Tariff systems
 Definition 1: Tariff
 A list or schedule of predetermined prices for services like trains, busses, and electrical usage.
Tariffs are predetermined rates that are used to determine how much money you owe for a given service. For example, the total amount of money you owe on your municipal bill is calculated using predetermined tariffs. A tariff usually includes a unit of payment with a unit of measurement  for example cell phone bills are charged in cents or rands per minute (R/minute) and electricity is charged in cents per kilowatt hour or c/kWh. Telkom charges monthly landline rental in Rands per month.
In this section you will work with different tariff systems, learn how to calculate costs using given tariffs and how to draw and interpret graphs of various tariff systems.
Municipal tariffs
Municipal tariffs include tariffs for electricity, water and refuse removal. They differ from one city to the next, but generally speaking, the more electricity or water you use in a month, the more expensive it is per unit. Sometimes municipal tariffs are also called “rates”. Your municipal bill also usually includes a monthly property rate based on the municipal value of your house  again, the more your house is worth, the more you will pay in rates.
Example 1: Working with tariffs
Question
The City of Durban's Ethekwini Municipality charges the following tariffs for domestic water usage in 2013, for properties valued at more than R 250 000:
Price per kilolitre, excluding VAT 

0 kl to 9 kl 
R 9,50 
from 9 kl to 25 kl 
R 11,22 
From 25 kl to 30 kl 
R 14,95 
from 30 kl to 45 kl 
R 23,05 
more than 45 kl 
R 25,36 
Megan's monthly water consumption is 12 kl. Calculate her monthly water costs, excluding VAT.
Gilbert's monthly water consumption is 32 kl.
 Calculate what his monthly water bill will be, including VAT.
 Calculate what Gilbert will pay on average, per kilolitre of water.
 Study the following graph, showing the relationship between the quantity of water used and the price per unit, as given in the above table:Why do you think the graph is a series of horizontal lines and sudden vertical rises?
Answer
 To calculate Megan's total water costs, we break up her usage into the different tariff categories:
In the 0 kl  9 kl category, she used 9 kl of water
In the 9 kl  25 kl category, she used 3 kl of water (9 kl + 2 kl = 12 kl)
Now we calculate how much the number of kilolitres used in each category cost: 9 kl \(\times\) R 9,50 = R 85,50
3 kl \(\times\) R 11,22 = R 33,66
Next we add these two subtotals together:
R 85,50 + R 33,66 = R 119,16
So her water usage for the month will cost R 119,16, excluding VAT  To calculate Gilbert's total water bill, we break the 32 kl up into the different tariff categories listed in the table above.
In the 0 kl  9 kl category, he used 9 kl of water
In the 9 kl  25 kl category, he used 16 kl of water (9 + 16 = 25 kl)
In the 25  30 kl category, he used 5 kl of water (25 + 5 = 30 kl)
In the 30  45 kl category, he used 2 kl of water (30 + 2 = 32 kl)
Now we calculate how much the number of kilolitres used in each category cost:
9 kl \(\times\) R 9,50 = R 85,50
16 kl \(\times\) R 11,22 = R 179,52
5 kl \(\times\) R 14,95 = R 74,75
2 kl \(\times\) R 23,05 = R 46,10
Next we add these subtotals together to get the total, excluding V.A.T:
R 85,50 + R 179,52 + R 74,75 + R 46,10 = R 385,87
Lastly, we calculate what the 14% VAT on the above total is, and add it to the total to get the final amount:
14% of R 385,87 = R 385,87 \(\times \frac{\text{14}}{\text{100}} =\)R 54,0218
R 385,87 + R 54,0218 \(\approx\) R 439,89
So Gilbert's total water costs will be R 439,89. (Note how we only round down at the last step, to make the calculation more accurate.)  Gilbert pays R 439,89 for 32 kl of water.
\(\frac{\text{R }\text{439,89}}{\text{32}\text{ kl}} \approx \text{R }\text{13,75}\) per kilolitre.
 To calculate Gilbert's total water bill, we break the 32 kl up into the different tariff categories listed in the table above.
 Because the price stays the same from 0 kl to 9 kl, for example, a horizontal line is drawn between 0 and 9. The dotted vertical lines show us how the price suddenly increases when we move into the next group of kilolitres (9 25 kl). The same applies for 9 
25 kl, 25  30 kl etc. We cannot join the points with a sloping line, because this suggests the price per unit rises with each kilolitre, which is not the case, rather the price is constant for a given range of kilolitres (and therefore the graph is flat) and
then increases suddenly for each new category of water usage.
Exercise 1: Calculating costs using given municipal tariffs
Domestic electricity in the City of Cape Town is charged for using the tariffs below, for households who use more than 450 kWh of electricity per month. They refer to each category of electricity usage as a block, and the tariffs are charged in cents per kilowatt hour (kWh). VAT is included in the tariff costs listed below.
Block number (kWh) 
cents per kWh (incl VAT) 
Block 1 (0  150 kWh) 
129,05 
Block 2 (150,1  350 kWh) 
134,65 
Block 3 (350,1  600 kWh) 
134,65 
Block 4 (\(>\) 600 kWh) 
159,81 

If Jason's household uses 140 kWh of electricity in a month, calculate what his electricity bill will be, in Rands.

Thomas uses 200,5 kWh of electricity in a month. What will his electricity costs be in Rands?

The City of Cape Town decides to introduce a fixed additional tariff for anyone who uses more than 350 kWh of electricity. This fixed cost is R 24,45 per month. (I.e. the cost is R 24,45 plus the price per units used). If Neil uses 423 kWh of electricity in a month, calculate what his total electricity cost will be.
140 \(\times\) 129,05 c = 18 067 c = R 180,67
Block 1:140 \(\times\) 129,05 c = 18 067 c = R 180,67. Block 2: 200,5  140 = 60,5 kWh. 60,5 \(\times\) 134,65 c = 8146,325 c = R 81,463 25. R 180,67 + R 81,4572 = R 262,133 25 \(\approx\) R 262,13
Basic cost: R 24,45. Block 1:140 \(\times\) 129,05 c = 18 067 c = R 180,67. Block 2: 350  150,1 = 199,99 kWh. 199,99 \(\times\) 134,65 c = 26 928,6535 c = R 269,286 535. Block 3: 423 kWh  350,1 kWh = 72,9 kWh. 72,9 \(\times\) 134,65 c = 9815,985 c = R 98,159 85. R 24,45 + R 180,67 + R 269,286 535 + R 98,159 85 = R 572,566 385 \(\approx\) R 572,57.
Telephone tariffs
Fixed phone line and cell phone service providers also commonly charge you for calls using tariffs. The tariff cost can be influenced by factors like the distance of the call (e.g. calls overseas are always more expensive than local calls) and the time of day (most service providers offer cheaper rates in the evenings, depending on which phone package you use).
Example 2: Working with telephone tariffs
Question
Telkom charges for normal, national landline calls according to the following tariffs:
Distance 
Basic, fixed charge in Rands (incl. VAT) (regardless of length of call or distance) 
Rands per second (incl. VAT) 
Rands per second (incl. VAT) 
All times 
Standard Time: Mon  Fri 07:00 to 19:00 
Callmore time: Mon to Fri 19:00  7:00 and Fri 19:00  Mon 7:00 

Local (0  50 km) 
0,570 
0,007 00 
0,003 44 
Long distance 
0,570 
0,009 50 
0,004 75 
 What two factors can influence the cost of a landline call, based on the information above?
Davina calls her mother, who lives 25 km away, during standard time.
 If she talks for 1000 seconds, what will she pay for the call, in Rands?
 For how many minutes did they talk? (Round your answer down to the nearest minute).
 Graham calls his aunt, who lives 100 km away, during Callmore time. They talk for 5 minutes. How much will the call cost him?
 Karabo calls his friend at 5 p.m. on a Saturday. They chat for 10 minutes. If his friend lives 5 km away, calculate how much the call with cost Karabo.
Answer
 The distance of the call, and the duration (number of seconds)

 Cost of call = minimum charge + (Rands per second) \(\times\) (number of seconds)
= R 0,570 + (R 0,007 00) \(\times\) 1000
=R 0,570 + R 7,00
=R 7,57  1000 seconds \(\div\) 60 seconds \(\approx\) 16 minutes
 Cost of call = minimum charge + (Rands per second) \(\times\) (number of seconds)
 Cost of call = minimum charge + (Rands per second) \(\times\) (number of seconds)
= R 0,570 + (R 0,004 75) \(\times\) (5 minutes)
= R 0,570 + (R 0,004 75) \(\times\) (5 minutes \(\times\) 60)
= R 0,570 + (R 0,004 75) \(\times\) (300 seconds)
= R 0,570 + R 1,425
= R 1,995
\(\approx\) R 2,00  5 p.m. on a Saturday is during Callmore time, therefore we use the tariffs given in the last column of the table.
They chat for 10 minutes which is 10 minutes \(\times\) 60 = 600 seconds.
Cost of call = minimum charge + (Rands per second) \(\times\) (number of seconds)
= R 0,570 + (R 0,004 75) \(\times\) (600 seconds)
= R 0,570 + R 2,85
= R 3,42
Exercise 2: Interpreting and comparing graphs of a tariff system
A local cellular provider charges the following for a standard contract:
Monthly subscription: R 100
Mandatory itemised billing: R 22
Calls and SMS's are charged for using the tariffs given below. This service provider uses per second billing, so the tariff is Rands per 60 seconds of call time, even if those 60 seconds are split over two or more calls. Assume that these tariffs, or rates, and the monthly subscription are VAT inclusive. (An SMS is a text message, and an MMS is a text and data message, that may include a photo, for example).
Rate per minute for the first 5 minutes of the day 
R 1,95 
Rate per minute (60 seconds) thereafter 
R 1,55 
Calls to same network 
R 0,99 per 60 seconds 
International SMS 
R 1,20 per SMS 
SMS 
R 0,60 per SMS 
MMS 
R 0,75 per MMS 

Alfred has a contract like the one above. Assuming he does not use more than R 140 airtime in a month and R 40 worth of SMS's, what will his monthly cell phone bill cost?

On the first day of the month, the first call Alfred makes is to his father (on a different network) and they talk for 2 minutes. How much will this call cost Alfred?

On the same day, Alfred then calls his friend, Ivan. They talk for 4 minutes. How much will this second call cost him?

Two weeks later, Alfred has made R 70,45 worth of calls, sent 25 local SMS's, 5 international SMS's and 2 MMS's. Calculate how much airtime he has left.

Alfred calls his sister for a big chat, and accidentally uses all of his R 140 airtime in one day (assume this happened at the beginning of the month and he did not send any SMS's). If she is on a different network, how long did they talk for? Round your answer to the nearest minute, and write it in hours and minutes.
Monthly subscription + itemised billing = R 122
R 1,95 \(\times\) 2 = R 3,90
After the first call, Alfred has used 2 of the first 5 minutes of calls. The next call will cost (3 \(\times\) R 1,95) + (1 \(\times\) R 1,55) = R 7,40
Alfred starts with R 140 worth of airtime. We know he has spent R 70,45 on calls. The local sms's are covered by his monthly contract so we do not deduct them from his airtime (he will have 15 free local sms's left.) The 5 international sms's cost 5 \(\times\) R 1,20 = R 6,00 and the 2 MMS's cost 2 \(\times\) R 0,75 =R 1,50. Now we deduct all these costs from the initial amount or R 140: R 140  R 70,45  R 6,00  R 1,50 = R 62,05. He has R 62,05 airtime left.
The first 5 minutes cost: R 1,95 \(\times\) 5 = R 9,75. This leaves him with R 140  R 9,75 = R 130,25 airtime. The remaining minutes will cost R 1,55 each. R 130,25 \(\div\) R 1,55 = 84 minutes = 1 hour and 24 minutes.
Transport tariffs
Transport tariffs work much the same as municipal and phone tariffs. They may apply to train travel, busses and taxis. Generally, transport tariffs are charged per unit of distance, and the further you need to travel, the higher the tariff and therefore the more expensive your ticket. Like some basic foods, transport fares for taxis, buses and trains in South Africa are exempt from VAT.
Example 3: Working with transport tariffs
Question
The municipal MyCiTi bus system in Cape Town charges the fares below. For all routes except the Airport route, you need to have a myconnect card. For the airport route, you can still use paper tickets, which you can buy at the bus stops at either end of the route.
Cost 
Route name 

R 23 
myconnect card 

R 5,30 
Gardens  Civic Centre  Waterfront Big Bay  Table View  Parklands East Parklands East  Table View  Blouberg Sands Marine Circle  Table View  Blouberg Sands 
F1 F14 F15 F16 
R 10,60 
Table View  Civic Centre 
T1 
Free 
Children under 1 m tall and under 4 years old 
(all routes) 
Airport Service
Cost 
Route name 

R 57 
Civic Centre  Airport route 
A1 
R 28,10 
Civic Centre  Airport route for children 4  11 years old 
A1 
Free 
Children under 1 metre tall and under 4 years old 
A1 
R 499,50 
Monthly ticket offering unlimited travel on the Airport  Civic Centre route (A1). This ticket cannot be used by anyone else and must have the name of the ticket holder on it. Write your name on the ticket as soon as buy it. 
A1 
 Marissa lives near the Waterfront and goes to school in Gardens. She decides to investigate the MyCiTi bus system, route F1.
 What will the bus ride from the Waterfront to Gardens cost her the first time, and why?
 After her first bus trip, how much will the journey cost, per trip?
 If she makes 10 trips a week (5 days to school and back again) what will she be paying per week?
Marissa's mother currently drives her to and from school and estimates that she spends R 13,62 on petrol per day, just for school lifts.
 How much is Marissa's mother spending on petrol for school lifts per week?
 How much money will Marissa save her mother by using the MyCiTi bus instead of travelling by car?
 Thuso's family wants to take the bus from the Civic Centre to Tableview, to go to the beach for the day. He is going with his parents and his aunt, his younger sister and his twin brothers who are 3 years old. How much will a one way trip to the beach cost the family, if both Thuso and his sister (who are both teenagers) have myconnect cards but no one else in the family does?
 Karen lives in Johannesburg but regularly travels to Cape Town for work. Her Cape Town office is based near the Civic Centre, so it is convenient to catch the MyCiTi bus from the airport to the Civic Centre in town.
 How much would a single trip from the airport to the Civic Centre cost her?
On average, Karen makes 5 round trips (from the airport to the Civic Centre and back again) in a month.
 Is there a cheaper bus fare or package she could use?
 If she buys the unlimited monthly Airport ticket, how much will each trip cost her? Is this cheaper than buying single trip tickets?
Answer

 She has to buy a myconnect card to use the bus, so the first trip will cost her R 23,00 (for the card) + R 5,30 (for the bus ride) or R 28,30 in total.
 Thereafter it will only cost her R 5,30 per trip.
 R 5,30 \(\times\) 2 trips per day \(\times\) 5 = R 53,00 per week

 R 13,62 \(\times\) 5 days = R 68,10 per week
 R 68,10  R 53,00 = R 15,10. Marissa will be saving her mother R 15,10 per week.
 Thuso's parents and aunt will all need to buy myconnect cards, so their fares will cost
3 \(\times\) (R 23,00 + R 10,60) = 3 \(\times\) R 33,60 = R 100,80
Thuso and his sister already have cards, so they only pay for the trip:
2 \(\times\) R 10,60 = R 21,20
His baby brothers travel for free because they are under the age of 4.
So the total cost is: R 100,80 + R 21,20 = R 122,00. 
 R 57,00

 Yes  she could investigate the monthly, unlimited travel option between the Airport and the Civic Centre.
 The monthly rate is R 499,50. 5 round trips = 10 one way trips.
R 499,50 \(\div\) 10 = R 49,95 per trip.
This is much cheaper than R 57 per trip!
Exercise 3: Working with transport tariffs
Metrorail in Gauteng charges the following tariffs for Metro (Standard) Class tickets for different zones:
Zone 
Single 
Return 
Weekly 
Monthly 
1  19 km 
4,00 
7,50 
22,00 
81,50 
20  29 km 
5,00 
9,50 
27,50 
97,00 
30  39 km 
6,00 
11,50 
32,50 
112,00 
40  49 km 
7,50 
14,50 
34,00 
123,00 
\(>\) 50 km 
9,50 
18,50 
38,50 
140,00 