208. Comparing Financial Options

Learning Intentions

• Model different financial options Use linear or quadratic functions.
• Compare outcomes over a given time period.
• Recommend an option using mathematical evidence.

Pre-requisite Summary

• Know that a linear function has a constant rate of change and can be written in the form .
• Know that a quadratic function includes a squared term and can be written in forms such as .
• Know how to substitute values into a function rule.
• Know that financial outputs should be interpreted using units such as dollars, weeks, months or years.
• Know that comparing financial options requires using the same time period for each option.
• Know that a recommendation should be supported by calculations, not only by opinion.

Worked Examples

Worked Example 1

A student compares two savings options over weeks.

Option A: Start with $ and save $ each week.

Option B: Start with $ and save an amount modelled by , where is the number of weeks.

a) Write a rule for Option A.

b) Calculate the amount saved in each option after weeks.

c) State which option gives more savings after weeks.

Worked Example 2

A company offers two payment plans for a short project.

Plan A:

Plan B:

where is pay in dollars and is hours worked.

a) Calculate the pay for each plan when .

b) Calculate the pay for each plan when .

c) Compare which plan is better at each time.

Worked Example 3

A subscription has two possible cost models.

Option A:

Option B:

where is total cost in dollars and is the number of months.

a) Complete a table for .

b) Decide whether each model is linear or quadratic.

c) Describe how the costs change over time.

Worked Example 4

A school fundraiser compares two collection models.

Option A:

Option B:

where is the total amount raised in dollars and is the number of days.

a) Calculate the total raised by each option after days.

b) Calculate the total raised by each option after days.

c) Explain why the better option changes over time.

Worked Example 5

A student is choosing between two savings plans for weeks.

Plan A:

Plan B:

where is savings in dollars and is weeks.

a) Calculate the savings for each plan after weeks.

b) Recommend one plan for weeks.

c) Justify the recommendation using mathematical evidence.

Worked Example 6

A customer compares two repayment options for a loan.

Option A: Amount owing is .

Option B: Amount owing is .

where is the amount owing in dollars and is weeks.

a) Calculate the amount owing after weeks for each option.

b) Calculate the amount owing after weeks for each option.

c) Recommend the option that reduces the debt faster over weeks.

Worked Example 7

A business compares two advertising options.

Option A costs .

Option B costs .

where is cost in dollars and is days.

a) Calculate the cost of each option after days.

b) Calculate the cost of each option after days.

c) Recommend an option if the aim is to spend less over days.

Problems

Problem 1

A student compares two savings options over weeks.

Option A: Start with $ and save $ each week.

Option B: Start with $ and save an amount modelled by , where is the number of weeks.

a) Write a rule for Option A.

b) Calculate the amount saved in each option after weeks.

c) State which option gives more savings after weeks.

Problem 2

A company offers two payment plans for a short project.

Plan A:

Plan B:

where is pay in dollars and is hours worked.

a) Calculate the pay for each plan when .

b) Calculate the pay for each plan when .

c) Compare which plan is better at each time.

Problem 3

A subscription has two possible cost models.

Option A:

Option B:

where is total cost in dollars and is the number of months.

a) Complete a table for .

b) Decide whether each model is linear or quadratic.

c) Describe how the costs change over time.

Problem 4

A school fundraiser compares two collection models.

Option A:

Option B:

where is the total amount raised in dollars and is the number of days.

a) Calculate the total raised by each option after days.

b) Calculate the total raised by each option after days.

c) Explain why the better option changes over time.

Problem 5

A student is choosing between two savings plans for weeks.

Plan A:

Plan B:

where is savings in dollars and is weeks.

a) Calculate the savings for each plan after weeks.

b) Recommend one plan for weeks.

c) Justify the recommendation using mathematical evidence.

Problem 6

A customer compares two repayment options for a loan.

Option A: Amount owing is .

Option B: Amount owing is .

where is the amount owing in dollars and is weeks.

a) Calculate the amount owing after weeks for each option.

b) Calculate the amount owing after weeks for each option.

c) Recommend the option that reduces the debt faster over weeks.

Problem 7

A business compares two advertising options.

Option A costs .

Option B costs .

where is cost in dollars and is days.

a) Calculate the cost of each option after days.

b) Calculate the cost of each option after days.

c) Recommend an option if the aim is to spend less over days.

Exercises

Understanding and Fluency

Exercise 1

For each financial rule, state whether the model is linear or quadratic.

a)

b)

c)

d)

Exercise 2

For each model, calculate the value when the input is .

a)

b)

c)

d)

Exercise 3

Complete the table for both options.

Option A:

Option B:

Option A
Option B

Exercise 4

Compare the two cost options after months.

Option A:

Option B:

a) Calculate Option A when .

b) Calculate Option B when .

c) State which option costs less after months.

Exercise 5

Compare the two payment options after hours.

Option A:

Option B:

a) Calculate Option A when .

b) Calculate Option B when .

c) State which option pays more after hours.

Exercise 6

For each pair of options, identify which option gives the larger value at the given input.

a) and , when

b) and , when

c) and , when

Exercise 7

A savings plan is modelled by .

a) Calculate the savings after weeks.

b) Calculate the savings after weeks.

c) Explain why the increase is not constant.

Exercise 8

A cost plan is modelled by .

a) Calculate the cost after months.

b) Calculate the cost after months.

c) Explain why the increase is constant.

Exercise 9

Match each financial situation to the better model type.

a) A subscription increases by the same amount each month.

b) A bonus increases according to the square of the number of sales.

c) A debt decreases by the same amount each week.

d) A fundraising total increases faster each day as more people join.

Model types:

• linear
• quadratic

Exercise 10

For each recommendation, state whether the evidence is sufficient.

a) “Choose Option A because I like it better.”

b) “Choose Option B because after weeks it gives $ , while Option A gives $ .”

c) “Choose the quadratic option because quadratic is always better.”

Reasoning

Exercise 11

Explain why could model a subscription with a joining fee and constant monthly cost.

Exercise 12

Explain why might model a savings plan where the amount saved increases faster over time.

Exercise 13

A student compares two options at different times.

Option A is calculated after weeks.

Option B is calculated after weeks.

Explain why this is not a fair comparison.

Exercise 14

A student says that a quadratic savings model is always better than a linear savings model.

Explain why this is incorrect.

Exercise 15

Two options are modelled by:

Option A:

Option B:

Explain why Option A may be better for small values of , while Option B may be better later.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) A recommendation should compare both options over the same time period.

b) A quadratic model always gives a smaller value than a linear model.

c) Mathematical evidence can include tables, substitutions and comparisons.

Problem-solving

Exercise 17

A student compares two savings plans.

Plan A:

Plan B:

a) Calculate the savings for each plan after weeks.

b) Calculate the savings for each plan after weeks.

c) Recommend a plan for weeks using mathematical evidence.

Exercise 18

A customer compares two phone plans.

Plan A:

Plan B:

where is cost in dollars and is months.

a) Calculate the cost for each plan after months.

b) Calculate the cost for each plan after months.

c) Recommend a plan for months if the customer wants the lower cost.

Exercise 19

A worker compares two pay options.

Option A:

Option B:

where is pay in dollars and is hours worked.

a) Calculate the pay for each option after hours.

b) Calculate the pay for each option after hours.

c) Recommend an option for hours and justify your choice.

Exercise 20

Create your own comparison of two financial options.

Your response must include:

• one linear model
• one quadratic model
• a chosen time period
• calculations for both options over that time period
• a recommendation supported by mathematical evidence
• appropriate units and context

Potential Misunderstandings

• Students may choose a model based only on whether the values increase, rather than checking whether the increase is constant or non-constant.
• Students may not recognise that linear financial models have constant repeated changes, while quadratic models have changing repeated changes.
• Students may assume that a quadratic model is always better because it can grow faster.
• Students may compare two options over different time periods, making the comparison unfair.
• Students may substitute the time value into one model correctly but use a different input in the other model.
• Students may forget that the aim matters: for savings or pay, a larger value may be better, while for cost or debt, a smaller value may be better.
• Students may make a recommendation without showing calculations.
• Students may give a mathematically correct recommendation that does not match the context.
• Students may forget to include units such as dollars, weeks, months or hours when interpreting results.