# CEIS Maths - Consumer Stuff by sofiaie

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55
A1
Budgeting by the month

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
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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
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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
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A2

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
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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
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
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B2

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

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
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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.

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
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C5
Water bills

Turn off that tap!
Answers to Questions 1 and 2 will vary.

Extension/revision

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

consumer affairs victoria                                                    www.consumer.vic.gov.au
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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
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
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D1
Credit cards

Extension/revision
They might include:
– 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,
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
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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
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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

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.

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

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

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
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.

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
(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 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

Further support material for teachers and students is available from Consumer
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:
– 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

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

Western Australia
www.docep.wa.gov.au

ACT

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

regarding the ‘Consumer Stuff’ resources.
Email consumerstuff@justice.vic.gov.au

Please complete the evaluation form at the back of this handbook and forward

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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?

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?

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

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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.

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!

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