Model practical problems with rational numbers

Pack A costs less than Pack B, but their quantities are different. Can the prices alone establish which has the lower cost per item?

Formulate, calculate, interpret and check

Formulate: state what must be found, list the given quantities and units, and identify assumptions or restrictions

Calculate: choose operations or a table that represents those quantities. Use exact fractions or decimals where practical

Interpret: answer the original question with the right units and explain what the number means

Check: test the result against the quantities and restrictions, estimate its size, and revise the model if an assumption is unsuitable

A calculator or table can carry out arithmetic, but it cannot decide whether your chosen calculation models the question

Decide what a fair price comparison assumes

Compare the same product or equal-sized items of the same quality. Use the listed prices and state whether fees, discounts or wastage are included

Use the same quantity unit for both offers. A price per litre cannot be compared directly with a price per hundred millilitres

The examples use fictional stated prices, equal quality, and no extra fees or discounts. They are mathematical comparisons, not claims about current shop prices

A model compares the price per roll in two paper packs and ignores roll length. Which assumption is needed for that comparison to measure value by quantity?

Divide to find the cost of one common unit

Six equal-sized items costing five dollars forty cents have this price per item

The unit price is ninety cents per item. Multiplying by six recovers the five-dollar-forty pack price

For a volume in millilitres, divide by the millilitre quantity and multiply by one thousand to find dollars per litre

One litre is one thousand millilitres, and one kilogram is one thousand grams. Write money answers to two decimal places in the price tasks

Find a rate from a fractional litre

A brand of fresh milk is sold in two sizes. All packs contain the same product of equal quality

Use the listed prices only, with no extra fees or discounts

Find the price of Pack A per 1 L. Write dollars to two decimal places

Use one common quantity to make the comparison meaningful

Divide the pack price by its quantity, then scale to 1 L

The rate for Pack A is $4.00 per 1 L

The inverse calculation recovers the listed pack price

Find the cost per equal-sized item

Compare rates, not the original pack prices

For each offer, calculate the price of the same quantity. Compare those rates and name the lower one

Pack A contains 500 g and costs $2.00. Pack B contains 1 kg and costs $3.50

Pack B has the lower price per kilogram even though its pack costs more

Check by imagining the same one-kilogram quantity: two Pack A bags would cost four dollars, while one Pack B bag costs three dollars fifty cents

Compare two mass offers and state the better rate

A brand of rice is sold in two sizes. All packs contain the same product of equal quality

Use the listed prices only, with no extra fees or discounts

Compare the price per kg. Which option is the better buy?

Compare the exact rates; the displayed prices are rounded only where needed

Pack A, in dollars per kg:

Pack B, in dollars per kg:

Pack B has the lowest price per kg, so it is the better buy

What is the unit price of the better option, per kg? Give your answer to the nearest cent

Pack B, in dollars per kg:

A smaller pack can have the lower rate

Convert the displayed quantities before comparing

When one label is in litres and another is in millilitres, convert before dividing

Do not divide one price by five hundred and the other by one while pretending those denominators are the same unit

Compare matching litres despite different labels

A bottle contains seven hundred and fifty millilitres and costs three dollars. Which calculation gives dollars per litre?

Decide whether the model multiplies or divides

For identical groups, multiply the amount per group by the number of groups

To find how many equal portions fit, divide the total amount by the portion size

State whether the result must be a whole number and whether leftovers are allowed. A fractional amount of liquid is meaningful, but a fractional person is not

Scale a fixed amount per group

Each of twelve groups receives 0.75 litres of water. Assume no water is lost. Find the total volume in litres

The same volume is needed for each group

Nine litres are needed in total

The inverse calculation recovers the stated volume per group

Calculate a practical total

Find a number of equal portions

A fraction or percentage selects part of a stated whole

Identify the reference whole before applying a fraction or percentage

If the whole counts people or objects, check that an exact stated proportion produces a possible whole-number count

Do not silently round a fractional person. Recheck whether the percentage is exact, rounded, or only a prediction

Select a fraction of a measured amount

Find the count selected by an exact percentage

Separate income, costs and the remaining balance

An event sells forty tickets at two dollars fifty cents each. Its only cost is sixty-eight dollars fifty cents

The event has thirty-one dollars fifty cents remaining after the stated cost

This model assumes every ticket is paid for, the given cost is complete, and no other money enters or leaves

Calculate income from the original quantities

Thirty-six tickets are sold and paid for at two dollars fifty cents each. Find the total ticket income in dollars

Multiply the ticket count by the price per ticket

The ticket income is ninety dollars

The inverse check recovers the ticket price

Build the complete two-step event calculation

An event receives forty dollars and pays forty-seven dollars fifty cents. Which conclusion matches income minus costs?

Whole packs change the question

The lowest unit price describes value per equal quantity. The lowest checkout cost depends on how much is needed and which purchases are allowed

Suppose Pack A is one litre for two dollars fifty cents and Pack B is two litres for four dollars

If only one litre is needed and only whole packs can be bought, one Pack A costs less than one Pack B

For these tasks, buy only one pack type, meet at least the required quantity, and allow unused contents. Mixing pack types is not permitted by this model

Compare feasible checkout totals

A brand of fresh milk is sold in two sizes. All packs contain the same product of equal quality

Use the listed prices only, with no extra fees or discounts

You need at least 1,000 mL. Buy whole packs of only one pack type, with no mixing of pack types. Unused contents are allowed

Find the minimum total cost in dollars. Write two decimal places

For each pack type, find the smallest whole number of packs that meets the requirement

Pack A: 1 whole pack is needed

Pack B: 1 whole pack is needed

Buying only Pack A gives the lower total cost: $2.50

The lower unit price need not give the lower checkout total, because you cannot buy part of a pack

Include every whole pack needed

Apply the whole-pack restriction to equal-sized items

Two feasible purchases can tie in total cost

Pack A contains one litre and costs $2.50. Pack B contains two litres and costs $4.00. The table buys whole packs of only one type and allows unused contents

At a requirement of three litres, which conclusion is correct under these conditions?

Describe a repeatable whole-pack procedure

For each pack type, start with zero packs and zero supplied quantity

While the supplied quantity is less than the requirement, add one pack and its quantity

When the requirement is met, multiply the pack count by the pack price

Repeat for the other pack type, then compare the two total costs and report any tie

A digital table can instead divide required quantity by pack size and round the pack count upwards to a whole number. This is not ordinary nearest-number rounding

Check that the chosen count supplies enough, while one fewer pack would not. Only then compare costs

A requirement is seven items and a pack contains six items. A procedure adds one whole pack at a time. When should it stop?

A calculation found the cheapest purchase when only one pack type could be bought. A shop now allows mixing pack types. What can be concluded?

Formulate and design your own best-buy model

Independent written and digital modelling task

Choose a fictional need for identical items or a measured quantity, then invent two pack sizes and positive prices. Write your own question asking for the lowest total cost that meets the need

State a common quantity unit, equal-quality assumption and whether unused contents are acceptable. Start with whole packs of one type only and no extra fees, so your model matches the procedure taught here

Choose a representation such as a labelled table, diagram or numbered procedure. Explain why it makes the quantities and buying restrictions clear

Write your own repeatable algorithm from inputs to conclusion. Include a starting state, an update, a stopping condition, a cost calculation and a comparison that can report a tie

Use hand calculations or a digital table to carry out your procedure. Keep the original inputs and intermediate results so another person can reproduce them

Test the procedure and communicate its limits

Test a requirement smaller than one pack, a requirement exactly divisible by a pack size, and a requirement that leaves a partial pack. Explain why ordinary nearest-number rounding can fail

For each pack type, verify that your chosen pack count supplies enough and that one fewer pack does not. Compare both feasible costs before naming a winner or tie

Check your arithmetic independently by multiplication or repeated addition. If using a digital table, compare at least one row with your hand calculation

Change the required quantity and explain whether the preferred pack changes. If you allow mixed pack types, fees or wastage, explain why the original two-cost comparison is no longer a complete model

Communicate the chosen purchase, its cost, any unused quantity and the assumptions that limit your conclusion. Justify your representation and explain any revision prompted by your tests

This topic automatically checks supplied calculations and selected modelling decisions. Your independently formulated problem, algorithm, digital work and written justification are not automatically checked; review them against this checklist or discuss them with a teacher

A mathematical answer must fit its model

State the question, givens, units and assumptions before calculating

Use a common unit for a fair rate comparison. Choose multiplication, division, a fraction, a percentage or a balance calculation to match the quantities

Keep whole-pack, whole-person and no-mixing restrictions visible. A lower unit rate does not automatically mean a lower feasible total cost

Use a clear calculation procedure, with digital tools where useful, then check by an inverse calculation, an estimate or the original constraints

Finish with a contextual conclusion and explain when a changed assumption requires recalculation