VIEWS: 22 PAGES: 32 POSTED ON: 7/28/2010 Public Domain
consumer affairs victoria www.consumer.vic.gov.au 55 A1 Budgeting by the month Achieving your goal 1. Nik’s income calculation 26 The average number of fortnights to a month is = 2.167. 12 Approximating this as 2.0 will needlessly introduce an 8.3% error into all the derived monthly figures. 26 $1050 × = $2275 12 365.25 365.25 $1050 × = $2282.81, in which = 2.174 . 12 × 14 12 × 14 Either answer is good enough for a budget estimate: they differ by less than 1%. The first is preferred because the calculation method is simpler. Where calculations have to be equal the 365.25 would have to be replaced with either 365 or 366. 2. Yearly to monthly expenses Leaving out a zero is a common mistake in dividing by ‘hand’ calculation. Some students would estimate by mental division by 12. Others might simply divide by 10 and obtain estimates that they knew were about 20% too high. A short-cut is to obtain the yearly total first, then $1877 ÷ 12 = $156.42 Annual Expenses Yearly fee Due date Monthly equivalents Car registration $277 April $23.08 Car insurance $600 April $50.00 Health Insurance $250 March $20.84 Gym membership $330 May $27.50 Further study fees $420 March $35.00 Totals $1,877 $156.42 3. Quarterly to monthly expenses Note: Dividing the total by 3 provides a check that is based on the distributive law. (a + b + c )÷ 3 = a ÷ 3 + b ÷ 3 + c ÷ 3 Service (Estimate) Quarterly Monthly estimate Monthly calculation Electricity $87 $30 $29 Telephone $102 $35 $34 Gas $45 $15 $15 Water $22 $7 $7.33 Total $256 $85.33 consumer affairs victoria www.consumer.vic.gov.au 56 A1 Budgeting by the month 4. Weekly to monthly expenses 52 1 Multiply each amount by = 4 ≈ 4.333 12 3 For the estimates, multiplying by 4 would give answers that were about 8% too low. If we round up the weekly figure this will compensate. Expense (Estimates) Weekly Monthly estimate Monthly calculation Rent $87 $90 x 4 = $360 $377 Kitty (shared items, food etc) $32 $35 x 4 = $140 $138.67 Car loan $49 $53 x 4 = $212 $212.33 Petrol $28 $30 x 4 = $120 $121.33 Fares $12 $12 x13/3 = $52 $52.00 Eating out/take away $20 $22 x 4 = $88 $86.67 Clothes $25 $108.33 Internet ISP $8 $34.67 Entertainment $45 $195.00 Computer rental $7 $30.33 Totals $313 $1,356.33 52 1 Check: The monthly total should be 4 = ≈ 4.333 weekly total. times the 12 3 5. The savings calculation Monthly Annual Net Income (Question 1) $2,275 $27,300 Expenses: (Question 2) $156.42 $1,877 Expenses: (Question 3) $626 $7,512 Expenses: (Question 4) $1,356.33 $16,276 Expenses: (Total) $2,138.75 $25,665 Savings = Expenses - Income $136.25 $1,635 Check: The annual total should be 12 times the monthly total. consumer affairs victoria www.consumer.vic.gov.au 57 A1 Budgeting by the month Extension/revision 1. Totalling the original column first provides a short cut, or check method. 2. Answers will vary to this question. For ‘impulse’ shoppers expenditure on clothes and entertainment can vary widely. Unanticipated expenses could arise in connection with car repairs or with medical expenses not covered by health insurance.’ 3. After two years Nik will have saved 2 x $1635 = $3270. Once air fares have been taken into account this might be enough for a short trip to a nearby overseas destination. 4. By 30th April, Nik should have paid all of his annual expenses except for the gym membership. Thus, for the first four months (17 weeks) of the year: Expenses: Annual charges to date $1548 First quarter service charges $256 Weekly expenses 17 @ $313 $5321 Total $7125 Income: 8 @ $1050 $8400 Savings: $8400 – $7125 $1275 5 & 6. Answers will vary but the data collected will be useful in the next lesson. consumer affairs victoria www.consumer.vic.gov.au 58 A2 Your budget My personal budget Answers to Questions 1, 2, and 3 will vary. Extension/revision 1. Answers will vary. Typical savings targets might be for Christmas, for a costly school camp, or for paying start of year fees. Making regular contributions to a special purpose savings account is an effective strategy for controlling spending on non-essentials. 2. The time scale for initial budget planning should be built around plans for meeting major expenses. Otherwise credit or ‘terms’ may need to be sought. 3. Discussion of spreadsheets should be focussed on setting up agreed design criteria. Detailed implementation could be given as alternative homework for some students. Points to make include: – Bank statements can be useful in providing checks on spreadsheet calculations. – Regular monitoring might mean by the month rather than by the week. – A schedule needs to be built in for payment of major quarterly and annual expenses. – Cumulative totals need to be built up for both projected and actual figures. – More detailed monitoring might involve the calculation of running totals and averages for particular key items, not just for income, expenditure and savings. 4. Answers again will vary. Use students’ findings for discussion at the start of the next lesson. consumer affairs victoria www.consumer.vic.gov.au 59 B1 Shopping around Value for money 1. Mental estimates mostly involve halving, doubling and multiplying or dividing by 10. Approximations can be improved by estimating and allowing for percentage errors. The following mental approximations are fairly easily made and are sufficient to determine the best buy in all cases except the Coca Cola. See also the answers to discussion Question 4. Orange juice per litre: $1.80 x 2 = $3.60 $4.90 ÷ 2 = $2.45 $6 ÷ 3 = $2 The best buy is 3 litres for $5.87 Sliced beet root per 400g: $0.67 x 2 > $0.95 > $1.51 ÷ 2 The best buy is 825 gram for $1.51 Coca Cola per litre: 3 x $1.00 = $3 $5 ÷ 2 = $2.50 $8.30 ÷ 4 = $2.10 0.8 x $1.32 ≈ $1.05 $2.09 ÷ 2 = $1.045 $9.00 ÷ 8 ≈ $1.12 The best buy is either 1.25 litre for $1.32 or 2 litres for $2.09. Tomato paste per 250g: $1.70 x 2 = $3.40 $2.74 ÷ 2 = $1.37 The best buy is 500 g for $2.74 Potato Crisps per 250g: $1.02 x 5 ≈ $5.10 $4.50 ÷ 2 = $2.2 The best buy is 415 g for $4.48. Toilet rolls: $1.59 ÷ 2 ≈ $0.80 $3.09 ÷ 4 ≈ $0.77 $4.50 ÷ 6 = $0.75 $5.76 ÷ 9 ≈ $0.60 The best buy is 9 rolls for $5.76. Perfume, per 50 ml: $25 x 2 = $50 $95 ÷ 2 = $47.50 The best buy is $44.95 for 50 ml. 2. Litre quantities are unrealistically large for comparing perfumes. 100 ml quantity comparisons are useful. They make further decimal comparisons easy. 50 ml quantity comparisons were the easiest for the mental comparisons in Question 1, since only doubling and halving were involved. 25 ml quantity comparisons are useful only if the quantities involved are all simple multiples of 25 ml. 3. Calculated values are: Orange juice, per litre: $3.58 $2.69 $2.45 $1.96 Sliced beet root, per 100g: $0.30 $0.22 $0.18 Coca Cola, per litre: $2.67 $2.22 $2.21 $1.06 $1.05 $1.12 Bottles are better buys than cans, and the best buy is the 2 litre bottle for $2.09. Tomato paste, per 100 g: $1.27 $0.94 $0.55 Toilet rolls, each: $0.80 $0.77 $0.75 $0.64 Perfume, per 100 ml: $99.80 $89.90 $94.95 consumer affairs victoria www.consumer.vic.gov.au 60 continued B1 Shopping around Extension/revision Factors to consider in choosing the unit quantity are: – ease of calculation for comparison purposes. This will depend on the method of calculation, which will be by mental approximation in most shopping situations. – amounts that accord with your experience. This could be 100 grams for potato crisps, kilograms for potato chips and megabytes for computer memory chips. 1. For the mental calculations: a) Doubling and halving could be used to find the best buy for the perfumes, for three of the orange juices and, with approximations, for the sliced beet root. b) For the Coca Cola: 3 375 ml is ths of a litre. Approximating 375 ml to one third of a litre involves 8 42 a percentage error of ≈ 11 % or 11 cents in each dollar. 6 x 375 ml = 2.25 litre. 375 The reciprocal of 1.25 is 0.8, so replace $1.32 ÷ 1.25 by $1.32 x 0.8 c) For the approximations for the different amounts of sliced beet root: Approximating 225 by 200 introduces a percentage error of approximately 12%. Approximating 425 by 400 introduces a percentage error of approximately 6%. Approximating 825 by 800 introduces a percentage error of approximately 3%. 2. Comparisons of highest and lowest unit prices are given in the table below. Item Highest unit price, h Lowest unit price, l Ratio hl /. % increase Orange juice $3.58 $1.96 1.83 83% Beet root 30c 18c 1.67 67% Coca Cola $2.67 $1.05 2.54 154% Tomato paste $1.27 $0.55 2.31 131% Potato crisps $2.04 $1.08 1.89 89% Toilet rolls 80 c 80c 64c 1.25 25% Perfume $99.80 $89.90 1.11 11% The 154% difference for Coca Cola could be partially due to the different packaging: cans versus bottles. The 131% difference for tomato paste needs a different explanation. 3. The two products for which the largest quantity did not have the lowest unit price were Coca Cola and perfume. This could be partly explained by packaging costs but another possible factor is marketing – a ‘special price’ for the Coca Cola or the brand name attached to different perfumes. 4. Larger packages may involve a greater amount of wastage due to deterioration after opening or through use of more than is needed. As far as food and drink are concerned individual needs are a matter of good diet. Smaller packages involve a greater amount of wastage in packaging. consumer affairs victoria www.consumer.vic.gov.au 61 continued B1 Shopping around Extension/revision - cont’d 5. a) Item Price Quantity Price per 100 ml Femme Thais fragrance $79.95 100 ml $79.95 Giorgiana fragrance $79.95 100 ml $79.95 Cherry Blossom fragrance $39.95 25 ml $39.95 x 4 = $159.80 Cherry Blossom fragrance $59.95 50 ml $59.95 x 2 = $119.90 Cherry Blossom fragrance $89.95 100 ml $89.95 Frank & Stein fragrance $74.95 100 ml $74.95 Sea wind fragrance $21.95 30 ml $21.95 ÷ 0.3 = $73.17 Sea wind fragrance $29.95 50 ml $29.95 x 2 = $59.90 Sea wind fragrance $41.95 100 ml $41.95 Samsona fragrance $89.95 100 ml $89.95 Thais Homme fragrance $79.95 100 ml $79.95 Frank & Stein Homme fragrance $79.95 100 ml $79.95 Spring dew fragrance $79.95 100 ml $79.95 b) Students should have ruled a number line marked in tens from $40 to $160. Dots should be neatly marked above the line at each of the 13 values in the right hand column of the table eg. five dots above each other at the $79.95 mark. c) Discuss differences in price levels – is more expensive always the best? consumer affairs victoria www.consumer.vic.gov.au 62 B2 Buying on terms What do you really pay? 1. a) Deposit = 10% of $3198 = $319.80 b) Total repayment = $319.80 + 60 x $79.95 = $319.80 + $4797 = $5116.80 c) Total interest charged = $5116.80 - $3198 = $1918.80 d) Principal, P = 90% of $3198 = $2878.20 I 1918.8 1 Flat rate of interest, r = = = 0.1333... = 13 % P × n 2878.2 × 5 3 2. Here P = $3198 – $500 I = P x rx t = $2698 = $2698 x 0.1333… x 3 = $1079.20 r= 1 A = P +I 13 % = 0.1333... 3 = $2698 + 1079.20 t =3 = $3777.20 Monthly payment, $3777.20 Q= = $104.92 36 3. Here P = $3198 – $1000 I = P x rx t = $2198 = $2198 x 0.1333… x 3 = $879.20 r= 1 13 % = 0.1333... A = P +I t =3 3 = $2198 + 879.20 = $3077.20 Monthly payment, $3077.20 Q= = $85.48 36 consumer affairs victoria www.consumer.vic.gov.au 63 B2 Buying on terms Extension/revision 1. With a simple or flat interest rate loan you repay the same amount of interest each month and its calculation is based on the amount you originally borrowed, not on how much you still have to pay. With a reducible or effective interest rate loan you only pay interest on the amount of the loan still outstanding. The amount of interest each month decreases as you pay off the loan. 2. Sophie should have been told that the comparison rate of interest being charged was approximately 26%. By comparing this with bank lending rates, Sophie may have decided to take out a personal bank loan and pay cash for the computer or shop elsewhere. 3. Computers become out of date very quickly. Sophie could need a new computer in three or four year’s time and would not want to be still paying off the old one. 4. The computer could be repossessed by the lender. Sophie could sell the computer and use some of the money to make the two outstanding payments. There might be a little money left over but no computer. Sophie could borrow enough money to cover the last two payments. If she really needs to keep the computer then this last option is what she would have to do. 5. a) Deposit = 25% of $3500 = $875 b) Total amount paid, A = $875 + 48 x $72 = $875 + $3456 = $4331 c) Interest, I = $4331 – $3500 = $831 d) Principal = 75% of $3500 = $2625 Flat rate of interest, e) This deal offers a much better rate of interest for Sophie - and a printer as well. The only disadvantage is the higher initial deposit required. consumer affairs victoria www.consumer.vic.gov.au 64 C1 Sharing the bill A fair split 1. Total usage charge = $207.50 - $74.65 = $132.85 Individual usages were Sophie: $54.25 Tye: $23.45 Nik: the rest: $55.15 If the total bill was divided up in the ratio of the usage charges, then the amounts each person should pay are: 54.25 23.45 55.15 Sophie: × $207.5 = $84.73 Tye: × $207.5 = $36.63 Nik: × $207.5 = $86.14 132.85 132.85 132.85 2. It could be argued that the costs for the party should be shared on the assumption that each host was paying for themselves and their friends. This would mean that, for the 12 people present, the bill should be shared in the ratio S: T: N = 3.5: 4.5: 4 = 7: 9: 8 3.5 4. 5 3.5 Sophie: × $78.50 = $22.90 Tye: × $78.50 = $29.44 Nik: × $78.50 = $26.17 12 12 12 3. Sophie and Tye were there for all 91 days but Nik was there for only 58 days. If the electricity bill was shared in the ratio of days present then the shares would be: 91 58 Sophie: × $280.65 = $106.41 Tye: $106.41 Nik: × $280.65 = $67.82 240 240 Alternatively, Sophie and Tye could halve the charges for the first 33 days and all three could share the charges for the remaining 58 days. On this basis: $280.65 Average daily charge = = $3.084 , which is more than the $2.80 quoted on the bill. 91 1 1 1 Sophie: × 33 × $3.084 + × 58 × $3.084 = $110 .51 Tye: $11 . 1 05 Nik: × 58 × $3.084 = $59.62 2 3 3 The weakness in this second calculation is that it assumes no increase in daily usage once Nik joins the household. 4. Total rent payment = 2 x $130 = $260. Equal shares for three people are $260 ÷ 3 = $86.67 consumer affairs victoria www.consumer.vic.gov.au 65 continued C1 Sharing the bill Extension/revision 1. Answers will vary. Ask students for a scenario and discuss alternative arrangements for payment. For example: Two people agree to share a bill of $131.50 (including GST) in the ratio 2: 3. For five meals that turns out to be between $25 and $30 per head. At $27.50 per head the payments would be $55 and $82.50, totalling $137.50. A tip of $6 is just under 5% of the bill. At $26 per head the payments would be $52 and $78, totalling $132. If the two people have their credit cards handy and are feeling generous then they could ask the waiter to add a 10% tip and then charge their credit cards in the ratio 40% to 60%. The actual charges should then be approximately 40% of $145 ≈ $58 and 60% of $145 ≈ $87. 2. Total number of equal parts = a + b + c. ax bx cx Shares of xin the ratio a: b: c are , and a+b+c a+b+c a+b+c 3. Answers will vary. Use the opportunity to ask students to collect data on phone bills and water bills in preparation for the upcoming lessons. As regards electricity charges raise the question of how much electricity would be used over a week in which all household members were away on holiday. (How much is saved if water heating is turned off?) 4. If Sophie, Tye and Nik fall behind with their rent payments then they need to quickly check their rights under the Residential Tenancy Act. Further advice and support is available through the Consumer Affairs website and the Tenants’ Union of Victoria. The reason behind their inability to pay is something they may need to explain. 5. a) 9 x $20 = $180, so one share should be about $2 less than $20 Dividing $159.95 by 9 gives $17.55, making the other share $140.40. b) Half of $160 is $80, so one share should be about $70 and the other about $90. Doubling $17.55 twice gives $70.20. By subtraction, the other share is $87.75 c) One tenth of $110 is $11. The shares are $11, $33 and $66. d) One tenth of $250 is $25. One tenth of 10 cents is one cent. The shares are proportionately just less than $25, $75 and $150, namely $24.99, $74.97 and $149.94 e) One sixth of $242 is about $40 and double that is about $80. One half of $242 is $141. Simple division gives shares of $40.33, $80.67 and $141. f) One tenth of $189 is $18.90. Doubling that gives $37.80 and then $75.60. The other share is found by multiplying $18.90 by 3. Answers: $18.90, $37.80, $56.70 and $76.60 g) The ration 2.4: 3.6: 4.8 = 2:3: 4, with a total of 9 equal parts. Approximations are $30, $45 and $60. Actual values are $29.93, $44.89 and $59.86 6. Parts c, d and f are easy because they involve dividing by 10. Halving and/or doubling can be used in parts c, d, e and f. Subtracting from the total makes parts a and b relatively easy. 7. Actual values are given above. Estimates given in Question 1 should be sufficiently accurate to show up any major mistakes in calculator work or written calculations. consumer affairs victoria www.consumer.vic.gov.au 66 C2 Home phone bills Talk time 1. Plan A New Price Usage (one month) Charge Locals Calls 20c per call 125 calls $25.00 Capped STD Calls $2.25 per call between 7pm 18 calls $40.50 (4pm Sat) and midnight Capped Calls $2.25 per call between 7pm 4 calls $9.00 to Same Company Mobiles (4pm Sat) and midnight Calls 33c /min 40 min $13.20 to Same Company Mobiles (24 hrs, 7 days a week) Calls 37c /min 35 min $12.95 to Other Company mobiles (24 hrs, 7 days a week) Monthly Line Rental $23.50 $23.50 Total $124.15 2. Plan B New Price Usage (one month) Charge Locals Calls 30c per call 125 calls $37.50 Capped STD Calls $3.00 per call between 7pm 18 calls $54.00 (4pm Sat) and midnight Capped Calls $3.00 per call between 7pm 4 calls $12.00 to Same Company Mobiles (4pm Sat) and midnight Calls 37c /min 40 min $14.80 to Same Company Mobiles (24 hrs, 7 days a week) Calls 42c /min 35 min $14.70 to Other Company mobiles (24 hrs, 7 days a week) Monthly Line Rental $17.50 $17.50 Total $150.50 a) Plan B would be cheaper provided less than 60 local calls were made in the month. Clearly Plan B is for people who wish to have a telephone but use it very little. This solution could be obtained by systematic trials. The algebraic solution is provided below in the solution to Question 3. b) For 60 calls the monthly charge would be $17.50 + 60 x $0.30 = $35.50. $35.50 Price per call = ≈ 59 cents. 60 c) Let n be the number of local calls made. For equal total charges 75 . 35 . 1 . + 0 3n < 2 . + 0 2n Subtract 6 + 0 2n from both sides: . . 0 1n < 6 Multiply both sides by 10: n < 60 The interpretation is that Plan B would be cheaper provided less than 60 local calls were made in the month. consumer affairs victoria www.consumer.vic.gov.au 67 continued C2 Home phone bills Extension/revision 1. If n is the number of minutes charged for calls to same company mobiles then Plan B will be cheaper for 1 . + 0 3 < 2 . + 0 3 7 5 . 7n 3 5 . 3n Subtract 1 . + 0 3 from both sides: 0 0 < 6 7 5 . 3n . 4n Multiply both sides by 25: n < 150 2. Answers will vary, but necessary calls are most likely to be calls to mobiles. For Plan B to still be cheaper these would have to be relatively few and the number of $3.00 calls would have to be very few indeed. As an extension question students could be asked: What if one third of all calls were local calls, one third were one minute calls to same company mobiles and one third were to other company mobiles? Algebraically this leads to 1 . + 0 3n + 0 3 + 0 4 < 2 . + 0 2n + 0 3 + 0 3 75 . . 7n . 2n 3 5 . . 3n . 7n Collect like terms: 7 5 . 9n 3 5 . 0n 1. +10 <2. +09 Subtract 1 . + 0 9n from both sides: 75 . . 9n 01 <6 Divide both sides by 0 1 : .9 n<3.16 Thus Plan B would be cheaper than Plan A for an average of one call per day of each type. 3. Answers will vary. Plan B is typical of the plans offered to concession card holders. Concessions given are often a partial reduction in the monthly line rental and a separate reduction in the cost of calls. 4. Answers will vary, with some involving plans that are very similar to Plan A. consumer affairs victoria www.consumer.vic.gov.au 68 C3 Mobile phone bills Calculate the bill Answers to questions 1 and 2 will vary. 3. Encourage students to enter their real data, such as from a recent bill of their own. Realistic reports could then recommend a change of service provider for the phone concerned. Extension/revision 1. To check the accuracy of the ‘Phonechoice’ bill calculator, students should develop simple usage scenarios and repeat the calculations themselves. The summary statistics can be used for calculating averages. 2 & 3. Answers will vary. The second question could focus attention on the situations in which possession of a mobile phone could be regarded as essential. 4. Answers will vary. Stimulate a class discussion. consumer affairs victoria www.consumer.vic.gov.au 69 C4 Mobile call charges Analyse the call charges Average time per call = 2hours:17 minutes: 30 seconds ÷ 89 = 1 minute: 33 seconds. Average call charge = $59.66 = $0.67 89 The use of these averages is limited, mainly because the data come from essentially different sub-populations, for example mobile calls and local calls to the National Grid. In a similar way giving separate average heights for adult males and females is more useful than for just giving one average height for adults. An initial analysis could involve sorting the data into categories according to Destination. Further analysis of Rates sub-categories might also be useful. Total and average costs and times for each category are: National Div-VM Mobile Weekend VM Totals Number 14 38 22 5 10 89 Cost, $y 15.73 1.95 37.73 1.35 2.9 59.66 tmin 27.5 19.5 71.5 6.5 12.5 137.5 / yn $1.12 $0.05 $1.72 $0.27 $0.29 $0.67 / tn 1.96 0.51 3.25 1.30 1.25 1.54 / yt $0.57 $0.10 $0.53 $0.21 $0.23 $0.43 More usefully, as percentages: National Div-VM Mobile Weekend VM Totals Number 15.7 42.7 24.7 5.6 11.2 100 Cost 26.4 3.3 63.2 2.3 4.9 100 Time 20.0 14.2 52.0 4.7 9.1 100 Analysis of costs ($y) in relation to time (tminutes) can be made for each major category, using either tables or scatter plots. Results are: – for calls to mobiles: y = 0 2 + 0 2 . 3t . 2 – for calls to the National grid: y = 0 23 + 0 2 . t .2 – for Div-VoiceMail: y = 0 1t . – for weekends, y=0.27, a constant charge for calls regardless of duration. For analysing the spread of call times it would be more useful to limit the analysis to calls to mobiles and the National Grid. Since times are measured in multiples of 30 seconds it is more appropriate to use a relative frequency histogram rather than a line plot or box plot. The summary table for National Grid and mobile calls combined is Time (min) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 – 9.0 9.5 Nat 2 3 1 2 3 1 1 1 Mob 5 3 1 1 1 2 2 1 0 2 1 – 2 1 Combined frequency 7 8 2 3 4 3 3 2 0 2 1 – 2 1 % age frequency 19% 17% 6% 8% 11% 8% 8% 6% 6% 3% 6% 3% consumer affairs victoria www.consumer.vic.gov.au 70 continued C4 Mobile call charges Extension/revision 1. In summary, approximately 50% of calls were less than 2 minutes, one third of all calls were between 2.5 and 4 minutes and 18% of calls – all to mobiles – were 5 minutes or more. Some recommendations for Sophie are: – Were those calls over 5 minutes to mobiles absolutely necessary? – Can you make more use of Weekend Rates? – Now that you know the pattern of your mobile use you can go to the Phonechoice website to see how your current plan compares with the alternatives. 2. Answers will vary 3. Sophie used her mobile phone 89 times during the month. 4. The total Amount Due for the month was $107.36 5. Therefore the average cost for each time the phone was used was $1.21 6. This average covers calls of very different types and costs. 7. The ratio of total charge to cost of calls is Hence all cost statistics and formulae should be scaled up by this factor. For example, the full cost of a call to a mobile would be given by y = 1.8(0.23t + 0.22) y = 0.414t + 0.40, i.e. 40 cents to make the call, and then 41.4 cents per minute. consumer affairs victoria www.consumer.vic.gov.au 71 C5 Water bills Turn off that tap! Answers to Questions 1 and 2 will vary. Extension/revision 1. Answers will vary. 2. A typical pensioner concession scheme is as follows: The maximum rebate per year is $135.00. The maximum of $33.76 per bill is granted on the Water and Sewerage Charges as follows: – A rebate of $10.55 on the Water Service Charge and a concession of up to 50% of the Water Usage Charges, up to a combined total of $16.88. – A rebate of $11.47 on the Sewer Service Charge and up to 50% of the Sewage Disposal Charge, up to a combined total of $16.88. 3. New homes are required to install water saving amenities. Support and subsidies are available for using rainwater and grey water. A more radical suggestion would be to raise the per kilolitre charge for households using large amounts of water. 4. Amount of water required to ‘pay for’ tank installation = 346.2 kilolitres. 5. Obtain approximate daily averages from the graph provided. Then obtain average weekly per person amounts by multiplying the daily figures (for two people) by 7 and then dividing by 2. Daily averages Weekly averages for two people per person April quarter 1150 litres approx 3925 litres July quarter 400 litres approx 1400 litres October quarter 550 litres approx 1925 litres January quarter 2350 litres 8225 litres Averages for the year 4450 ÷ 4 = 1110 litres 3885 litres 6. Answers will vary. 7. Answers will vary. consumer affairs victoria www.consumer.vic.gov.au 72 D1 Credit cards Credit is debt 1. a) Tye owed $144.35 at the beginning of the statement. b) Total purchases for the period were $262.03. c) To avoid credit charges the full $262.03 would need to be paid by the due date. If payment is any later then interest is calculated from the Statement issue date. 2. Amount still owing = $262.03 - $25 = $237.03 Time = 32 days a. Interest charged = $237.03 x 0.0004397 x 32 = $3.34 b. For the next statement: Opening balance $262.03 New charges $230.00 Interest $3.34 FID etc $0.73 Payments $25.00 Closing balance $471.10 3. For the cash advance: a) Interest charge (27 days) = $100 x 0.0004397 x 27 = $1.19 b) Interest charged (52 days to 29 Aug) = $100 x 0.0004397 x 52 = $2.29. consumer affairs victoria www.consumer.vic.gov.au 73 D1 Credit cards Extension/revision 1. Answers will vary. They might include: – Avoid impulse buying. – The card’s credit limit is not the same as your own credit limit. – Take advantage of the ease of payment and the interest-free period. – Report quickly if your credit card is lost or stolen. 2. Answers will vary as some are individual judgments. For instance, some students will see credit cards as an advantage. 3. For Tye’s credit card: a) Minimum amount due = 2.5% of $1225.00 = $30.63 b) 2.5% of $1000 = $25.00, the minimum amount due on $1000. c) Minimum amount due = 2.5% of $2000 + $15 = $65.00. 4. Credit card interest calculations: a) $250 x 0.0004452 x 45 = $5.01 b) $350 x 0.0004452 x 33 = $5.14 5. Bank interest calculations: a) A 6% annual rate is equivalent to a 0.01644% daily rate. $250 x 0.0001644 x 45 = $1.85 b) An 8% annual rate is equivalent to a 0.02192% daily rate. $350 x 0.0002192 x 33 = $1.90 6. Major problems with credit cards could include: a) paying for transactions made on a lost card in the time before the loss is reported b) the ease of use makes it all too easy to overspend c) being persuaded to spend more on rewards programs d) banks frequently offering higher spending limits e) interest charges are at high rates and can mount up, particularly if, for example, you change your address and fail to receive statements f) annual charges, government charges and other credit card charges can be significant g) Interest on cash advances is calculated immediately and at the high credit card rate. consumer affairs victoria www.consumer.vic.gov.au 74 E1 Interest calculations Flat or reducing? 1. Principal, P = $800 Rate per annum, r= 6% = 0.06 Time, t= 4 year a) I= Prt= $192 b) Compounding rate, R = 106% = 1.06 Number of time periods, n = 4. A = P + I= PRn, so I= P R – 1 = $800 (1.064 -1) = $209.98 (n ) c) Compounding rate, R = 103% = 1.03 Number of time periods, n = 8. I= P(Rn – 1) = $800 (1.038 - 1) = $213.42 Some students may not have used the formulas in answering parts a, b and c. The key part of their explanations should be that the amount of simple interest is proportional to the constant principal, whereas the new amount (total value) in a compound interest investment grows at a constant factor ( R ) times the previous amount. 2. The solution to Part a. is provided in full: a) Starting equation I=P rt I Pr t Divide both sides by rt = =P rt rt I Transpose from a = b to b = a P= rt I b) r= Pt I c) t= P×r I 200 d) r= = = 0.05 = 5% Pt 800 × 5 3. The relationship is R = 1 + r or R = 100% + r. Thus, if r = 5%, then R = 105%. , A = P + I or I = A – P. The interest is the difference between the final amount owing (A) and the initial amount owing, which is called the principal (P). 4. a) Starting equation A = PRn A Divide both sides by Rn =P Rn A Transpose from a = b to b = a P= Rn consumer affairs victoria www.consumer.vic.gov.au 75 continued E1 Interest Calculations b) Starting equation A = PRn A Divide both sides by P = Rn P A Take nth root of both sides n =R P A Transpose from a = b to b = a R = n P A 5 1000 c) R = n = = 1.0456 = 104.56%, so the annual rate is 4.56%. P 800 5. a) Law of indices: ( m ) = amn a n Here ( 00.02531) = 100.02531 x n 1 n Law of indices: am x an = a(m + n) Here 102.9031 x 1 0.02531 x n =102.9031 + 0.02531 x n 0 b) Here 103.0802 = 102.9031 + 0.02531 x n Equal indices (or logs base 10): 3.0802 = 2.9031 + 0.02531 x n Subtract 2.9031 from both sides: 0.1771 = 0.02531 x n Divide both sides by 0.02531 and 0.1771 transpose from a = b to b = a n= = 6.997 ≈ 7 0.02531 The interpretation is that it will take 7 years for an initial $800 to amount to $1202.90 when it is invested at 6% compounded annually. c) Starting with: A = PRn By definition: o 10l gA=10logP x ( 0logR) 1 n n n oR logR) = 10n x l g Law of indices: ( m ) = am x n a n 1 R = (0 Law of indices: am x an = a(m + n) Here 10logP x 10nlogR = 10logP + nlogR Equal indices (or logs base 10): Subtract logP from both sides: o o o l gA = l gP + nl gR Divide both sides by logR o o o l gA – l gP = nl gR and transpose from a = b to b = a n = log A − log P log R A 1000 = d) Using R = n gvs R = 4 ie R = 1.25 = 107 (by square root of .53 P 800 square root.) 1.0573 = 105.73% = 100% + 5.73%, so the annual rate of compound interest = 5.73% consumer affairs victoria www.consumer.vic.gov.au 76 continued E1 Interest calculations Extension/revision 1. The answers to Question 1 on the worksheet ‘Flat or reducing’, illustrate that, at the same annual rate, the lowest amount of interest is simple interest and the highest amount is compound interest that is calculated on the most frequent basis. The differences arise from the fact that compound interest is calculated on the increasing value of the investment, which increases most quickly when the interest is calculated most frequently. In Question 4d the 5th root could be calculated by using logarithms, by using the key or by using the xn key (with n = 0.2). In Question 5b students could use logarithms as suggested or they could use systematic trials of n values with the xn key. In Question 5d the 4th root could be found by using x. 2. Answers will vary, but students should be able to identify all 11 formulas. 3. By mental estimation a) 0.05 x 4 x $1000 = $200: approximately 10% + 10% = 20% under the true value. An improved approximation would be $240. b) With compound interest the value should be slightly higher, about $250. c) At 5% simple interest money would be doubled in 20 years. At 5% there would be an approximately 10% difference, giving about 18 years. d) With compound interest the value would double more quickly. There is a ‘rule of 70’ that gives a good approximate answer, namely ‘rate x doubling time = 70’. In this case that would give a doubling time of 70 ÷ 5 = 13 years. 4. By calculator: a) Simple interest = $1100 x 0.055 x 4 = $242 b) Compound interest = $1100(1.0554 – 1) = $262.71 c) To double the value by simple interest solve r x t = 1, in this case giving t = 18.18 d) To double the value by compound interest solve 2 = 1.055n. Here years 5. Mental estimates should be within about 20% of the calculated answers, close enough to act as checks against major mistakes in the calculations. consumer affairs victoria www.consumer.vic.gov.au 77 E2 Buying a car The hidden costs 1. a) The phrase ‘8% per annum, calculated quarterly’ is a commonly used way of trying to say 2% calculated quarterly. This is despite the fact that 1.024 is not quite equal to 1.08. b) The calculator check should ensure that the spreadsheet has been correctly developed c) An extract from the spreadsheet shows: Time Amount owing Plus interest Minus repayment 15 $359.35 $366.54 $181.54 16 $181.54 $185.17 $0.17 17 $0.17 $0.17 -$184.83 This shows that after 16 quarters (4 years) the loan is paid off (all but 17 cents). d) Total payments = 16 x $185 = $2960 Total interest = $2960 - $2512 = $448 I 448 Flat rate of interest r = = = 0.0446 = 4.46% P × t 2512 × 4 e) Set the values $2500 and $755 into the appropriate cells in the spreadsheet and try different values for R until the amount owing at the end of the time = 4 row is zero. The spreadsheet should eventually show: A B C D 1 Principal, P = $2,500.00 2 Rate, R = 1.08 3 Repayment, Q = $755.00 4 5 Time Amount owing Plus interest Minus repayment 6 at the start 7 1 $2,500.00 $2,700.00 $1,945.00 8 2 $1,945.00 $2,100.60 $1,345.60 9 3 $1,345.60 $1,453.25 $698.25 10 4 $698.25 $754.11 -$0.89 11 5 -$0.89 -$0.96 -$755.96 Thus the effective rate of interest is very close to 8% per annum. f) Total payments = 4 x $755 = $3020 Total interest = $3020 - $2500 = $520 I 520 Flat rate of interest r = = = 0.052 = 5.2% P × t 2500 × 4 consumer affairs victoria www.consumer.vic.gov.au 78 continued E2 Buying a car 2. Using the formulas gives Value of n Actual E Actual F E by formula F by formula Questions 1c, 1d 16 8% 4.46% 8% 8 × 17 % = 4.25% 32 Questions 1e, 1f 4 8% 5.20% 5.2 × 8 5.20% % = 8.32% 5 Extension/revision 1. Comparing effective rate with flat rate: a) The formulas are more accurate for loans that are paid off over a long time in a large number of separate payments. b) For the salesperson the comparison percentage rate quoted needs to be accurrate to at least the first decimal place. Providing the other charges on the loan are very small the comparison rate will be close to the effective rate. 2. For Nik’s car loan: a) The amount Nik can pay back, A = 48 x $250 = $12,000 We need to solve for P in A = P(1 + rt). A $12000 P= = = $9090.91 1 + r × t 1 + 0.08 × 4 b) Nik should be advised that the bank’s effective rate is lower than that offered by the car trader. He should take out the personal loan of $4000, make allowance for repayments to the bank and then use some or all of the rest of his monthly $250 to decide on price and terms with the car trader. A spreadsheet calculation would show that the bank would require a monthly repayment of $89, so Nik would be able to use some or all of the remaining $161 to borrow from the car dealer an amount of up to A $161 × 48 P= = . = $5854 1 + r × t 1 + 0.08 × 4 Together with the $4000 this would allow Nik to buy a car with a sale price of up to $9854. Nik also needs to be given a few warnings, in particular about the conse- quences of falling behind with his monthly repayments to either the bank or the car trader. He also needs to be reminded that he is committing himself to monthly repayments of $250 per month for four years and a further $89 per month for a fifth year. Nik also needs advice about the costs of running a car. c) An initial approach to answering this important question is to refer to the Car Challenge at www.moneystuff.net.au. Students’ answers can be collated and then checked against the reference material on the website. consumer affairs victoria www.consumer.vic.gov.au 79 continued E2 Buying a car 3. Having this information, Nik could have expected to be told that the 2 × 48 × 8 comparison rate being offered by the car dealer was approximately % = 15.7% 48 + 1 This might have persuaded him to be more serious about trying to get a bank loan at a lower rate of interest. 4. a) Total payment is 36 x $620 = $22,320 (and she would give up the old car.) The flat rate of interest would be b) Using approximate formulas The 9% effective rate of interest would be equal to a flat rate of slightly more than 4.5%. The amount of interest saved in three years would be approximately This more than the covers the extra $500 offered on the old car as a trade-in. Using a spreadsheet: An extract from the spreadsheet, with P = 18500, R = 1.0075, Q = 620 shows: Time Amount owing Plus interest Minus repayment 32 $1,774.55 $1,787.86 $1,167.86 33 $1,167.86 $1,176.62 $556.62 34 $556.62 $560.80 -$59.20 Thus the $620 monthly payments would end with a final payment of $560.80 in the 34th month. Compared with 4a, this alternative would save 2 x $ 620 + $59.20 = $1299.20. Using the Savings Calculator A calculation of 33.9 months for the period of the loan, is consistent with the spreadsheet answer. Using this figure, the amount saved is (36 – 33.9) x $620 = $1302. consumer affairs victoria www.consumer.vic.gov.au 80 E3 Home loan Paying the mortgage 1. a) (n + 1) F 301 × 6 By the approximate method E = = = 3.01 2n 600 Then A = P (1 + r × n) = $180000(1 + 0.0301 × 25) = $315450 $315450 For which the monthly repayment would be Q = = $1051.50 300 Using 3% instead of 3.01% would give Q = $1050. b) By the more accurate methods Enter P = $180000, R = 1.005 into the spreadsheet and adjust Q until the amount owing at the end of 300 months is zero. This occurs for a value Q = $1159.74 c) The Savings Calculator can be used as a loans calculator if negative amounts are entered for the monthly payment. This value can be adjusted until the time is as close as possible to 25 years: How much do I start with? ($) 180000 How much can I save? ($) -1159.74 saved each month Interest rate (% per annum) 6 compounded/paid monthly How long do I have to save for? 25.0001259313341 years How much will I have at the end? ($) 0 d) By the annuities formula PR n (R − 1) $180000 × 1.005 300 × 0.005 Q= = = $1159 .74 Rn −1 1.005 300 − 1 The annuities formula does not require computer access and so is the most convenient of the accurate methods. Its disadvantage is that it relies on the mathematics of geometric sequences, which some people do not meet in their senior maths courses. The spreadsheet method is more easily understood and can be easily adjusted to deal with changes to the parameters, such as a change in interest rates. consumer affairs victoria www.consumer.vic.gov.au 81 continued E3 Home loan Extension/revision 1. If Nina can determine what monthly payment she can afford then the amount she could borrow under the same terms can be calculated. She needs to be aware of the fact that a small rise in bank interest rate can cause a significant increase in the monthly repayment required. 2. In Question 1 the approximate method gave a monthly payment that was an underestimate of approximately 10% of the correct amount. This approximate method gives reasonably accurate answers when the number of payments, n, is a relatively large number. This will be true if a loan is repaid using a large number of instalments. 3. Discussions will vary. 4. Information placed in the Venn diagram will vary. 5. Good reports could include: a) a discussion about the cost of loans b) a calculation of what size loan could be serviced by the amount that a renter is currently paying c) the advisability of investing in property d) the greater ease of shifting between rented properties. consumer affairs victoria www.consumer.vic.gov.au 82 Additional Resources Further support material for teachers and students is available from Consumer Affairs Victoria (CAV) and can be viewed and downloaded at www.consumer.vic.gov.au under ‘Publications’. Alternatively, an order may be placed by fax on (03) 8684 6333 using a general order form and/or the teacher order form provided on the website. Following is a list of useful fact sheets and other publications which CAV is continually updating and adding to. Check regularly to see what is new. IMPORTANT: If you have visited our website previously, please click the ‘refresh’ button on your toolbar. This will make sure that any new information added or updated since your last visit, becomes available to you. Fact sheets A range of fact sheets provide detailed information on many consumer issues including: – Your basic consumer rights – Shopping tips – Bag seachers – Lay-by – Credit and finance – Renting – E-commerce – Product safety Booklets and brochures Stuff magazine ‘Stuff’ contains advice for school leavers on many topics including: mobile phone contracts, managing money, consumer rights, car maintenance, careers and lifestyle. Better Car Deals - your guide for buying a new or used car Renting a Home A guide for tenants and landlords. Little black book of scams How to recognise scams and scammers and avoid being ripped-off. consumer affairs victoria www.consumer.vic.gov.au 83 Additional Resources Websites Victoria Consumer Affairs Victoria www.consumer.vic.gov.au Maths Association of Victoria www.mav.vic.edu.au Victorian Commercial Teachers Association www.vcta.asn.au Victorian Association for the Teaching of English www.vate.org.au youthcentral www.youthcentral.vic.gov.au Department of Sustainability and Environment www.dse.vic.gov.au/dse The Consumer Credit Legal Service www.ccls.org.au Victorian Legal Aid (What’s the Deal?) www.legalaid.vic.gov.au Victorian Civil and Administrative Tribunal www.vcat.vic.gov.au New South Wales www.moneystuff.net.au South Australia www.b4usplashcash.ocba.sa.gov.au Queensland www.fairtrading.qld.gov.au Western Australia www.docep.wa.gov.au ACT www.fairtrading.act.gov.au Northern Territory www.nt.gov.au/justice Commonwealth Financial Literacy Foundation www.understandingmoney.gov.au Australian Consumers Association (Choice Magazine) www.choice.com.au Australian Communications and Media Authority www.acma.gov.au Australian Securities and Investment Commission (Fido) www.fido.asic.gov.au/fido Other resources Shopsafe CD-ROM Advice about shopping safely online. Your feedback Consumer Affairs Victoria welcomes your comments and suggestions regarding the ‘Consumer Stuff’ resources. Email consumerstuff@justice.vic.gov.au Please complete the evaluation form at the back of this handbook and forward to the address shown. consumer affairs victoria www.consumer.vic.gov.au 84 Resource book evaluation sheet Consumer Affairs Victoria thanks you for using the Mathematics resource book, and would appreciate you taking a few minutes to complete this evaluation sheet. Your feedback is very important to us. Any comments and suggestions for improvement will be appreciated. 1. Tick the resource book you are providing feedback on? (Please tick) Consuming Planet Earth English Mathematics Health & Wellbeing Commerce 2. Which sections of the resource book have you used with your students? (Please tick) SECTION A SECTION D SECTION B SECTION E SECTION C 3. Which of the sections have your found most useful and relevant to your course? (Please tick) SECTION A SECTION D SECTION B SECTION E SECTION C 4. Which Year level have you used the material with? (Please tick) Year 7 Year 10 Year 8 Year 11 Year 9 5. How would you rate the relevance of the curriculum material to the Victorian Essential Learning Standards? (Please tick) Very relevant Relevant Partly relevant Not relevant at all consumer affairs victoria www.consumer.vic.gov.au 85 Resource book evaluation sheet 6. How would you rate the ‘user friendliness’ of the resource book in terms of: a) photocopiable worksheets Excellent Very good Good Passable Poor b) teacher notes Excellent Very good Good Passable Poor 7. Suggestions for improvement You may wish to suggest some new topics which could be added to the online version of the handbook. Please outline these below. 8. Any other suggestions (for example, additional teacher notes, activities, resources). Contact details (optional): Name School Phone Fax Email Please send us this evaluation by fax or post. Should you have any other suggestions in the future, please contact us: Email consumerstuff@justice.vic.gov.au Phone (03) 8684 6042 or (03) 8684 6043 Fax (03) 8684 6440 Mail Consumer Affairs Victoria Education and Information Branch GPO Box 123A Melbourne, VIC 3001 Thank you for your time and valuable feedback! consumer affairs victoria www.consumer.vic.gov.au 86