204. Modelling Simple Financial Growth

Learning Intentions

• Draw linear models for simple financial situations.
• Calculate future values Use a linear rule.
• Interpret results using appropriate units and context.

Pre-requisite Summary

• Know that a linear model has a constant rate of change.
• Know that a linear rule can often be written in the form .
• Know that the gradient represents the repeated increase or decrease.
• Know that the intercept represents the starting value when the input is .
• Know how to substitute values into a rule.
• Know that financial answers should usually be written in dollars and cents, such as $ .

Worked Examples

Worked Example 1

A student starts with $ and saves $5 each week.

a) Write a linear rule for the total savings after weeks.

b) Calculate the savings after weeks.

c) Interpret the answer in context.

Worked Example 2

A bank account starts with $ and increases by 25$ each month.

a) Write a linear rule for the account balance after months.

b) Calculate the balance after months.

c) State the meaning of the gradient and intercept.

Worked Example 3

A phone plan costs $ per month plus a $ setup fee.

a) Write a linear rule for the total cost after months.

b) Calculate the total cost after months.

c) Interpret the answer using units.

Worked Example 4

A gym membership costs $ to join and $ per week.

a) Write a linear rule for the total cost after weeks.

b) Calculate the total cost after weeks.

c) Explain why this situation is linear.

Worked Example 5

A student owes $ and repays $ each week.

a) Write a linear rule for the amount owing after weeks.

b) Calculate the amount owing after weeks.

c) Interpret the gradient in context.

Worked Example 6

A worker earns $ per hour and receives a $ bonus.

a) Write a linear rule for total pay after hours.

b) Calculate the total pay for hours of work.

c) Interpret the intercept in context.

Worked Example 7

A school fundraiser has already collected $ and collects another $ each day.

a) Write a linear rule for the total amount after days.

b) Calculate the total amount after days.

c) Interpret the result in context.

Problems

Problem 1

A student starts with $ and saves $ each week.

a) Write a linear rule for the total savings after weeks.

b) Calculate the savings after weeks.

c) Interpret the answer in context.

Problem 2

A bank account starts with $ and increases by $ each month.

a) Write a linear rule for the account balance after months.

b) Calculate the balance after months.

c) State the meaning of the gradient and intercept.

Problem 3

A phone plan costs $ per month plus a $ setup fee.

a) Write a linear rule for the total cost after months.

b) Calculate the total cost after months.

c) Interpret the answer using units.

Problem 4

A gym membership costs $ to join and $ per week.

a) Write a linear rule for the total cost after weeks.

b) Calculate the total cost after weeks.

c) Explain why this situation is linear.

Problem 5

A student owes $ and repays $ each week.

a) Write a linear rule for the amount owing after weeks.

b) Calculate the amount owing after weeks.

c) Interpret the gradient in context.

Problem 6

A worker earns $ per hour and receives a $ bonus.

a) Write a linear rule for total pay after hours.

b) Calculate the total pay for hours of work.

c) Interpret the intercept in context.

Problem 7

A school fundraiser has already collected $ and collects another $ each day.

a) Write a linear rule for the total amount after days.

b) Calculate the total amount after days.

c) Interpret the result in context.

Exercises

Understanding and Fluency

Exercise 1

Write a linear rule for each savings situation.

a) Starts with $ and saves $ per week.

b) Starts with $ and saves $ per week.

c) Starts with $ and saves $ per month.

d) Starts with $ and saves $ per fortnight.

Exercise 2

Calculate the future value for each savings rule.

a) when

b) when

c) when

d) when

Exercise 3

For each financial rule, state the gradient and intercept.

a)

b)

c)

d)

Exercise 4

For each situation, write a rule and calculate the requested value.

a) $ setup fee and $ per month. Find the cost after months.

b) $ starting balance and $ saved per week. Find the balance after weeks.

c) $ debt and $ repaid per week. Find the amount owing after weeks.

Exercise 5

Complete each table.

a)

b)

Exercise 6

Match each financial situation to its linear rule.

a) Starts with $ and increases by $ each week.

b) Starts with $ and decreases by $ each week.

c) Costs $ per month with no setup fee.

d) Costs $ plus $ per ticket.

Rules:

Exercise 7

Calculate the future balance for each account.

a) after months

b) after weeks

c) after months

d) after days

Exercise 8

Calculate the future cost for each rule.

a) after months

b) after weeks

c) after uses

d) after days

Exercise 9

For each rule, interpret the gradient in context.

a) , where is savings after weeks.

b) , where is cost after months.

c) , where is amount owing after weeks.

Exercise 10

For each rule, interpret the intercept in context.

a) , where is balance after weeks.

b) , where is total cost after months.

c) , where is total pay after hours.

Reasoning

Exercise 11

Explain why a person saving the same amount every week can be modelled using a linear rule.

Exercise 12

A student writes the rule for a savings account that starts with $ and increases by $ each week.

Explain the mistake and write the correct rule.

Exercise 13

A subscription costs $ per month plus a $ joining fee.

A student says the gradient is because it is the first cost paid.

Explain the mistake.

Exercise 14

A student owes $ and repays $ each week.

Explain why the rule should have a negative gradient.

Exercise 15

Explain the difference between these two rules in financial context:

a)

b)

Exercise 16

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

a) In , the starting amount is $ .

b) In , the cost increases by $ each month.

c) In , the amount owing decreases by $ each week.

Problem-solving

Exercise 17

A student has $ and saves $ each week.

a) Write a linear rule for the total savings after weeks.

b) Calculate the savings after weeks.

c) Interpret the answer in context.

Exercise 18

A music subscription charges a $ joining fee and $ per month.

a) Write a rule for the total cost after months.

b) Calculate the total cost after months.

c) Interpret the gradient and intercept.

Exercise 19

A loan starts at $ and is repaid by $ each week.

a) Write a rule for the amount owing after weeks.

b) Calculate the amount owing after weeks.

c) Explain what the result means in context.

Exercise 20

Create your own simple financial linear model.

Your response must include:

• a financial situation
• a linear rule
• a calculation of a future value
• appropriate units
• a sentence interpreting the answer in context

Potential Misunderstandings

• Students may reverse the gradient and intercept, such as treating the starting amount as the repeated change.
• Students may forget that the intercept represents the amount when the input is .
• Students may think every financial situation is linear, even when interest or percentage growth is involved.
• Students may substitute the future time value into the wrong part of the rule.
• Students may forget to multiply before adding when calculating future values.
• Students may make sign errors in repayment situations where the balance decreases.
• Students may give a numerical answer without units such as dollars, weeks or months.
• Students may interpret the gradient as a total amount rather than an amount per unit of time.
• Students may interpret a future cost, saving or debt without explaining what the input value represents.

Next: 205. Recognising Quadratic Change