Model lengths, areas and volumes with ratios

A usable length of twenty metres is divided into A and B in the ratio two to three. What fraction of the total belongs to A?

Turn a practical design into a checkable model

Formulate the question: what must be found, and which conditions must the design satisfy?

List the measured or supplied quantities, their units, the ratio order and the usable whole

State assumptions such as negligible cutting loss or divider thickness. These connect the real object with a simpler mathematical representation

Represent the situation with a ratio strip, labelled plan or geometric formula. Choose the representation because it exposes the relevant relationship

Calculate, then interpret the answer with its unit and role in the situation

Check the total, the ratio and any geometric constraints. If measurements or conditions disagree, revise the model or collect more information rather than hiding the mismatch

Use equal ratio parts for an additive allocation

If A to B is two to three, the entire allocation contains five equal ratio parts

T denotes the whole usable quantity. It could be a length, an area or a volume

The ratio is unitless when both quantities use the same unit. The calculated portions retain the unit of T

A ratio strip shows the relative portions without pretending to show the physical shape

A labelled rectangle or box can establish the total area or volume before the ratio is applied

Allocate a usable ribbon length

A ribbon has usable length 20 m. It will be cut into pieces A and B in the length ratio 2 to 3

Use all the ribbon and assume no length is lost at the cut

Calculate the length assigned to A in m. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use the stated usable ribbon length

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Find the second length rather than the first

Keep an exact decimal length

A ribbon has usable length 7.5 m. It will be cut into pieces A and B in the length ratio 2 to 3

Use all the ribbon and assume no length is lost at the cut

Calculate the length assigned to A in m. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use the stated usable ribbon length

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Apply the fraction without rounding

The usable whole may be smaller than the starting material

Suppose a ribbon starts at ten metres, but the design reserves one metre for testing and trimming

Allocate the remaining nine metres in the ratio two to three

The two finished pieces total nine metres, not ten. The reserved metre is accounted for separately

Do not silently ignore a stated loss or subtract an invented allowance. Ask for missing information when it changes the result

A rectangular garden is 8 m long and 5 m wide. Its entire area is allocated to A and B in the ratio three to two, with no paths. Which model finds the area of A?

Find total area before dividing it in a ratio

A rectangular plan gives total area as length multiplied by width

For A to B in the ratio a to b, use fractions of that complete area

The letters a and b count ratio parts; they are not side lengths

These area allocations do not by themselves determine the shape of either region. More layout conditions are needed to locate a dividing line

Allocate a rectangular plan area

A rectangular garden plan has usable length 10 m and width 4 m. It assigns regions A and B in the area ratio 2 to 3

The entire rectangle is allocated. Ignore path or divider area. Treat the stated dimensions as fixed model values

Calculate the area assigned to A in m². Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width for the complete rectangular area

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Use the complete rectangle and the requested portion

Combine decimal area with a part-whole fraction

A rectangular garden plan has usable length 7.5 m and width 4 m. It assigns regions A and B in the area ratio 2 to 3

The entire rectangle is allocated. Ignore path or divider area. Treat the stated dimensions as fixed model values

Calculate the area assigned to A in m². Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width for the complete rectangular area

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Keep the dimensions and the ratio separate

The rectangle has width three metres. A occupies a equal length parts and B occupies three. Change a to change the ratio, or change k while holding a fixed to scale both portions

When a is 2 and k is 3, which statement describes the area allocation?

An area allocation determines a strip length only with a layout condition

Suppose both regions are rectangular strips across the same full width W, with the dividing line perpendicular to the total length

After calculating the assigned area A, divide by that common width

In a ten-metre by four-metre rectangle with area ratio two to three, A has area sixteen square metres

The other strip is six metres long. Since both strips have the same width, their length ratio and area ratio agree

Without the common-width condition, an area ratio alone does not imply the same ratio between arbitrary side lengths

Recover the full-width strip length

A rectangular garden plan has usable length 10 m and width 4 m. It assigns regions A and B in the area ratio 2 to 3

The entire rectangle is allocated. Ignore path or divider area. Treat the stated dimensions as fixed model values

Each region is a rectangular strip extending across the full width. The dividing line is perpendicular to the stated length. Find the length of A, measured in the same direction as the total length

Calculate the partition length for A in m. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width for the complete rectangular area

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

The strip area is its length times the full width

Check this partition length against the total length

Check the portions in the stated A-to-B order

Divide out the common width

Rectangle A measures 2 m by 6 m and rectangle B measures 3 m by 2 m. Their named first side lengths are in the ratio two to three. Are their areas in that ratio?

Use internal dimensions for a volume allocation

A rectangular box is modelled with usable internal length L, width W and height H

If two compartments fill that entire volume with negligible divider thickness, their volumes add to T

Use cubic units for the allocated volumes. External dimensions can overstate usable space when walls or fittings take up room

If both compartments extend through the same width and height, the shared cross-sectional area is WH

This is an additive compartment model. It does not assert that scaling all three dimensions by a factor scales volume by that same factor

Allocate the complete usable box volume

A rectangular display box has usable internal length 20 cm, width 4 cm and height 3 cm. Its compartment volumes A and B must be in the ratio 2 to 3

Both compartments use the full width and height. Assume a divider with negligible thickness, no gaps and fixed model dimensions

Calculate the volume assigned to A in cm³. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width times height for the complete usable box volume

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Find the second compartment volume

Keep exact decimal internal dimensions

A rectangular display box has usable internal length 12.5 cm, width 4 cm and height 3 cm. Its compartment volumes A and B must be in the ratio 2 to 3

Both compartments use the full width and height. Assume a divider with negligible thickness, no gaps and fixed model dimensions

Calculate the volume assigned to A in cm³. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width times height for the complete usable box volume

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

Allocate an exact decimal-model volume

Recover compartment length from its assigned volume

A rectangular display box has usable internal length 20 cm, width 4 cm and height 3 cm. Its compartment volumes A and B must be in the ratio 2 to 3

Both compartments use the full width and height. Assume a divider with negligible thickness, no gaps and fixed model dimensions

Each compartment extends across the full width and height. The dividing plane is perpendicular to the stated length. Find the length of A, measured in the same direction as the total length

Calculate the partition length for A in cm. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

Use length times width times height for the complete usable box volume

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

The compartment volume is its length times the common width and height

Check this partition length against the total length

Check the portions in the stated A-to-B order

Divide by width times height, not by one side alone

A known portion is not a known total

If A to B is two to three and A is eight metres, the eight metres represent two ratio parts

Do not multiply eight by two fifths: that would treat a known part as the whole

The same reasoning works with a supplied area or volume portion, provided the ratio compares those same quantities

Check by applying the fraction to the recovered whole and obtaining the known portion again

Recover total length from one piece

Two ribbon lengths are labelled A and B

Their length ratio A to B is 2 to 3. A measures 8 m. Together A and B make the entire usable length

Calculate the total usable length in m. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

8 m represents 2 ratio parts, not the whole

Check that applying the requested fraction to this total recovers the supplied portion

Check the portions in the stated A-to-B order

Use the ratio term belonging to the known second piece

Recover total area from its first portion

Two areas of a rectangular display are labelled A and B

Their area ratio A to B is 2 to 3. A measures 18 m². Together A and B make the entire usable area

Calculate the total usable area in m². Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

18 m² represents 2 ratio parts, not the whole

Check that applying the requested fraction to this total recovers the supplied portion

Check the portions in the stated A-to-B order

Keep the recovered total in square units

Recover usable volume from one compartment

Two compartment volumes in a rectangular box are labelled A and B

Their volume ratio A to B is 2 to 3. A measures 36 cm³. Together A and B make the entire usable volume

Calculate the total usable volume in cm³. Give an exact whole number or decimal

The ratio compares A with B, not a part with the whole

36 cm³ represents 2 ratio parts, not the whole

Check that applying the requested fraction to this total recovers the supplied portion

Check the portions in the stated A-to-B order

Recover cubic units without guessing missing side lengths

A total check and a ratio check are different

Two positive portions can fill the whole but still have the wrong ratio

A two-hundred-and-forty-cubic-centimetre box with target ratio two to three needs volumes ninety-six and one-hundred-and-forty-four cubic centimetres

A draft with A equal to ninety and B equal to one hundred and fifty has the right total but the wrong ratio

Increase A by six and decrease B by six to keep the total fixed

A numerical correction belongs to the stated model. A real build still needs measurement checks and allowance for any non-negligible walls or divider

Correct a portion below the target

A rectangular display box has usable internal length 20 cm, width 4 cm and height 3 cm. Its compartment volumes A and B must be in the ratio 2 to 3

Both compartments use the full width and height. Assume a divider with negligible thickness, no gaps and fixed model dimensions

A draft allocation gives compartment A volume 90 cm³. Compare that allocation with the requested ratio, keeping the total volume fixed

Calculate the target volume of A, compare it with the draft and state how both compartments must change while keeping the total fixed

The ratio compares A with B, not a part with the whole

Use length times width times height for the complete usable box volume

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

The draft A portion is below the ratio target by 6 cm³

Increase A by this amount and decrease the other compartment by the same amount. Recheck the ratio and the unchanged total

This comparison does not identify a cause for any physical mismatch. Check dimensions and divider assumptions before making a real object

Increase the deficient portion and offset the change

Correct a portion above the target

A rectangular display box has usable internal length 15 cm, width 4 cm and height 3 cm. Its compartment volumes A and B must be in the ratio 2 to 3

Both compartments use the full width and height. Assume a divider with negligible thickness, no gaps and fixed model dimensions

A draft allocation gives compartment A volume 80 cm³. Compare that allocation with the requested ratio, keeping the total volume fixed

Calculate the target volume of A, compare it with the draft and state how both compartments must change while keeping the total fixed

The ratio compares A with B, not a part with the whole

Use length times width times height for the complete usable box volume

Apply the selected part as a fraction of the whole

The other portion is the remainder, so the two portions still use the whole

Check the portions in the stated A-to-B order

The draft A portion is above the ratio target by 8 cm³

Decrease A by this amount and increase the other compartment by the same amount. Recheck the ratio and the unchanged total

This comparison does not identify a cause for any physical mismatch. Check dimensions and divider assumptions before making a real object

Decrease the excess portion rather than increasing it further

A box model allocates its entire internal volume between two compartments while treating the divider thickness as zero. The chosen divider actually occupies a measurable volume. Which action is justified?

Design your own practical ratio-allocation model

Independent written and digital modelling task

Choose a problem about dividing a usable length, a rectangular area or a rectangular box volume between two purposes. State your own question, dimensions and positive whole-number ratio in a named A-to-B order

Decide whether your dimensions describe the usable interior or the outside. State assumptions about cutting loss, borders, walls and divider thickness rather than silently treating them as zero

Draw a labelled geometric plan with the dimensions needed to calculate the usable whole. Add a separate ratio strip if it helps show equal parts, and explain why each representation is useful

Calculate the usable whole and both target portions with appropriate units. If your question asks for partition lengths, choose a full-width or full-cross-section layout and use the unchanged perpendicular dimensions

Use hand calculations or a digital table, retaining inputs and intermediate results so another person can reproduce your model

Check the ratio, total and physical fit separately

Check that both portions add to the usable whole and that their ratio simplifies to the stated A-to-B ratio. A correct total alone does not establish a correct allocation

Check any partition lengths against the dimensions of the chosen layout. Do not infer an area or volume ratio from arbitrary side lengths when perpendicular dimensions differ

Estimate the sizes, verify units and repeat at least one calculation using a different method or a hand check of a digital result

Change one assumption, such as allowing for a measurable divider or a lost strip, and explain which usable measurement and calculations need revision

Report the allocation in context, justify your chosen representations and distinguish a calculation from a physically measured or constructed result

Supplied calculations and selected model decisions are automatically checked here. Your own formulated problem, drawing, digital work, physical design and written justification are not automatically checked; review them against this checklist or discuss them with a teacher

Make the representation and the checks explicit

Identify what is compared, in what order, and which quantity is the usable whole

A ratio strip makes equal parts visible. A geometric plan or box model establishes area or volume from relevant dimensions

For a known portion, divide by its ratio term and multiply by the sum to recover the whole

To find a partition length, divide the assigned area or volume by the unchanged perpendicular dimensions

Report the answer with its correct role and unit. Check the total, the ratio and the physical constraints separately

State assumptions and revise them when observations or design conditions make them unsuitable