Model practical situations with linear relations

A fictional order costs twelve dollars once plus four dollars for each item. What changes when one extra item is added?

Connect a practical question to a mathematical model

Formulate the question and identify what must be found. Define each variable with its unit and permitted values

Record the supplied quantities and assumptions. A constant-rate model assumes the same change for each equal input step

Represent the relationship with an equation, table or graph. Choose the form that makes the required comparison or calculation clear

Calculate within the stated domain, then interpret the result with its unit and practical meaning

Check the arithmetic and compare the assumptions and predictions with the situation. Revise the model when conditions change

A correct calculation does not by itself establish that a practical assumption is reasonable

Separate an initial charge from a per-item rate

A fictional order has a twelve-dollar setup charge and a four-dollar charge per item, with no other charges. The setup is paid even if no items are produced

Let n be the whole number of items and C the total cost in dollars. The order is limited to twenty items

In the template C equals b plus m n, b is the initial cost twelve and m is the rate four dollars per item

Seven items cost forty dollars under the stated rule

The initial value is not zero because the fixed setup cost remains

Construct and use a fixed-charge model

A fictional order has a fixed charge of $12 plus $4 for each item. There are no other charges

Let \(n\) be the number of items and \(C\) the total charge in dollars. The stated rule includes the fixed charge when the item count is zero

Use this model only for \(0\le n\le 20\), with whole-number item counts only

Input for prediction, measured in items: \(n=7\)

Separately test whether \(n=5/2\) is allowed by the stated input domain

Construct \(C=b+mn\) by entering its initial value \(b\) and signed rate \(m\)

Initial amount in dollars: \(12\). Signed change in dollars per item: \(4\)

Add the constant change once for every input unit, rather than treating it as a total change

Predicted output, in dollars: \(40\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Calculate the output at the stated input, in dollars

Initial amount in dollars: \(12\). Signed change in dollars per item: \(4\)

Add the constant change once for every input unit, rather than treating it as a total change

Predicted output, in dollars: \(40\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(n=5/2\) allowed by this model?

The input must be a whole count as well as lie within the bounds

This input fails the whole-item count condition

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Keep the one-off and per-item amounts in their correct roles

Model, with C in dollars and n a whole item count

\(\displaystyle C=18+3n\)

What does the coefficient of n mean in the stated model?

Use different representations for different purposes

A table lists selected input-output pairs. Each step of two items raises the cost by eight dollars, giving four dollars per item

The equation gives an exact rule for any allowed count. A graph makes trends and comparisons visible

For whole-item counts, the permitted graph consists of separate points. A connecting line can show the pattern but does not make fractional counts valid

A nonzero initial value means the total is not directly proportional to the count. Doubling the count does not double the fixed charge

Recognise the proportional special case with no fixed charge

Use a required output to find the input

For the same twelve-dollar setup and four-dollar per-item model, find the count giving a total cost of forty-four dollars

Eight is a whole count within the twenty-item limit, so the model answers the practical question

If a requested total gives a fractional count, do not silently round and claim exact equality. Check whether an exact whole-count solution exists or whether the question asks for an affordable maximum

Recover an item count from a stated cost

A fictional order has a fixed charge of $12 plus $4 for each item. There are no other charges

Let \(n\) be the number of items and \(C\) the total charge in dollars. The stated rule includes the fixed charge when the item count is zero

Use this model only for \(0\le n\le 20\), with whole-number item counts only

Required output, measured in dollars: \(C=44\)

Separately test whether \(n=21\) is allowed by the stated input domain

Construct \(C=b+mn\) by entering its initial value \(b\) and signed rate \(m\)

Initial amount in dollars: \(12\). Signed change in dollars per item: \(4\)

Add the constant change once for every input unit, rather than treating it as a total change

Subtract the initial value from both sides, then divide by the signed nonzero rate

Required input, measured in items: \(n=8\). This is within the stated domain and is a whole item count

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Find the input giving the required output, in items

Initial amount in dollars: \(12\). Signed change in dollars per item: \(4\)

Add the constant change once for every input unit, rather than treating it as a total change

Subtract the initial value from both sides, then divide by the signed nonzero rate

Required input, measured in items: \(n=8\). This is within the stated domain and is a whole item count

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(n=21\) allowed by this model?

The input must be a whole count as well as lie within the bounds

This input fails the upper bound

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Solve a new exact cost target

Continuous time permits fractional inputs

A tank starts with twenty litres and receives six litres per minute at a constant rate, with no other flow. The rule is used only for the first ten minutes

V is volume in litres and t is elapsed time in minutes. Fractional times are meaningful

After two and a half minutes, the model gives thirty-five litres

The positive rate means volume increases. The initial twenty litres is present at time zero, before the additional inflow

Predict at an exact fractional time

A tank initially contains 20 litres. Water enters at a constant rate of 6 litres per minute, with no other flow

Let \(t\) be elapsed time in minutes and \(V\) the remaining water volume in litres

Use this model only for \(0\le t\le 10\), allowing fractional times

Input for prediction, measured in minutes: \(t=5/2\)

Separately test whether \(t=5/2\) is allowed by the stated input domain

Construct \(V=b+mt\) by entering its initial value \(b\) and signed rate \(m\)

Initial amount in litres: \(20\). Signed change in litres per minute: \(6\)

Add the constant change once for every input unit, rather than treating it as a total change

Predicted output, in litres: \(35\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Calculate the output at the stated input, in litres

Initial amount in litres: \(20\). Signed change in litres per minute: \(6\)

Add the constant change once for every input unit, rather than treating it as a total change

Predicted output, in litres: \(35\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(t=5/2\) allowed by this model?

Fractional elapsed times are permitted, but the stated bounds still apply

This input meets all stated domain conditions

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Calculate a volume and test the stated time limit

Find the exact time for a target water volume

A decreasing quantity needs a negative rate

A tank begins with ninety litres and loses six litres per minute at a constant rate, with no inflow

The initial value is ninety litres and the signed rate is negative six litres per minute

The tank becomes empty after fifteen minutes. Use this draining rule only from zero through fifteen minutes

The algebra gives negative six outside that domain, but negative remaining water is not a valid physical prediction

After emptying, the constant outflow assumption fails. A new rule would be needed to describe later times

Reach the empty-tank boundary without extending past it

A tank initially contains 90 litres. Water leaves at a constant rate of 6 litres per minute, with no other flow

Let \(t\) be elapsed time in minutes and \(V\) the remaining water volume in litres

Use this model only for \(0\le t\le 15\), allowing fractional times

Input for prediction, measured in minutes: \(t=15\)

Separately test whether \(t=16\) is allowed by the stated input domain

Construct \(V=b+mt\) by entering its initial value \(b\) and signed rate \(m\)

Initial amount in litres: \(90\). Signed change in litres per minute: \(-6\)

Subtract the amount lost each minute, so the rate coefficient is negative

Predicted output, in litres: \(0\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Calculate the output at the stated input, in litres

Initial amount in litres: \(90\). Signed change in litres per minute: \(-6\)

Subtract the amount lost each minute, so the rate coefficient is negative

Predicted output, in litres: \(0\)

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(t=16\) allowed by this model?

Fractional elapsed times are permitted, but the stated bounds still apply

This input fails the upper bound

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Use a negative rate at a fractional time

Recover the time when half the initial water remains

Draining model in litres

\(\displaystyle V=90-6t,\quad0\le t\le15\)

The model is stated only for the given time interval. What can be concluded about its value at sixteen minutes?

Build revenue and cost before subtracting

A fictional sale has a forty-dollar setup cost, a three-dollar production cost per item and an eight-dollar selling price per item. Every produced item sells, with no other costs or taxes

Let n be the whole number produced and sold. Revenue R, cost C and signed profit P are all in dollars

The initial signed profit is negative forty dollars because setup has been paid before any sales. Each additional sold item contributes five dollars after its own production cost

Twelve sales give a twenty-dollar profit under these assumptions

Construct profit from separate per-item and fixed costs

A fictional small-batch sale has a fixed setup cost of $40, a production cost of $3 per item and a selling price of $8 per item

Every produced item is sold. There are no other costs or taxes. Let \(n\) be the number of items produced and sold, and \(P\) the signed profit in dollars

Use this model only for \(0\le n\le 20\), with whole-number item counts only

Input for prediction, measured in items: \(n=12\)

Separately test whether \(n=5/2\) is allowed by the stated input domain

Construct \(P=b+mn\) by entering its initial value \(b\) and signed rate \(m\)

The initial signed profit is the negative fixed cost. Each extra sold item contributes selling price minus production cost

Predicted signed profit, in dollars: \(20\)

This is a profit of 20 dollars

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Calculate the output at the stated input, in dollars

The initial signed profit is the negative fixed cost. Each extra sold item contributes selling price minus production cost

Predicted signed profit, in dollars: \(20\)

This is a profit of 20 dollars

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(n=5/2\) allowed by this model?

The input must be a whole count as well as lie within the bounds

This input fails the whole-item count condition

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Interpret a negative signed result as a loss

Break-even means revenue equals cost

Let n be the whole number of items sold at eight dollars each, with a forty-dollar setup cost and a three-dollar cost per item

Eight sales recover all stated costs, giving zero profit. This is neither a positive profit nor a loss

On revenue and cost graphs, break-even occurs where their outputs agree. On a profit graph it occurs where profit is zero

The item count is the horizontal input. The common revenue and cost are different quantities, not another break-even count

Verify the zero-profit boundary

Find the sales count needed for a specified profit

A fictional small-batch sale has a fixed setup cost of $40, a production cost of $3 per item and a selling price of $8 per item

Every produced item is sold. There are no other costs or taxes. Let \(n\) be the number of items produced and sold, and \(P\) the signed profit in dollars

Use this model only for \(0\le n\le 20\), with whole-number item counts only

Required output, measured in dollars: \(P=20\)

Separately test whether \(n=0\) is allowed by the stated input domain

Construct \(P=b+mn\) by entering its initial value \(b\) and signed rate \(m\)

The initial signed profit is the negative fixed cost. Each extra sold item contributes selling price minus production cost

Subtract the initial value from both sides, then divide by the signed nonzero rate

Required input, measured in items: \(n=12\). This is within the stated domain and is a whole item count

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Find the input giving the required output, in items

The initial signed profit is the negative fixed cost. Each extra sold item contributes selling price minus production cost

Subtract the initial value from both sides, then divide by the signed nonzero rate

Required input, measured in items: \(n=12\). This is within the stated domain and is a whole item count

Check by evaluating the original model at the calculated input

This verifies the arithmetic under the stated constant-rate assumptions, not whether those assumptions describe another real situation

Is the separate test input \(n=0\) allowed by this model?

The input must be a whole count as well as lie within the bounds

This input meets all stated domain conditions

A formula can be evaluated outside its domain, but that value is not a valid prediction from this model

Reverse a different profit model

A whole-count condition changes a threshold decision

Let n be a nonnegative whole item count and P the signed profit in dollars

For a positive profit, solve the strict inequality rather than only the break-even equation

Seven and a half is the continuous boundary. Whole items cannot be split, so the least permitted count giving a positive profit is eight

Seven sales still lose two dollars; eight give a two-dollar profit. Rounding without checking the condition could give the wrong practical answer

Signed profit in dollars

\(\displaystyle P=6n-50\)

Permitted item count

\(\displaystyle 0\le n\le20,\quad n\text{ whole}\)

The fictional profit rule is valid for whole counts from zero through twenty. Find the least count giving a strictly positive profit

Solve P greater than zero, then test the adjacent whole counts

Compare revenue and cost at the same input

For the same item count, revenue above cost means a positive vertical difference and therefore a profit

Revenue below cost means a loss. Equal graph heights give break-even

The break-even count is six items, not thirty-six. Thirty-six dollars is the common revenue and cost at that count

Revenue in dollars

\(\displaystyle R=6n\)

Cost in dollars

\(\displaystyle C=24+2n\)

Domain

\(\displaystyle 0\le n\le12,\quad n\text{ whole}\)

Move the count n. The solid line is revenue and the dashed line is cost. Q and S show the two outputs at that same count

Which condition gives a positive profit within this whole-count domain?

A cost rule applies only to whole item counts. What does drawing a straight line through its plotted points establish?

Review the assumptions, not only the arithmetic

Items sell for eight dollars each, with a forty-dollar setup cost and a three-dollar cost per item produced. The profit model assumes every produced item is sold. If twelve are produced but only ten sell, revenue and production cost use different counts

If all twelve sell, profit is twenty dollars. With only ten sales, it is four dollars

A bulk discount, changed unit price, extra fee, capacity limit or changing flow rate can also break a single constant-rate rule

Compare predicted outputs with observations. A systematic mismatch may require revised parameters, a restricted domain or more than one rule rather than a forced straight line

A fictional batch costs forty dollars to set up and three dollars per item produced. Items sell for eight dollars each. Twelve are produced, ten sell and unsold items have no resale value. Which calculation gives the batch result?

Construct, test and revise a practical linear model

Independent written and digital modelling task, not automatically marked

Choose a practical constant-rate situation or a fictional small sale with fixed and per-item costs. State a question that requires a prediction or target input

Define variables, units, supplied values and assumptions. State a sensible domain and explain whether its inputs are whole counts or continuous quantities

Construct an equation, an input-output table and a graph with labelled scales. Explain which representation best answers your question and why

Use hand calculations and a digital table or graphing tool to solve the problem. Retain formulas and exact values so another person can reproduce the result

Interpret the result with its unit and check it by substitution, contextual limits and any relevant nearby whole counts

Change one realistic assumption, such as the sold proportion, fee, rate or capacity. Revise the model and explain how the conclusion changes

For a financial model, distinguish revenue, total cost, signed profit and the input at break-even. Review the model and explanation against these instructions or with a teacher

Keep the model connected to its practical meaning

An initial value and constant signed rate determine a linear rule; the rate has output units per input unit

An equation supports exact calculation, a table records selected pairs and a graph displays relationships. None changes the stated practical domain

To find an output, substitute the input. To find an input, set the model equal to the target and solve, then check the domain

Decreasing quantities need negative rates, and physical restrictions can stop a linear rule before arbitrary algebraic inputs

Revenue minus all stated costs gives signed profit. Break-even means zero profit, while a positive-profit whole count must satisfy the strict condition

Check the original relationship and interpret the answer in context. Revise assumptions or rules when observations and conditions no longer support the model