Ratios, Proportions, and Percentages PDF

Chapter 2 Ratios, Proportions and Percentages 2.1 Introduction Understanding ratios, proportions, and percentages forms the bedrock of mathematical literacy in various fields, including computing. However, mathematical literacy is for all, everyone should be able to tell if their pay check is...

Chapter 2 Ratios, Proportions and Percentages 2.1 Introduction Understanding ratios, proportions, and percentages forms the bedrock of mathematical literacy in various fields, including computing. However, mathematical literacy is for all, everyone should be able to tell if their pay check is correct! Ratios represent the relationship between two quantities, proportions refer to the distribution or allocation of a quantity among different parts or shares, and percentages denote ratios expressed as fractions of 100. Mastery of these concepts not only enables students to analyse data and solve problems but also equips them with essential tools for interpreting and manipulating numerical information, making them indispensable skills for computing students navigating the complexities of data analysis and algorithmic thinking. 2.2 Ratios Definition 2.2.1. A ratio is a comparison of two or more quantities by division. It is often expressed in the form a : b, where a and b represent numbers or quantities. Ratios can involve any number of quantities and can be written a in various forms, such as , a/b, or as a list. b Definition 2.2.2. Proportions refer to the distribution or allocation of a quantity (total) among different parts or a b shares. Therefore in a ratio of a : b, the proportions are and. Note that the sum of the proportions is a+b a+b always 1. Example 2.2.3. The ratio of petrol to oil is typically 50 : 1 for 2 stroke engines. In a petrol oil mixture what proportion of the mixture is oil and what proportion of the mixture is petrol. 1 50 Solution: The proportion of oil in the mixture is and the proportion of petrol in the mixture is. 51 51 Two ratios are equivalent if they result in the same proportions. Example 2.2.4. Show that the ratios 2 : 5 and 6 : 15 are equivalent. 2 5 Solution: Using the ratio 2 : 5, the proportions are and. However, using the ratio 6 : 15, the proportions are 7 7 6 2 15 5 = and =. Therefore the ratios are equivalent. 21 7 21 7 10 2.3. PERCENTAGES 11 Multiplying (scaling) each quantity in ratio by any nonzero number results in an equivalent ratio. Therefore the proportions can be expressed as an equivalent ratio. Example 2.2.5. Ann, Barry and Cath share a prize of e140 in the ratio of 2 : 1 : 4 respectively. How much money does each person receive? Solution: We shall calculate the proportions and express them as a ratio (equivalent to 2 : 1 : 4). We will then scale up by a factor of the amount (in this case 140). There are 2 + 1 + 4 = 7 shares in total. Thus the proportions are 2 1 4 2 1 4 : : , which is equivalent to (140) : (140) : (140) = 40 : 20 : 80. Therefore Ann receives e40, Barry receives 7 7 7 7 7 7 e20 and Cath receives e80. 1 In the previous example we essentially multiplied (scaled) the ratio first by and then by 140 which the the same as 7 multiplying by 20. We can often use this ”shortcut”. However, it is important to understand why it works. Example 2.2.6. In a bag of marbles, the ratio of red marbles to blue marbles is 3:5. If there are 25 blue marbles, how many red marbles are there? Solution: Multiplying the ratio 3 : 5 by 5 gives the equivalent ratio 15 : 25. Therefore there are 15 red marbles. Exercises 2.2.7. (a) Divide e800 into the ratio 3 : 1 : 4. (b) Divide e1000 into the ratio 4 : 5 : 7. 1 1 1 (c) Find x such that 2 : 3 is equivalent to x : 7. (d) Divide 221 into the ratio : :. 2 3 4 2.3 Percentages A percentage is a way of expressing a number as a fraction of 100. It is denoted by the symbol %. It represents a portion or share of a whole, where 100% represents the entire quantity or total. Example 2.3.1. Suppose that you have 30 out of a total of 100 marbles. Then you have 30 marbles. The proportion 30 30 of the marbles that you have is. The percentage of the marbles that you have is 30%. Note that 30% =. 100 100 1 Example 2.3.2. Suppose that there 80 marbles on a table. Ann takes 20 marbles from the table. You then take 2 of the remaining marbles. What percentage of the marbles do you have. 1 Solution: There are 60 marbles on the table after Ann takes 20. You take of these 60 marbles which means that 2 30 you have 30 out of the 80 marbles. Therefore you have × 100% = 37.5% of the marbles. 80 1 Note that at first glance it may appear that the proportion and the percentage 37.5% don’t seem to match up. This 2 is the most important learning outcome for percentages. A percentage is a proportion expressed in a particular way, but it is a proportion nonetheless. Therefore, just like proportions, percentages depend on the total amount, i.e. what 1 is the 100% value? In the example above the total amount for the was 60, the amount of marbles remaining on the 2 table after Ann had removed 20. The total amount for the 37.5% was 80, the total amount of marbles. When working with percentages and amounts there are essentially 4 possible questions, which are listed below. When we know the 100% value we use multiplication and when we don’t we use division. When the percentage is an increase we use a plus and when it is a decrease we use a minus. 12 CHAPTER 2. RATIOS, PROPORTIONS AND PERCENTAGES 1. Increase an known amount x by r%. Then the result is x(1+r%). 2. Decrease an known amount x by r%. Then the result is x(1−r%). x 3. Increase an unknown amount by r%, where the result is x. Then the original amount is. (1+r%) x 4. Decrease an unknown amount by r%, where the result is x. Then the original amount is. (1−r%) Example 2.3.3. In a sale at pair of shoes were reduced by 15% to e120.70. What was the price of the shoes before the sale. Solution: We do not know the 100% value so we use division. The percentage lead to a reduction, i.e. a decrease and so we use a minus sign. That is, we use number 4 in the list above. The price of the shoes before the sale is 120.7 120.7 = = e142. (1 − 15%) 0.85 Definition 2.3.4. The percentage error is a measure of the accuracy of a measurement or estimate. It is calculated as the absolute difference between the measured or estimated value and the true or accepted value, divided by the true value, and multiplied by 100%. Estimated (measured) Value − True Value The formula for calculating the percentage error (PE) is given by: PE = × True Value 100%. Example 2.3.5. A share price (in euros) opened at 2.70 and closed at 2.25. However, in a newspaper the closing price was printed as 2.52. What was the percentage decrease in the share price and what was the percentage error in the printed value. 0.45 Solution: The share price dropped by 2.70 − 2.25 = 0.45. The percentage decrease is × 100% = 16.66%. 2.70 Estimated Value − True Value 2.52 − 2.25 The percentage error, PE = × 100% = × 100% = 12%. True Value 2.25 Exercises 2.3.6. (a) A student got 23 marks out of 40 in a test. What percentage did they get. (b) An exam has 55 marks. How many marks do you need to pass (40%). (c) The cost of a car increases by 5% to e27,298.95. What was the cost before the increase. (d) The attendance at a football game was estimated at 35,000. The actual attendance was 36,153. Calculate the percentage error of the estimate. (e) A share price increases by 20% in the year 2022 but decreases by 20% in 2023. Comparing the price at the start of 2022 to the end of 2023, has the price increased, decreased or is it the same. 2.4 Weighted averages To calculate the mean to two numbers x and y we simply add the two numbers x + y and then divide by 2 to get x+y 1 1 1 1 = x + y. Written like this we can see that the fractions and are proportions (they sum to 1). Indeed 2 2 2 2 2 1 this is true in the general case of taking the mean of n numbers. The n fractions sum to 1. n 2.4. WEIGHTED AVERAGES 13 The only difference in a weighted average (mean) is that any set of proportions can be used. That is, any set of values between 0 and 1 that sum to 1 can be used. The set of proportions are called the weights. These weights are usually given as proportions but could also be given as a ratio.   0 1 2 3 4 Example 2.4.1. let S = {2, 4, 2, 6, 7} and let W = , , , ,. Calculate the mean of the data in the set 10 10 10 10 10 S. Also, using the set W as weights, calculate the weighted mean of the data in the set S. 2+4+2+6+7 21 Solution: The mean of the data in S is = = 4.2. 5 5 0 1 2 3 4 The weighted mean is given by (2) + (4) + (2) + (6) + (7) = 0.4 + 0.4 + 1.8 + 2.8 = 5.4. 10 10 10 10 10 Exercises 2.4.2.   1 1 2 3 4 1 (a) Calculate the weighted mean of {1, 5, 6, 6, 7, 8} using the set of weights , , , , ,. 12 12 12 12 12 12 (b) A student gets 80% in CA which is worth 30% and 50% in the final exam which is worth 70%. Calculate the students overall percentage for the module. (c) A student gets 24 out of 40 marks in the CA which is worth 30%. What is the minimum percentage that the student needs in the final exam (worth 70%) in order to pass the module overall. 40% or above is a pass.