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PERCENTAGE Calculating percentage change Lesson Learning Outcomes and Assessment Standards Learning Outcome 1 4 Number and Operations in Context The learner is able to use knowledge of numbers and their relationships to investigate a range of different contexts, which include financial aspects of personal, business and national issues Assessment Standard AS 11.1.1 In a variety of contexts find ways to explore and analyse situations that are numerically based, by: • Estimating efficiently • Working with formulae by hand and with a calculator • Showing awareness of the significance of digits • Checking statements and results by doing relevant calculations (The range of problem types includes percentage, ratio, rate and proportion.) Overview In the previous lessons we have looked at the following calculations: ● Changed fractions or ratios into percentages; ● Calculated a percentage of an amount; ● Added a percentage of an amount to an amount; and ● Subtracted a percentage of an amount from an amount. In this lesson we will consider another type of percentage calculation. We will: ● Calculate a percentage change. Lesson A common example of percentage change is the year-on-year inflation. If inflation DVD is said to be 6%, this gives an indication of the percentage change in the price of a basket of goods from one year to the next. This means that the price of the goods has increased by 6%.The way that inflation is measured is as follows: Every month, let’s say January 2006, statisticians find out the cost of a list of 1 500 goods and services. For our example let us say this comes to R5 000. 11 One year later, in January 2007, statisticians find out the cost of that same list of 1 500 different goods and services. This time they find that the cost is R5 650. Note: they use the same list of goods and services each time to ensure that they are comparing the same things with one another. The basket of goods and services has increased by R650. This increase, expressed as a percentage of the original amount, is called the year-on-year inflation or percentage change. Note: when we added a percentage of an amount to an amount, in a previous lesson, we knew the percentage and had to find the actual amount to add it on. In this case we know the amount that has been added so we need to express it as a percentage of the original amount. Methods and worked examples For the inflation problem above we can write down the steps on how to calculate percentage change as follows: Step 1: find the difference between the final value and the original value. Final value – original value = R5 650 – R5 000 = R650 Step 2: find what percentage this is of the original amount. This takes us back to the first lesson when we learnt to express a fraction as a percentage. The calculator keys are: 650/5000% This gives an answer of 13% We can write the above two steps as a formula: final value – original value % change = ____ original value (expressed as a %) The inflation example was an example of where the amount had increased. The following example is an example of where the amount decreases. Worked example: Last year 150 incidents of theft were reported at our school. This year 117 incidents were reported. What is the percentage change? Solution: final value – original value % change = ____ (expressed as a %) original value 117 – 150 = __ 150 (expressed as a %) 33 _ = –150 (expressed as a %) = –22% The key strokes on the calculator are: 117-150/150% The negative sign in the answer shows that the original amount decreased or got smaller. Note: this calculation reminds us of subtracting an amount from an amount. In this example we know the whole, i.e. the original amount. We worked out 12 what was subtracted, i.e. the difference between the final and the original amounts, and we express this as a percentage of the whole. In the case of subtracting an amount from an amount, we knew the percentage change so we first calculated what the amount to be subtracted was and then we subtracted it from the amount. In this case we know the amount that was subtracted and our job is to calculate what percentage it is of the original amount. Consider the following additional examples: Additional example 1 1. What is the percentage change in the price of a can of cold drink if the price went from R2,50 to R2,75? Additional example 2 2. What is the percentage change in the volume of water in a tank that goes from 1 600 l of water to 1 150l of water? Solutions: final value – original value (3) % change = ____ (expressed as a %) original value 2,75 – 2,50 = __ 2,50 (expressed as a %) 0,25 _ =2,50 (expressed as a %) = 10% The answer is positive because there was an increase in the price of the cold drink. (4) % change = final value - original valueoriginal value (expressed as a %) = 1 150 – 1 600/1 600 (expressed as a %) = – 450/1 600 (expressed as a %) = –28,125% ≈ –28% The answer is negative because there was a decrease in the volume of water. Activity InDIVIDual Practice examples 1 Calculate the percentage increase in the price of bread if the price summative increased from R5,20 to R5,90. assessment 2 Calculate the percentage of VAT on goods and services if a pair of sunglasses costs R250, excluding VAT, and R285, including VAT. (VAT: value added tax ) 3 The table below shows how the population of South Africa grew from 1996 to 2001. 1996 40,5 million 2001 44,8 million Calculate the percentage growth in population from 1996 to 2001. 4 The table below shows how the urban population of four centres in South Africa has grown from 1991 to 2007. 1991 (census) 1996 (census) 2007 (estimate) Johannesburg Cape Town 4 180 897 2 175 951 4 508 915 2 775 677 7 372 314 4 709 212 13 George 63,895 94,121 197,056 Vryheid 25,099 53,228 175,119 Information from World Gazetter 4.1 Calculate the percentage increase in population for Johannesburg from 1991 to 1996. 4.2 Calculate the percentage increase in population for Johannesburg from 1996 to 2007. 4.3 Calculate the percentage increase in population for Cape Town from 1996 to 2007. 4.4 Calculate the percentage increase in population for George from 1996 to 2007. 4.5 Calculate the percentage increase in population for Vryheid from 1996 to 2007. 4.6 Which of the four centres had the biggest percentage growth in urban population? 4.7 Which of the four centres had the most people moving to the centre since 1996? 4.8 Calculate the percentage increase in population for Vryheid from 1991 to 1996. 5 The following advertisement appeared in the local newspaper: One-Year Business Computer Diploma EARN WHILE YOU LEARN!! Was R18 999 now only R11 999 What percentage discount is this company offering for this course? 6 A sheep farmer had 78 new-born lambs. After unexpected cold weather he had only 56 lambs left because many of them died. What percentage of his flock of lambs did he lose? 7.1 Rashid’s salary went up from R4 500 a month to R4 800 a month. What is his percentage increase? 7.2 Tshapelo’s earns R5 000 a month and his salary also went up by R300 a month. What percentage increase did he get? 14