Chapter 7-Growth Rates and Percentages

Transcription

1 Chapter 7-Growth Rates and Percentages Percentages 7.1 The danger with using percentages is that the percentage sign is easily detached from the percentage value leading to incorrect statements. For example we know that 25% is a quarter but writing: 25 = ¼ or 25 = 0.25 is incorrect. E7.1 Correct the statements below for 25% is a quarter: 25 = ¼ or 25 = A7.1 25% = ¼ or 25% = We know that 10% of is 10. We can write this mathematically as: 10% x = 10. If we replace the % sign the statement becomes: Note that 10 i. is the fractional form for 10%. ii. of is represented by the x operation. E7.2 i.write the statement, 25% of is 25 in mathematical form without using the % sign. ii. what is the fractional form for 25% A i ii. 4 is the fractional form of 25% 7.3 We can also write, 10% of is 10, mathematically as: i.e. replace the % form with the decimal form i.e is the decimal form for 10%. Another example is : 17.5% (the rate of VAT) = To convert to decimal form remove the % sign and move the decimal point two places to the left. E7.3 Complete the Table: The first row has been done for you. For 10% we use 10 1 or

2 For 20% We use or 19 we use or For 50% we use For % we use For % we use 0 For % we use 0.01 A7.3 For 10% 1 we use 10 or 0.1 For 20% For 19% We use 20 1 or we use or 0.19 For 50% 50 1 we use 2 or 0.50 For % we use 1 For 0 % we use 0 For 1 % we use 0.01 Solving % problems using their equation form 7.4 Sometimes solving a percentage problem is best done by casting it into an equation form and then solving it. This may seem a sledgehammer method in easy cases. It will help with not so easy problems. Example: Find 20% of 50. Write statement: n is 20% of 50 in equation form using the decimal form of 20%: n = 0.20(50) is the equation and its solution is n = 10. The word is is replaced with the = sign and of is the multiplication operation. So 10 is 20% of 50 E7.4 Find 17.5% of 350 by casting the statement into an equation in the decimal form: A ( 350) = n So n = Here is a slightly more difficult problem. 36 is 12% of what quantity? Change the Question to a statement with n replacing what 79

3 quantity : 36 is 12% of n. Cast this as an equation in decimal form. 36 = 0.12n Solve: n 0.12n 36 n So 36 is 12% of 300. E is 2% of what quantity? A = 2% of n Can be written as 0.2n = 25. n = 25/0.2 = Another problem: 24 is what percentage of 1200? Statement with n: 24 is n% of Equation in fractional form: n 24 (1200) 24() n So 24 is 20% of E is what % of 1440? A = (x%)(1440) Replacing the % sign and solve: 1440x 36 = 36() x x So 36 is 3% of Back percentages 7.7 Consider the problem: 80% of nurses interviewed said they did not mind doing night 80

4 shifts. If this number is 36 how many nurses were interviewed? We want to find total number of nurses so call this number n. Then we get the statement: 80% of n nurses is 36. Equation in decimal form: 0.80n = Solve equation: n E7.7 In a class 24 students received an A grade. If this is 30% of the class what is the size of the class? A7.7 We want to find total number of students, so call this number n. Then we get the statement: 30% of n students is 24. Equation in decimal form: 0.30n = Solve equation: n Percentage growth 7.8 In finance we are very interested in percentage growth. For Example if your normal pay is per annum and you get an increase of 10% what is the new pay level? We can do this as a 2 stage calculation: Stage 1: Find the change, we use the symbol (pronounced Delta) for the increase. in = 10% of 30k = 0.10(30 000) = 3000 Note the is +ve implying an increase. Stage 2: Add to find new pay level. So the new pay level = = E7.8 Increase a salary of 50,000 by 7% using a 2 stage calculation. Working in Stage 1: in = Stage 2: Add to find new pay level. So the new pay level in =. A7.8 Stage 1: in = 7% of 50k = 0.07(50 000) = 3500 Stage 2: Add to find new pay level. So the new pay level = = Because in finance we are dealing with percentage growths over 81

5 many periods we need a more efficient (shorter) way of dealing with growth problems. Suppose 0 is increased by 10%. What is the new value? New value = original value + = (0) Now this expression is made up of two terms. Factorising we get New value = 0( ) = 0(1.01) = So we can go straight to the new value in one step by multiplying the original value by A growth rate of 10% has been transformed to a growth factor of E7.9 Find the new value in one step if 250 is increased by 10%. Growth rate = 10% ; growth factor = 1.10 New value = 250( ) = A7.9 Growth rate = 10% ; growth factor = 1.10 New value = 250(1.10) = So if we want to increase by a growth rate of 10% we multiply the original value by the growth factor, Now if we wanted to increase the original value by 20% we would multiply by the growth factor of 1.20 Example: A plant of 320cm grows by 7% in a week. What is its height at the end of the week? The growth rate of 7% growth factor of 1.07 so: New height = 320(1.07) = cm E7.10 A plant of height 120cm grows by 11% in a week. What is its height at the end of the week? growth rate = 11%, growth factor = so: height at end of week =120( )cm = A7.10 A plant of height 120cm grows by 11% in a week. What is its height at the end of the week? growth rate = 11%, growth factor = 1.11 so: height at end of week =120(1.11) cm = cm In general if the growth rate is r% then the growth factor is (1 + r per cent in decimal form) Examples: Normal pay %growth growth factor new pay pd 50% = 1.50 (1.50)=150p 82

6 d Share Price %growth growth factor new share price 75p 25% (1.25)p = 93.75p Price ( ) before Vat rate growth factor price after Vat Vat % (1.175) =29.38 E7.11 Complete the Table: The first row has been done for you. Normal pay %growth growth factor new pay pd 50% = 1.50 (1.50)= pw pa 5% Share Price %growth growth factor new share price % S 10% Price before Vat Vat rate growth factor price after Vat % X 17.5% A7.11 Normal pay %growth growth factor new pay pd 50% = 1.50 (1.50)= pw 25% (1.25)= pa 5% (1.05)= Share Price %growth growth factor new share price % (1.08) = % (1.1)= % (1.5)=7.35 S 10% 1.1 S(1.1) = 1.1S Price before Vat Vat rate growth factor price after Vat % (1.175) = X 17.5% X(1.175) =1.175X Back Percentages using growth factors 7.12 Now consider this problem. A plant grows by 20% to a height of 120cm at the end of the week. What was the height at the beginning of the week? Using h for the original height and (20% giving) a growth factor of 83

7 1.20 we can write the equation: h(1.20) 120cm h 120 cm 1.20 cm E7.12 The population of Woodgreen grew by 12% to 6048 in a decade. What was the population at the beginning of the decade? Growth rate = 12% growth factor = Equation using P for original population: A7.12 The population of Woodgreen grew by 12% to 6048 in a decade. What was the population at the beginning of the decade? Growth rate = 12% growth factor = 1.12 Equation using P for original population: 1.12P = P = 1.12 P = 5400 Multiperiod growth 7.13 An investment of 0 grows by 5% in the first year and by 4% in the second year. What is the value of the investment at the end of the two years? Value at end of year 1 = 0(1.05) = 1050 Value at end of year 2 = 1050(1.04) = However we can do this in one stage: Value at end of two years by linking the growth factors. = 0(1.05)(1.04) = The linking is by multiplying the growth factors. E7.13 An investment of 500 grows by 5% in the first year and by 7% in the second year. Obtain the value of the investment at the end of the two years in one stage? A7.13 Value at end of two years by linking the growth factors. = 500(1.05)(1.07) = An investment grows by 5% in the first year and by 6% in the second year. What is the whole period growth factor and growth 84

8 rate? The yearly growth rates are 5% and 6% so the yearly growth factors are 1.05 and The whole period growth factor = (1.05)(1.06) = So the whole period growth rate = = = 11.3 % E7.14 An investment grows by 6% in the first year, 6% in the second year and remains unchanged in the third year. What is the whole period growth rate? The yearly growth rates are.%,..% and..% so the yearly growth factors are.., and... The whole period growth factor = ( )( )( ) = So the whole period growth rate = = % A7.14 The yearly growth rates are 6%, 6% and 0% so the yearly growth factors are 1.06,1.06 and 1. The whole period growth factor = (1.06)(1.06)(1) = So the whole period growth rate = = 12.36% Percentage decrease 7.15 We will do this directly using growth factors. An investment of 0 falls by 5% in a year. Find the value of the investment at the end of the year in one stage (without finding )? The growth is now negative = -5% So the growth factor is (1 0.05) = 0.95 So value at end of the year = 0(0.95) = 950. Note the terminology growth factor was used even though the value fell. The fall was reflected in the growth factor being below 1. E7.15 Complete the Table: The first row has been done for you. Change Growth rate growth factor Fall by 5% -5% = % 0.88 Increase by 2% 4% 85

9 Drop by 12% Increase by 1 basis point 1.03 A7.15 Change Growth rate growth factor Fall by 5% -5% = 0.5 Fall by 10% -10% = 0.90 Fall by 12% -12% 0.88 Increase by 2% 2% 1.02 Increase by 4% 4% 1.04 Increase by 3% 3% 1.03 Drop by 12% -12% 0.88 Increase by 1 basis point 0.01% Multiperiod growth including negative growth 7.16 An investment of decreases in value by 5% in the first year and decreases by a further 4% in the second year. What is the end period value of the investment? What is the whole period percentage change in value? We will do this directly without the use of deltas. The period of investment is 2 years with yearly growth rates of -5% and -4%. The corresponding growth factors are 0.95 and 0.96 (below 1). End period value = (0.95)(0.96) = The whole period percentage change is obtained as follows: Whole period growth factor = (0.95)(0.96) So whole period growth rate = (0.95)(0.96) 1 = = -8.8% E7.16 An investment of 2000 decreases in value by 5% in the first year and decreases by a further 3% in the second year. What is the end period value of the investment? What is the whole period percentage change in value? 86

10 A7.16 An investment of 2000 decreases in value by 5% in the first year and decreases by a further 3% in the second year. What is the end period value of the investment? What is the whole period percentage change in value? The period of investment is 2 years with yearly growth rates of -5% and -3%. The corresponding growth factors are 0.95 and 0.97 (below 1). End period value = 2 000(0.95)(0.97) = The whole period percentage change is obtained as follows: Whole period growth factor = (0.95)(0.97) So whole period growth rate = (0.95)(0.97) 1 = = -7.85% 87

11 Exercise 7 1. A football team wins 18 of its 42 games in the season. What is the percentage of the games won? students, which is 80% of a class, passed their exam. What is the size of the class? 3. (i) A plant of height 10cm increases by 5%. What is the new height? (ii) A plant increases in height by 4% in a week and grows to a height of 108 cm. What was the height at the beginning of the week? (iii) A investor invests 00 for 20 years at a compounded rate of 7% per annum. What is the value of the investment after the 20 years? 88