Thinking Skills: Critical Thinking and Problem Solving

120 Unit 3 Problem solving: basic skills

mathematics. What effect would these

changes have on someone earning $50,000

a year?

As an exercise, consider other possible tax

structures which might give similar results.

Plotting graphs of tax paid against earnings

gives a clearer representation of how the

various models of taxation work. The graph

below shows the tax paid under the current

system of tax in the example above.

01 0,00020,00030,00040,00050,000

0

2000

4000

6000

8000

10,000

12,000

Annual earnings

Tax due

Add lines for the proposed new system and

for any other model you may think of.

The next activity, below, requires you to go

some way towards developing a mathematical

model of a new situation in order to solve the

question.

Commentary

The model of tax used here is quite simple,

consisting of a fixed amount of income on which

no tax is paid and a single standard rate on

earnings above this amount. Currently, those

earning $26,000 pay (26,000−2000) × 0.2 dollars,

or $4800. If they are to pay the same under the

new regime, they will pay tax on $16,000, and

the total will be the same as before, i.e. $4800.

The new tax rate will be 30¢ on each dollar

earned over $10,000 ($16,000 × 0.3 = $4800).

This could be done by algebra, but the process

is no simpler than that given above, which

requires no more than elementary school

The government is determined to reduce

the tax burden on lower-paid people and

intends to bring in a new system, which will

mean that the threshold for paying tax will

rise to $10,000. They intend that all those

below the average earnings of $26,000 will

pay less tax, and all those earning more than

this will pay more tax. What will be the tax

rate on earnings over $10,000?

Activity

My company regularly uses a taxi service to

take staff to the airport. If there are several

passengers needing to travel from our town

at similar times, they combine this into a

single journey. They divide the total cost by

the number of passengers and invoice each

passenger separately. The distance is always

the same and the time only varies by a small

amount, but I do not know how they work out

the charge for the journey.

There are a number of different charging

structures they could use. All taxis charge a

fixed price per kilometre and per minute of

journey time. In addition they may charge a

fixed hire fee and an additional charge

depending on the number of passengers

carried.

What is the charging structure used by this

taxi company? What limitations are there to

the conclusions we can derive?

Number of passengers 1 2 3 4 5

Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38