Model rational quantities, percentage changes and financial outcomes

A school event receives $300 in ticket sales. Is that enough information to decide whether it made a profit?

Formulate, calculate, interpret and review

Formulate a precise question. Identify the quantities, units, time period and restrictions; distinguish supplied facts from assumptions

Choose a calculation, labelled table or diagram that represents the relationships. Decide which percentage base applies at each stage

Use efficient exact mental or written work where practical, and a calculator or spreadsheet when repeated calculations or awkward values make it useful

Interpret the numerical output in the original context, with units and any necessary whole-number or rounding condition

Review the model as well as the arithmetic. Test a boundary, vary an assumption and explain whether the original conclusion remains justified

Choose an origin and a sign convention

Temperature values can lie below zero. A positive temperature change is a rise and a negative change is a fall

For a vertical coordinate measured from the water surface, choose upward as positive. A position below the surface then has a negative coordinate, but its physical depth is a non-negative distance

Do not swap a coordinate with a distance or a final value with a change. State which quantity the question asks for

Interpret a rise from a negative temperature

A measured temperature starts at -2.5 °C and rises by 3.75 °C. Treat both readings as exact for this model. Find the final temperature in degrees Celsius

A rise is a positive change added to the initial temperature

The final temperature is 1.25 °C, above zero

The reverse comparison recovers the stated rise

Distinguish a vertical coordinate from depth

An average rate need not describe each interval

A total signed change divided by the number of equal intervals gives the average signed change per interval

Using that average as the change on every day is an additional constant-rate assumption. Real daily measurements may differ even when their total matches

Calculate an average fall without losing its sign

An average fall of 0.9 °C per day was observed over four days. A model predicts the next ten days by subtracting 0.9 each day. What additional assumption does it make?

Define the financial boundary before calculating

Consider a fictional event selling forty tickets at $7.50 each. Supplies cost $180 and the venue costs $60. Assume all forty tickets are paid for and those are the only costs included

Ask whether the event covers those stated costs, and calculate its profit as a percentage of them

Revenue is not profit. State whether unpaid labour, equipment already owned, unsold items and other expenses are excluded before interpreting the result

Tickets sold

\(\displaystyle 40\)

Price per ticket in dollars

\(\displaystyle 7.50\)

Forty tickets are all sold and paid for at $7.50 each, with no refunds or other income. Calculate ticket revenue in dollars

Multiply the paid ticket count by the price per ticket

Supplies in dollars

\(\displaystyle 180\)

Venue in dollars

\(\displaystyle 60\)

The only costs in an event model are $180 for supplies and $60 for the venue. Calculate the total included cost in dollars

Add the two included expenses

Use costs as the percentage-profit base

A positive result is profit, a negative result is loss and zero is break-even for the included costs

The signed percentage uses cost as its base. A percentage of revenue answers a different question

A negative signed profit can also be reported in words as a positive loss magnitude, provided the convention is stated

Complete the stated event financial calculation

The complete cost for a supplied sales record is $240 and the total revenue is $300. Both amounts refer to the same sales record; there are no other costs or receipts

All quantities and rates are exact. Keep every intermediate quantity exact; no rounding is needed or permitted in this model

Calculate the signed profit or loss amount

Use a negative answer for a loss or decrease and zero for no net change

Give the exact result in $; do not round

Subtract the complete stated cost from the revenue or final selling price

The positive result is a profit

Calculate the signed percentage profit or loss on cost

Use a negative answer for a loss or decrease and zero for no net change

Give the exact result in %; do not round

Subtract the complete stated cost from the revenue or final selling price

The positive result is a profit

Use the complete stated cost as the percentage base, not the revenue or selling price

Keep this result exact; a negative percentage represents a decrease or loss

Report a loss using the signed convention

Test the break-even boundary

A mark-up and a discount use different bases

For one item, first apply the mark-up to its cost. Then apply the discount to the marked selling price, not to the original cost

Subtract the original cost from the final selling price to find the profit on that item. This model assumes the item is sold and there are no other costs or taxes

A twenty percent mark-up and a twenty percent discount do not cancel because the second percentage acts on an increased base

Carry a mark-up through a later discount

An item costs the seller $80. The seller marks up that cost by 25%, then discounts the marked price by 10%. The item is sold at that final price. Ignore other expenses, taxes and receipts

All quantities and rates are exact. Keep every intermediate quantity exact; no rounding is needed or permitted in this model

Calculate the marked price

Use the updated quantity as the base of the second change when needed

Give the exact result in $; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

Calculate the final selling price

Use the updated quantity as the base of the second change when needed

Give the exact result in $; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

The second percentage uses the quantity after the first change as its base

Subtract this change from its own base

Calculate the signed profit or loss amount

Use a negative answer for a loss or decrease and zero for no net change

Give the exact result in $; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

The second percentage uses the quantity after the first change as its base

Subtract this change from its own base

Subtract the complete stated cost from the revenue or final selling price

The positive result is a profit

Calculate the signed percentage profit or loss on cost

Use a negative answer for a loss or decrease and zero for no net change

Give the exact result in %; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

The second percentage uses the quantity after the first change as its base

Subtract this change from its own base

Subtract the complete stated cost from the revenue or final selling price

The positive result is a profit

Use the complete stated cost as the percentage base, not the revenue or selling price

Keep this result exact; a negative percentage represents a decrease or loss

A discount can turn the planned profit into a loss

A batch model estimates total profit by multiplying profit per item by twenty, assuming every item sells. Only sixteen items sell, and the four unsold items have no resale value. What must be revised?

Update the percentage base at every period

For successive percentage changes, calculate the first new value and use that result as the next base

A combined signed percentage change compares the final value with the original value, not the intermediate one

Keep intermediate values exact unless the model explicitly requires rounding at a particular stage

Compare equal-sized opposite percentage changes

The original quantity is 1000 units. First increase it by 10%, then decrease the resulting quantity by 10%. Apply the second percentage to the updated quantity

All quantities and rates are exact. Keep every intermediate quantity exact; no rounding is needed or permitted in this model

Calculate the quantity after the first change

Use the updated quantity as the base of the second change when needed

Give the exact result in units; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

Calculate the final quantity after both changes

Use the updated quantity as the base of the second change when needed

Give the exact result in units; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

The second percentage uses the quantity after the first change as its base

Subtract this change from its own base

Calculate the signed net percentage change

Use a negative answer for a loss or decrease and zero for no net change

Give the exact result in %; do not round

Use the stated original quantity as the first percentage base

Add this change to its own base

The second percentage uses the quantity after the first change as its base

Subtract this change from its own base

Find final minus original, keeping the sign

Compare the signed net change with the original quantity, not the intermediate quantity

Keep this result exact; a negative percentage represents a decrease or loss

Determine the combined change instead of adding the rates

Construct two increases and the net percentage

Apply two stated periods in a population model

A fictional recycling model starts with 200 litres collected per period and increases that quantity by 10% each period. What amount does the second increase use as its base?

P is the starting quantity in litres and r is the signed percentage change per period. The model applies the same rate to each updated value, with no other changes or capacity limit. Table displays are rounded to two decimal places; calculations retain the full expression

Why can this table not by itself establish a real long-term recycling forecast?

Interpret a fictional marginal tax table

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments

A rate applies only to the income portion in its row, not to the entire income. Allocate each dollar to one band, then add the tax from those portions

These invented rules teach table interpretation. They are not current Australian tax law or a personal tax calculation

Annual income in dollars

\(\displaystyle 10000\)

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments. Find the tax on an annual income of exactly $10000

Income equal to the first threshold has no portion above it

Tax only the portion above the threshold

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments

For annual income of $20000, the first $10000 is untaxed and the remaining $10000 is in the ten percent band

The tax is $1000, leaving $19000 after this tax alone

The overall percentage of income paid in tax is five percent, although the highest applied marginal rate is ten percent

Annual income in dollars

\(\displaystyle 25000\)

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments. Find the tax on annual income of $25000

Subtract the untaxed first ten thousand before using ten percent

Add the tax from two taxed portions

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments

For annual income of $40000, fill the ten percent band before applying twenty percent to the portion above $30000

The tax is $4000 under this model. Applying twenty percent to all $40000 would wrongly tax earlier portions at the higher rate

Annual income in dollars

\(\displaystyle 35000\)

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments. Find the total tax on annual income of $35000

The complete middle band contributes two thousand dollars, then tax the remaining five thousand at twenty percent

Annual income in dollars

\(\displaystyle 35000\)

Total tax in dollars

\(\displaystyle 3000\)

Under a supplied fictional model, annual income is $35000 and its total tax is $3000. What exact percentage of income is paid in tax? Enter an exact fraction of a percent, not a rounded decimal

Divide total tax by total income and multiply by one hundred

Annual income in dollars

\(\displaystyle 35000\)

Tax in dollars

\(\displaystyle 3000\)

A fictional annual model has income of $35000 and total tax of $3000, with no other adjustments. Find the income remaining after that tax

Subtract the total tax from the income

Use this fictional annual income-tax model only: the first $10000 is untaxed, the portion above $10000 up to $30000 is taxed at 10%, and only the portion above $30000 is taxed at 20%. There are no deductions, levies or other adjustments. An income increases from $30000 to $30100. Does the entire income suddenly become taxed at twenty percent?

Rounding timing is part of the model

A fictional price model starts at $19.95, adds 7.5%, then discounts that updated amount by 10%. Its rule says to round only the final selling price to cents

The required final price is $19.30. Rounding the intermediate marked price to $21.45 would instead give $19.305 before final rounding

Real checkout rules may require intermediate rounding. State the rule rather than silently switching between models

Original price in dollars

\(\displaystyle 19.95\)

Increase then decrease percentages

\(\displaystyle 7.5\%,\quad10\%\)

A fictional model starts at $19.95, increases it by 7.5%, then decreases the updated amount by 10%. Keep intermediate values exact and round only the final amount to cents. Enter exactly two decimal places

Use both multipliers before rounding: 1.075 followed by 0.9

A constant-percentage population model predicts a fractional number of people after many periods. What is an appropriate review?

Formulate and solve your own financial question

Independent written and digital modelling task, not automatically marked

Invent a small event or sale with positive costs, stated quantities and prices. Formulate a question about whether it covers costs and how a mark-up, discount or changed sales count affects profit

Define revenue and the full costs you choose to include. State which items are sold, what happens to unsold stock, and whether fees or taxes are excluded

Choose and justify a representation: a labelled calculation, a table or a spreadsheet. Identify the percentage base at each stage and state any rounding rule

Carry out the calculation and check at least one case independently by mental or written work. Report profit or loss in dollars and as a signed percentage of positive cost

Communicate a conclusion that answers your question and names the assumptions needed for it

Test your model and revise one assumption

Test a break-even case, a loss case and a change in the number sold. If a cost changes, update both profit and its percentage-on-cost denominator

Vary one meaningful assumption, such as adding a venue fee or retaining unsold stock with no resale value. Recalculate and explain whether the conclusion changes

For an alternative environmental investigation, choose a signed starting quantity and observations over equal intervals. Compare a constant amount-change model with a stated constant percentage model only where a percentage base is meaningful

For a fictional tax investigation, use this rule: the first $10000 is untaxed, income above $10000 up to $30000 is taxed at 10%, and income above $30000 is taxed at 20%. Test positive incomes below, at and just above each threshold. Calculate total tax and the overall percentage, and explain why the marginal and overall rates differ

Keep the original and revised calculations so another person can reproduce them. Explain any mismatch with practical limits and what further information would improve the model

Review your independent formulation, digital work and written justification against these questions or with a teacher; the supplied calculation checks do not automatically assess that work

Return to the original question after calculating

Define signs, units, quantities, time periods and assumptions before choosing operations

Profit compares revenue with the costs included in the model; its signed percentage uses positive cost as the base

Successive percentage changes use successive bases. State when rounding occurs

In a marginal tax model, allocate income to the supplied bands before adding their tax contributions; distinguish the overall percentage from a marginal rate

Use exact arithmetic and digital tools appropriately, then interpret the result in context

Check boundaries and changed assumptions. A reasonable model is revisable, and its conclusion is limited by its stated conditions