is called proportional. “a” is the 1st proportional. “b” is the 2nd proportional. “c” is the 3rd proportional. “d” is the 4th proportional. Example: Find the 3rd proportional in 2 : 3 = Solution: Let 2 : 3 = : 15 i.e. : 15 3x 3 = 2 x 15 (by the principle of proportion) = 30 Example: Find the missing value in Solution: Let : 8 = 9 : 12 12 =9x8 12 = 72 : 8 = 9 : 12 (by the principle of proportion) Example: Find the 1st proportional in 18, 8 and 6. Solution: Let be the 1st proportional : 18 = 8 : 6 x 6 = 18 x 8 = = 24 Example: Find the 2nd proportional in 4, 20, 30. Solution: Let be the 2nd proportional 4 : =20 : 30 x 20 = 4 x 30 = =6 Direct Proportion Types of Proportion: There are four types of proportion. 1. 2. 3. 4. Direct Proportion Inverse Proportion Compound Proportion Continued Proportion Direct Proportion: Suppose the price of one piece of soap is 20 Rs. If a person wants to buy one dozen pieces of soap, then he has to pay 240 Rs. If he wants to buy two dozen pieces of soap, he has to pay 480 Rs and so on. We can easily see that if the person buys more pieces, he has to pay more or he has to pay less if he buys less pieces. That is, as pieces of soap are increased total price also increased, conversely, if pieces of soap are decreased total price also decreased. In such situation, we say that pieces and price are directly related. In other words, If increase in one quantity causes increase in other quantity or decrease in one quantity causes decrease in other quantity, then we say that they are related directly (They are direct proportion). If and are in direct proportion, then division of and will be constant. i.e. In the above example, we see that each ratio is the same. Hence, if we are dealing with quantities, which are related directly, (which are in direct proportion), then we shall use the follow rule. 24 x 240 = 12 x 480 In general Principle of Direct Proportion Example: If 30 dozens of eggs cost 300 Rs. Find the cost of 5-dozens of eggs. Solution: Let be the required price of 5 dozens eggs Since quantities are in direct proportion, so we use the above principle. x 30 = 5 x 300 = 50 Rs. Example: A car travel 81 km in 4.5 liters. How far will it goes by 20 liters of petrol. Solution: Let be required required distance travelled by car in 20 liters. Since quantities are related directly, so by the above principle 4.5 x = 20 x 81 = 360 km Inverse Proportion Inverse Proportion: Suppose that 20 men build a house in 6-days. If men are increased to 30 then they take 4-days to build the same house. If men become 40, they take 2-days to build the house. i.e. It can be seen that as the no. of men is increased, the time taken to build the house is decreased in the same ratio. In other words, If increased in one quantity causes decrease in other quantity or decrease in one quantity, then we say that both quantities are inversely related. More explicitly, If two quantities and are in inverse proportion, then their product will be constant. i.e. where = constant In the above example, we see that 20 x 6 = 120 30 x 4 = 120 40 x 3 = 120 Shows each product is constant or same. Therefore, if we are dealing with quantities, which are related inversely, the we can use the following rule. 20 x 6 = 30 x 4 In general, Example: Four pipes can fill a tank in 70 minutes. How long will it take to fill the tank by 7 pipes? Solution: By the principle of inverse proportion, we have 4 x 70 = 7 x = 40 minutes Example: Thirty-five workers can build a house in 16-days. How many days will 28 workers working at the same rate take to build the same house? Solution: By the principle of inverse proportion, we have 28 x = 35 x 16 = 20 days Example: Imran brought 40 toys each cost Rs.14. How many toys Imran can buy at Rs.8 each from the same amount? Solution: By the principle of inverse proportion, we have 14 x 40 = 8 x = 70 toys Compound Proportion “The proportion involving two or more quantities is called Compound Proportion” for Solving Compound Proportions Rules CASE-1 If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are also directly related. Then we use the following rule CASE-2 If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are inversely related. Then we use the following rule CASE-3 If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are directly related. The we use the following rule CASE-4 If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are also inversely related. Then we use the following rule Example: 195 men working 10 hour a day can finish a job in 20 days. How many men employed to finish the job in 15 days if they work 13 hours a day. Solution: Let be the no. of men required 20 x 10 x 195 = 15 x 13 x = 200 men Example: A soap factory makes 100 units in 9 days with help of 20 machines. How many units can be made in 12 days with the help of 18 machines? Solution: 20 x 9 x = 600 x 18 x 12 = 720 units Continued Proportion Two or more quantities are said to be in continued proportion if 1st is related to 2nd , 2nd is related to 3rd , 3rd is related to 4th and so on. i.e. if a, b, c, d, e are in continued proportion then Example: Three persons A, B and C earned a profit of 70000 Rs in a business. Their share in profit is as follows A:B=4:2 B : C = 10 : 5 Find share of each person. Solution: Total amount of profit = 70000 Ratio of A, B and C is Sum of ratio = 40 + 20 + 10 = 70 Share of A = x 70000 = 40000 Share of B = x 70000 = 20000 Share of C = x 70000 = 10000 Example: Divide 2562 among the three friends Waqas, Usman and Shakeel. Such that ratio between their shares as Waqas : Usman = 4 : 5 Usman : Shakeel = 8 : 10 Solution: Let amount to be divide = 2562 Ratio of their share Sum of ratios = 32 + 40 + 50 + 122 Share of Waqas = x 2562 = 672 Share of Usman = Share of Shakeel = Concept of Percentages x 2562 = 840 x 2562 = 1052 Comparison of fractions is not an easy task, especially when the two fractions have different denominators. For example; if you are asked which of the fractions compare, whether is greater than or less than and is greater than the other. i.e. we want to . Since, denominators of and are different. To compare these fraction, first we make their denominators same Now, we have two fraction as Since numerator of , with same denominator. is greater than numerator of . is greater than But comparison becomes more easier if common denominator is 100. The fraction with denominator 100, is called a percentage, denoted by a% or a/100. The sign % is called percent. e.g., = 3%, = 5% The term percent is a short form of the Latin word “Per Centum” that means “Out of Hundred”. e.g. In a paper of math, out of total marks 50, Waqas got 35 marks, Usman got 43 marks and Shakeel got marks 32. Waqas got 35 out of 50 marks i.e. Usman got 43 out of 50 i.e. Shakeel got 32 out of 50 i.e. Examples: 15% = (replace % by 1/100) 48% = Example: Express in decimal (1) (2) (3) Changing a Fraction into Percentage: We can change a fraction into percentage by multiplying the fraction by 100% and simplify it, if possible. Examples: Express as percentages Expressing One Quantity as Percentage of Another Number In a school, 56 out of 70 teachers are female. What percentage of the teachers are females? What percentage of them is male? The fraction of female teachers in the school is 56/70 changing this fraction to percentage we have x 100% = 80% 80% of the teachers are female and percentage of male teachers (100%-80%) =20% In general to express one quantity “ ” as a percentage of the other quantity “ ”, we have 1. Write the fraction 2. Multiply fraction by 100% to convert into percentage. Example: 108 students out of 150 are passed in Math and 96 out of 160 are passed in English. Find percentage of passed students. Solution: Percentage of passed in Math = x 100% = 72% Percentage of passed in English = Finding a Percentage of a Number: In order to find percentage ( x 100% = 60% ) of another number “ ”. We have the following method. 1. Multiply with the number “ ” i.e. 2. Simplify it, if possible. Example: If 75% of the students in a class of 40 passed a Mathematics test. How many of them failed? Solution: Total Students = 40 Percentage of passed students = 75% No. of students who passed the test = 75% of 40 students = No. of failed students = 40 – 30 = 10 Example: Find (1) 25% of 21.60 (2) Solution: % x 1.60 x 40 = 30 o 25% of 21.60 = o % x 1.60 = Percentage Changing x 21.60 = 5.40 % x 1.60 = x = = 0.60 The change in the value of an item can be expressed as a percentage increase or decrease of the original value. An increase of 5% in the salary of a men who earns Rs.500 per month. That means, there is an increase of Rs.5 For every hundred (100). i.e. after increase, Rs.100 becomes 105. Therefore, New Salary = x Original Salary = x 500 = 525 On the other hand, a decrease of 5% in his salary means that for every Rs.100 in the original salary, there is a decrease of Rs.5 i.e. each Rs.100 becomes Rs.95 i.e. New Salary = x Original Salary = In general, If an amount “ ” as increased by If an amount “ ” as decreased by x 500 = 475 Profit and Loss Manufacturer produces goods at a certain cost. If goods are sold at a higher price than the cost price, then the manufacturer makes a profit or gain. But, if for some reason, the manufacturer sells goods at lower price than the cost price, then he suffers a loss. Thus Profit = Selling Price – Cost price Loss = Cost Price – Selling Price Suppose, a person bought a book of Rs.100 and sold it for Rs.120. He makes a profit of Rs.20 i.e. Profit = 120 – 100 = Rs.20 Similarly if a person bought a machine for Rs.500 and sold it for Rs.400, then he make a loss of Rs.100 i.e. Loss = 500 – 400 = Rs.100 Percentage Profit and Percentage Loss: For the purpose of comparison, we usually express the actual profits or loss as a percentage of cost price. e.g. a shopkeeper sold an article for Rs.60 which costs Rs.50 and another article sold for Rs.110 which cost Rs.100. In each case, shopkeeper makes a profit of Rs.10. It seems that in both cases profit is equally gained. But in percentage profit, we have Percentage Profit = x 100% = 20% Percentage Profit = x 100% = 10% Hence, in first case he made a more profit than in 2nd case, this is better and accurate method for comparison. Example: A bag costing Rs.28 sold for Rs.35. Find percentage profit. Solution: Let Cost Price = Rs.28 Selling Price = Rs.35 Profit = Selling Price – Cost Price = 35 – 28 = Rs.7 Percentage Profit = x 100% = 25% Example: A machine costing Rs.60 and sold for Rs.50. Find percentage loss. Solution: Let Cost Price = Rs.60 Sold Price = Rs.50 Loss = Cost Price – Sold Price = 60 – 50 = Rs.10 Percentage Loss = x 100% = % Example: Let a bookseller gain 30% by selling a book for Rs.65. Find the cost price of the book. Solution: Let be the cost price Profit = 30% of = But Profit = Selling Price – Cost Price Concept of Discount It is usually seen that the retailers cannot sell defective items, old items, etc at the retail-selling price. If these items are sold at lower price, then it is called sale price. The difference between marked price (selling price) and the sale price, is called discount. In other words, a reduction in the price of an article for payment in cash is called discount. Hence Discount is usually given as a percentage of the original price. Example: A watch priced at Rs.160 is sold for Rs.140. Find percentage discount. Solution: Let Marked Price = Rs.160 Sale Price = Rs.140 Discount = Marked Price – Sale Price = 160 – 140 = Rs.20 Percentage Discount = x 100 = Example: The marked price of a machine is Rs.600. A discount of 6% is given during a sale. What is the sale price of the washing machine. Solution: Let Marked Price = Rs.600 Discount Rate = 6% Discount = 6% of the marked price = x 600 = Rs.36 Sale Price = Marked Price – Discount = 600 – 36 = Rs.564 Concept of Commission A commission is the payment, an agent gets for selling or buying something on behalf of another person. It is usually written as a percentage of cost price or selling price. Example: An agent got 2% commission on selling a plot for 1000000. Find his commission. Solution: Let Sale Price = 1000000 Commission Rate = 2% Amount of Commission = 2% x 1000000 = x 1000000 = 20000 Example: An agent received Rs.6000 as commission for selling a house. If his commission rate is 2%. What is the selling price? Solution: Let Selling Price = Commission Rate = 2% Commission = (Selling Price)(Rate) 6000 = x