The most common Problem-Solving and Data Analysis type (4.9% of the bank): proportional reasoning and unit conversion. Covers setting up a proportion with units attached, cancelling units like algebra as a check on the setup, chained conversions, scale drawings, the difference between part-to-part and part-to-whole ratios, and telling direct from inverse proportion — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 9 of 19
4.9% of the question bank — 82 of 1675 questions
Objectives
A proportion is two equal ratios, and a rate is a ratio with different units on top and bottom. The arithmetic is the easiest on the Math section. The errors are not rare, and essentially all of them happen before any arithmetic starts, in deciding which number goes on top.
One habit carries the whole type: attach the units to every number, and treat them as algebra. They will tell you whether to multiply or divide before you have to decide.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 82 tagged questions of this type in the site's bank
Section
Section 1
Concept
A proportion is two equal ratios, and a rate is a ratio with different units on top and bottom. The arithmetic is the easiest on the Math section. The errors are not rare, and essentially all of them happen before any arithmetic starts, in deciding which number goes on top.
You will see it phrased in these ways:
The signature words are per, for every, at this rate, and in the ratio of. Any of them tells you a proportion is coming.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Ratios, rates, proportions, and units: the tell, the move, and the trap
The green panel describes a check that runs before the arithmetic rather than after it. That is unusual and it is why this type is so improvable.
Prediction
Proportional means a constant ratio, and it is stricter than merely increasing together.
Predict first
Three of these are proportional. Which is not?
Correct: A taxi charges 4 dollars plus 2 dollars per mile
Why: A proportional relationship passes through the origin: zero of one thing means zero of the other, and the ratio is the same at every point. The taxi has a 4 dollar flag drop, so a zero-mile trip still costs 4 dollars and the ratio of cost to distance changes with every mile. It is linear but not proportional. The other three all have a fixed ratio and give zero when the input is zero.
Concept
Four shapes, all solved by the same setup with the units attached.
| variant | what is constant | the setup |
|---|---|---|
| Rate problem | the rate itself | quantity over time, or cost over amount |
| Unit conversion | the equivalence, such as 12 inches per foot | multiply by a fraction equal to 1 |
| Scale drawing | the scale factor | drawing over real, on both sides |
| Inverse proportion | the product, not the ratio | first times second equals a constant |
The last row is the odd one out and the least practised. In inverse proportion, doubling one quantity halves the other, so the usual proportion setup gives exactly the wrong answer.
Definition probe
Units decide the setup, so name them first.
Sort into buckets
What units does each quantity carry?
Discrimination
Six relationships. Which way do they move?
Sort into buckets
Is the relationship direct or inverse?
Warm-up
Do not think about whether to multiply or divide. Look at the units.
Discussion prompt
A car uses 8 litres of petrol per 100 km. How much petrol does a 250 km journey need?
Hint: Write the rate as a fraction with its units, and see which unit cancels.
Answer:
Write the rate with units: 8 litres over 100 km.
Multiply by the journey: 250 km times (8 litres over 100 km). The km cancels top and bottom, leaving litres — which is what was asked for.
So the answer is 250 times 8 divided by 100, which is 20 litres.
Notice you never decided whether to multiply or divide. The units decided it: to end up with litres, the km had to cancel, which forced the rate to sit that way up.
The crossed setup, 250 km times (100 km over 8 litres), leaves km squared per litre — visibly meaningless, and it fails before you compute.
Pattern
Three steps, and the check happens in the middle rather than at the end.
Write every number with its units, including the rate as a fraction.
Why: A rate is a fraction with different units above and below, and writing it that way is what makes the next step possible.
Arrange the multiplication so the unwanted units cancel, and confirm the survivors are what the question asked for.
Why: This is the setup check. If the surviving units are wrong, the arrangement is wrong, and you know before any arithmetic.
Compute, then state the answer with its units.
Why: The units are part of the answer, and on a grid-in question they tell you which number to type.
Step 2 is the whole method. It replaces a decision about multiplying or dividing with an observation about what cancels.
Section
Section 2
Concept
Set up both sides with the same quantity on top and the same quantity underneath.
Consistency of order is exactly the same discipline as consistency when computing a slope. Mixing the orders is the classic setup error.
Concept
Treat a unit as a symbol: the same unit above and below cancels, and what survives is the unit of the answer.
This is the highest-value habit in the deck. It converts a judgement call into an observation.
Prediction
Which arrangement leaves the right unit standing?
Predict first
A rope costs 3 dollars per metre. Which computation gives the cost of 12 metres?
Correct: 12 m times (3 dollars over 1 m)
Why: The metres cancel, leaving dollars, which is the unit the question asked for. The answer is 36 dollars. The second arrangement leaves metres squared per dollar, which is meaningless. The third gives dollars but only by accident, since the metres cancel against each other rather than against the rate. Letting the units decide removes the guesswork.
Concept
Any equivalence, such as 12 inches equals 1 foot, gives two fractions that both equal 1.
| equivalence | fraction to use | when |
|---|---|---|
| 12 inches = 1 foot | 12 inches over 1 foot | to turn feet into inches |
| 12 inches = 1 foot | 1 foot over 12 inches | to turn inches into feet |
| 1000 m = 1 km | 1 km over 1000 m | to turn metres into km |
| 60 min = 1 hour | 60 min over 1 hour | to turn hours into minutes |
You never have to remember whether to multiply or divide by 12. You choose the fraction that cancels what you want gone.
Prediction
Add the parts first.
Predict first
Red and blue marbles are in the ratio 3 to 7, and there are 40 marbles. How many are red?
Correct: 12
Why: The parts add to 10, so red is 3 out of 10 of the whole. Three tenths of 40 is 12. Answering 3 gives the ratio number rather than a count, and 28 is the number of blue marbles — the right method applied to the wrong colour. The 17 comes from adding 3 and 7 and then subtracting from 20, which does not follow from anything.
Concept
Several conversions multiply together, and the intermediate units cancel in sequence.
Writing the whole chain before computing anything is safer than converting in stages, because every cancellation is visible at once.
Concept
A ratio of 3 to 5 means 3 parts to 5 parts, and therefore 3 out of 8 of the whole.
| the ratio 3 to 5 means | value |
|---|---|
| first part to second part | 3 to 5 |
| first part to the whole | 3 out of 8 |
| second part to the whole | 5 out of 8 |
| total parts | 8 |
Whenever you see a ratio with a colon, immediately write down the total number of parts. It is the number the question usually needs and never states.
Prediction
Fewer workers means more days.
Predict first
If 6 machines fill an order in 10 hours, how long do 4 machines take?
Correct: 15 hours
Why: The total work is 6 times 10, which is 60 machine-hours, and that total is fixed. With 4 machines the time is 60 over 4, which is 15 hours. The sanity check confirms it: fewer machines must take longer, and 15 is longer than 10. Answering 6.7 sets up a direct proportion and gets the direction backwards, which the sanity check would have caught.
Concept
Write drawing over real on both sides of the proportion, then solve.
Scale questions are ordinary proportions with a longer story attached. The setup discipline is identical.
Concept
When one quantity doubles and the other halves, multiply them rather than dividing.
The sanity check is free: fewer workers must take MORE days. If your answer moved the wrong way, you used the wrong kind of proportion.
Prediction
Two fractions, both equal to 1.
Predict first
Which chain converts 5 metres per second into kilometres per hour?
Correct: 5 times (1 km over 1000 m) times (3600 s over 1 hour)
Why: Starting from metres over seconds, the metres must cancel against a metres underneath, so the km fraction goes with 1 km on top. The seconds must cancel against a seconds on top, so the time fraction goes with 3600 s on top. What survives is km over hour. The result is 5 times 3600 over 1000, which is 18 km per hour — a sensible walking-to-cycling speed, confirming the direction.
Two truths and a lie
Three statements about ratios and rates are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: A proportional relationship is a linear one that passes through the origin, so proportional relationships are a subset of linear ones rather than the same thing. A taxi fare of 4 dollars plus 2 per mile is linear but not proportional: doubling the distance does not double the fare, because the flag drop does not double. The distinction matters because only proportional relationships can be solved with a simple ratio.
Check
Let the units pick the operation.
Check your understanding
A tap delivers 18 litres per minute. How many litres flow in 2.5 hours?
Answer: A
Why: Convert the time first: 2.5 hours is 150 minutes. Then 150 minutes times 18 litres per minute gives 2,700 litres, with the minutes cancelling and litres surviving.
Choice D is the intermediate quantity — a genuinely useful number on the way to the answer, and the wrong one to select. Convert first, then compute once.
Section
Section 3
Worked example
A printer produces 14 pages per minute. How long does it take to print 350 pages?
Figure (svg): A number line with the answer of twenty-five minutes marked
Write the rate with units: 14 pages over 1 minute.
Why: A rate is a fraction, and writing it as one makes the cancelling visible.
The answer must be in minutes, so pages must cancel — which means the rate goes underneath.
Why: 350 pages divided by (14 pages per minute) leaves minutes standing.
Compute 350 divided by 14, which is 25 minutes.
Why: The arithmetic is the last and easiest step.
The crossed setup would give 350 times 14, which is 4,900 — and its units would be page-minutes, which is not a time. The units flag it before the arithmetic.
Verify: check forwards: 25 minutes at 14 pages a minute is 350 pages.
Why: Multiplying back reproduces the quantity given, so the direction was right.
Answer: 25 minutes
Worked example
Convert 72 kilometres per hour into metres per second.
Figure (svg): A table showing the two conversion fractions and which units cancel
Write the chain: 72 km over 1 hour, times 1000 m over 1 km, times 1 hour over 3600 s.
Why: Each fraction equals 1, so the value is unchanged and only the units move.
Check the cancelling: km against km, hour against hour, leaving metres over seconds.
Why: This confirms the setup before any arithmetic is attempted.
Compute: 72 times 1000 over 3600, which is 72000 over 3600, or 20.
Why: Doing the arithmetic once at the end avoids rounding at intermediate stages.
The useful shortcut worth remembering: to convert km per hour into m per second, divide by 3.6. Here 72 over 3.6 is 20.
Verify: sanity-check the size: 72 km per hour is a fast road speed, and 20 m per second is about the same thing.
Why: A person walks at roughly 1.5 m per second, so 20 is plausibly vehicular.
Answer: 20 metres per second
Worked example
A mixture of sand and cement is in the ratio 5 to 2. A batch weighs 63 kg. How much cement is in it?
Figure (svg): Bars showing the batch split into five parts sand and two parts cement
Add the parts: 5 plus 2 is 7 parts in total.
Why: The ratio is part-to-part, and the whole is only available after adding.
One part is 63 divided by 7, which is 9 kg.
Why: Finding the value of a single part converts the ratio into quantities.
Cement is 2 parts, so 2 times 9, which is 18 kg.
Why: Multiply the part value by the number of parts that quantity has.
The one-part method is worth adopting generally. Once you know a single part is 9 kg, every question about that mixture is one multiplication away.
Verify: check the total: sand is 5 times 9, which is 45, and 45 plus 18 is 63 kg.
Why: The two quantities reconstruct the given batch weight, confirming the split.
Answer: 18 kg of cement
Step zero
Before deciding to multiply or divide.
Discussion prompt
A question says a machine fills 240 bottles per hour and asks how long it takes to fill 1,800 bottles. What do you write first, and how do you know whether to multiply or divide?
Hint: Write the rate as a fraction with its units.
Answer:
Write the rate as a fraction: 240 bottles over 1 hour.
Then ask what unit the answer needs. The question asks how long, so the answer is in hours.
For hours to survive, bottles must cancel — which means dividing by the rate rather than multiplying by it.
So it is 1800 bottles divided by (240 bottles per hour), giving 7.5 hours.
You never made a decision about multiplying or dividing. You observed which arrangement leaves hours standing, and the arrangement followed.
Worked example
A map has a scale of 2 cm to 15 km. Two towns are 7 cm apart on the map. How far apart are they really?
Figure (svg): A number line comparing the two map distances with their real distances
Write the proportion with matching orders: 2 cm over 15 km equals 7 cm over x km.
Why: Map distance on top on both sides, real distance underneath on both sides.
Cross-multiply: 2x equals 105.
Why: Seven times fifteen is 105, and the cm units cancel on the cross-multiplication.
Divide: x equals 52.5 km.
Why: Solving the resulting linear equation gives the real distance.
A quick proportional sanity check like that catches a crossed setup instantly, because a crossed setup moves the answer the wrong way.
Verify: check the direction: 7 cm is more than three times 2 cm, and 52.5 is more than three times 15.
Why: The two sides scaled by the same factor, so the proportion held.
Answer: 52.5 km
Worked example
Eight painters finish a job in 15 days. How long would 12 painters take, working at the same rate?
Figure (svg): Bars showing the constant total of 120 painter-days against the incorrect direct-proportion answer
Compute the fixed total: 8 painters times 15 days is 120 painter-days.
Why: The job itself does not change, so the product of workers and time is constant.
Divide by the new number of painters: 120 over 12, which is 10 days.
Why: The same total work spread across more painters takes less time each.
Sanity-check the direction: more painters, fewer days. Ten is less than fifteen.
Why: An inverse relationship must move the answer the opposite way to the input.
The direct-proportion answer would have been 22.5 days — more painters taking longer, which is obviously absurd. The sanity check is free and decisive.
Verify: check the product: 12 times 10 is 120, matching the original total.
Why: The invariant is preserved, which is the defining property of inverse proportion.
Answer: 10 days
Faded example
From memory. The habit that carries this type.
Fill in the blanks
Write the units on every number. Arrange the multiplication so the unwanted units cancel. If the surviving units are what the question asked for, the setup is right. And a conversion works by multiplying by a fraction equal to 1.
Why: The power of the third line is that the check happens BEFORE the computation rather than after it. On most question types you verify an answer; here you verify a setup, which is where all the errors on this type actually live.
Worked example
Shop A sells 5 kg of rice for 12 dollars. Shop B sells 8 kg for 18.40 dollars. Which is cheaper per kilogram?
Figure (svg): Two points plotted with a line through the origin, showing which lies below
Reduce both to a common unit: shop A is 12 over 5, which is 2.40 dollars per kg.
Why: Rates are only comparable when expressed per the same amount.
Shop B is 18.40 over 8, which is 2.30 dollars per kg.
Why: The same division applied to the second shop.
Compare: 2.30 is less than 2.40, so shop B is cheaper.
Why: The lower price per unit is the better value.
Always reduce to a common unit before comparing. Comparing 12 dollars with 18.40 dollars directly is meaningless, since they buy different amounts.
Verify: check at a common amount: 40 kg would cost 96 dollars at A and 92 at B.
Why: Scaling both to the same quantity confirms B is cheaper, independent of the per-unit arithmetic.
Answer: Shop B, at 2.30 dollars per kg against 2.40
Fill the middle
Part-to-part and part-to-whole.
Fill in the blanks
A ratio of 4 to 5 has 9 parts in total. The first quantity is 4 out of 9 of the whole. If the whole is 45, one part is worth 5, so the second quantity is 25.
Why: The one-part method in the third blank is the most useful move on ratio questions. Once you know what a single part is worth, every quantity in the problem is one multiplication away, and you never have to build a fraction of the whole for each one separately.
Estimation
A rough answer catches a crossed setup instantly.
Predict first
A car travels 468 km using 36 litres. Roughly what is its fuel consumption in km per litre?
Correct: about 13
Why: Round to 450 over 36, or roughly 480 over 40, which is 12. The exact value is 13. The answer 0.08 is the crossed setup, litres over km, which is a valid quantity but not the one asked for. Having a rough expectation before dividing means a crossed setup is caught immediately, because it lands two orders of magnitude away.
Check
Add the parts before taking any fraction.
Check your understanding
Two classes have students in the ratio 4 to 7. Together they have 88 students. How many are in the larger class?
Answer: A
Why: The parts add to 11, so one part is 88 divided by 11, which is 8 students. The larger class has 7 parts, so 7 times 8, which is 56. Checking: the smaller class has 4 times 8, which is 32, and 56 plus 32 is 88.
Choices A and B are the two classes, and the only thing separating them is the word larger. Underline it before computing.
Section
Section 4
Trap
The trap. A car uses 8 litres per 100 km and you want the petrol for 250 km, so you write 250 times 100 over 8.
That gives 3,125 — a wildly wrong number, and the units are km squared per litre rather than litres.
The arithmetic is correct. The fraction was simply the wrong way up.
The fix. Write the units on every number and arrange the multiplication so the unwanted unit cancels.
If the surviving units are not the ones asked for, the setup is upside down and you know immediately.
Trap
The trap. A ratio of 3 to 5, with a total of 40, and you compute three fifths of 40, getting 24.
But 3 to 5 is part-to-part: the parts total 8, so the first quantity is three eighths of 40, which is 15.
Both 24 and 15 will be among the choices, and the wrong one comes from a completely reasonable-looking calculation.
The fix. Add the parts the moment you see a ratio, and write the total down.
Then every fraction of the whole uses that total as its denominator.
Error analysis
A student converting a rate. The arithmetic is fine and the answer is not.
Annotate
On: \( \text{8 L per 100 km, for 250 km} \;\Rightarrow\; 250 \times \frac{100}{8} = 3125 \)
Setup errors on this type are never small. They land orders of magnitude away, which means a five-second estimate catches every one of them.
Trap
The trap. Six machines take 10 hours, so you reason that 4 machines take 4 over 6 times 10, about 6.7 hours.
Fewer machines finishing faster is impossible. The relationship is inverse: the total work is fixed at 60 machine-hours, so 4 machines take 15 hours.
The setup felt like every other proportion question, which is exactly why it slipped through.
The fix. Ask whether there is a fixed total being shared out. A fixed job, distance, or amount signals inverse proportion.
Then multiply the two quantities to find the constant, and divide to find the new value.
Elimination
A tap fills a tank at 15 litres per minute. How long to fill 450 litres?
Eliminate the wrong options
Which computation gives an answer in minutes?
Survives elimination: a
Why: Only the first arrangement cancels litres and leaves minutes standing, giving 30 minutes. Notice that this elimination was done entirely on units, with no arithmetic at all — and that three of the four choices produce quantities that are not times, which no amount of correct computation could fix. This is the units check working exactly as intended.
Trap
The trap. A rate is given in litres per minute and the question asks about hours, and you answer as though the units matched.
Or a length is given in centimetres and an area in square metres, and the conversion is skipped entirely.
Area and volume conversions are worse: 1 square metre is 10,000 square centimetres, not 100.
The fix. Scan the stem for every unit before starting and check they are consistent.
For area, the conversion factor is squared; for volume, it is cubed.
Counterexample
A rule that holds for lengths and not for everything.
Discussion prompt
A student says: if 1 metre is 100 centimetres, then 1 square metre is 100 square centimetres. Give the correct figure and explain the error.
Hint: How many lengths are being converted?
Answer:
The correct figure is 10,000 square centimetres. A square metre is a square 100 cm by 100 cm, so its area is 100 times 100.
The error: an area conversion needs the linear factor applied TWICE, because an area is a product of two lengths.
For volume it is worse: 1 cubic metre is 100 cubed, which is 1,000,000 cubic centimetres.
The pattern is the scaling rule again, seen from the units side: linear factor k, area factor k squared, volume factor k cubed.
So the same fact underlies the geometry scaling questions and the unit conversion questions, and knowing it once covers both.
Edge cases
Setting up a proportion is the standard move. When is it wrong?
Discussion prompt
When does a straightforward proportion give the wrong answer, and what should you check before setting one up?
Hint: Two different situations break it.
Answer:
First: when the relationship is inverse. More workers means fewer days, so the ratio is not constant and a direct proportion reverses the answer.
Second: when the relationship is linear but not proportional. A taxi fare of 4 dollars plus 2 per mile does not double when the distance doubles, because the flag drop does not scale.
The check before setting up: ask what happens at zero. If zero of one thing gives zero of the other, a proportion is safe. If not, you need the full linear model.
And ask about direction: if increasing one quantity should decrease the other, the product is constant rather than the ratio.
Those two questions take five seconds and they separate the three cases — direct proportion, inverse proportion, and linear-with-an-intercept — which need three different methods.
Check
Check which way the answer should move.
Check your understanding
A tank drains in 12 hours through 3 open valves. How long would it take through 4 valves, at the same rate each?
Answer: A
Why: The total draining work is 3 times 12, which is 36 valve-hours, and it is fixed. With 4 valves the time is 36 over 4, which is 9 hours. The direction check confirms it: more valves must drain the tank faster, and 9 is less than 12.
The direction check eliminates B and D without any arithmetic. On inverse-proportion questions it is the fastest filter available.
Section
Section 5
Matching
Six situations, six setups. No arithmetic.
Match the pairs
Why: The first two rows are the same rate used in opposite directions, and the units alone decide which. That pairing is the heart of the type: whether you multiply or divide by a rate is not something to remember, it is something to read off the units.
Sorting
Which factor does each need?
Sort into buckets
Is the conversion factor linear, squared, or cubed?
This is the geometry scaling rule wearing different clothes. Anywhere a factor is applied to lengths, an area gets it squared and a volume gets it cubed.
Comparison
Fill the blanks from memory.
Comparison matrix
| direct proportion | inverse proportion | |
|---|---|---|
| What stays constant | the ratio, first over second | the product, first times second |
| Double the first, and the second | doubles | halves |
| Typical story | cost per kilogram, distance per hour | workers sharing a fixed job |
| The tell | a per, and zero gives zero | a fixed total being shared out |
The bottom row is the one to internalise. Looking for a fixed total takes two seconds and it is what separates the two methods, which otherwise look identical on the page.
Trade off
Fill in what each method costs.
Comparison matrix
| method | when it is fastest | the risk |
|---|---|---|
| Find the value of one part | when the whole is given | none — it answers every follow-up too |
| Take a fraction of the whole | when only one quantity is wanted | using the wrong denominator |
| Set up a cross-multiplied proportion | when scaling a ratio up or down | crossing the two orders |
| Guess from the numbers | never | the part-to-part answer is always offered |
The one-part method wins most often because it front-loads the work. Once you know one part is 8, every question about that ratio is a single multiplication.
Real world
One minute on why dimensional analysis is worth having.
Discussion prompt
Engineers and chemists use the units-cancel method constantly, and it has caught real errors. Why is it more reliable than reasoning about whether to multiply or divide?
Answer:
Because it replaces a judgement with an observation. Deciding whether to multiply or divide requires holding the situation in your head; checking which units cancel does not.
It catches errors before the arithmetic, which is the only point at which catching them is free.
It scales to long chains. Converting a fuel consumption from litres per 100 km into miles per gallon involves four conversions, and no one can reason through that directly — but the cancellations are mechanical.
A famous cautionary case: a NASA orbiter was lost in 1999 because one team supplied a quantity in imperial units while the software expected metric. A units check on the interface would have caught it.
The habit is exactly the one this deck teaches, applied to something more expensive than a test question.
Ranking
Order from cheapest per kilogram to most expensive.
Put in order
Why: Dividing each price by its weight gives: (b) 2.20, (a) 2.50, (c) 2.60, and (d) 2.50 per kilogram. So (a) and (d) tie at 2.50 and either order between them is defensible. The point of the exercise is that the totals tell you nothing — the largest total, 20 dollars, is neither the best nor the worst value — and only reducing to a common unit makes the four comparable at all.
Concept
This type is 4.9 per cent of the section and the arithmetic is trivial, so essentially all the gain is in the setup habit.
| session | what you do | why |
|---|---|---|
| 1 | Twenty rate questions, writing the units on every number and cancelling before computing. | Builds the habit that turns the setup decision into an observation. |
| 2 | Fifteen ratio questions, writing the total number of parts immediately. | Part-to-part versus part-to-whole is the most common conceptual error in the type. |
| 3 | Ten conversions including at least three chained ones and two area conversions. | Chains and squared factors are the two variants students prepare least. |
| 4 | Ten questions mixing direct and inverse, predicting the direction of the answer first. | The direction check is the fastest filter on inverse-proportion questions. |
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Explain it to yourself
Close the deck.
Discussion prompt
Without looking, explain how to decide whether to multiply or divide by a rate, and how a unit conversion works.
Hint: Neither answer involves remembering a rule about multiplying.
Answer:
Write the rate as a fraction with both units, such as 8 litres over 100 km.
Ask what unit the answer must carry. Then arrange the multiplication so every other unit cancels.
If the surviving units are the ones asked for, the setup is right — before any arithmetic is done.
A conversion multiplies by a fraction equal to 1, such as 1000 m over 1 km. Choose whichever way up makes the unwanted unit cancel.
If you explained it without ever saying multiply when or divide when, you have the method rather than a memorised rule.
Explain it
Two minutes, out loud.
Discussion prompt
A friend sees a ratio of 3 to 5 with a total of 40 and computes three fifths of 40. What do you say?
Answer:
Ask them what the 5 refers to. It is the second quantity, not the total — so five is not the denominator of anything.
Draw it: three boxes and five boxes side by side. Count them: eight boxes altogether. That is the whole.
So the first quantity is three out of eight, which is 15, not 24.
Check it together: three eighths of 40 is 15 and five eighths is 25, and 15 plus 25 is 40. Their version gives 24 and 40 minus 24 is 16, and 24 to 16 is not 3 to 5.
Give them the habit: the moment a ratio appears, add the parts and write the total beside it. Then the denominator is never in doubt.
Commit first
Commit before you check.
Predict first
If 10 pumps empty a reservoir in 6 hours, how long do 15 pumps take?
Correct: 4 hours
Why: The total work is 10 times 6, which is 60 pump-hours and is fixed. With 15 pumps the time is 60 over 15, which is 4 hours. The direction check confirms it: more pumps must be faster, and 4 is less than 6. The answer 9 comes from a direct proportion, which has more pumps taking longer and is impossible on its face.
Connect it up
Blank paper.
Draw it
Draw a decision tree for proportional reasoning. At the top ask: does zero of one give zero of the other? If no, it is linear with an intercept and needs the full model. If yes, ask: when one goes up, does the other go up or down? Up means direct, so the ratio is constant; down means inverse, so the product is constant. Under the direct branch, draw the units-cancelling method with a worked fraction showing km cancelling to leave litres. Under the inverse branch, write multiply to get the fixed total, then divide. In a box at the side, write the ratio rule: add the parts to get the whole, find one part, multiply up. At the bottom write the conversion rule: multiply by a fraction equal to 1, and square it for areas, cube it for volumes.
Exit ticket
One question before you close the deck.
Predict first
You have written a proportion and are unsure whether the fraction is the right way up. What is the fastest way to check?
Correct: Cancel the units and see what survives
Why: The surviving units either are or are not what the question asked for, and that is a fact you can read off in about two seconds without any arithmetic. Computing both versions doubles the work and the nicer number is not evidence of anything. An integer answer is not evidence either — plenty of correct answers are decimals. And re-reading asks you to repeat the judgement that went wrong the first time, rather than replacing it with a check.
Recap
One type, one habit: write the units, cancel them, and let them choose the operation.
| never do this | do this instead |
|---|---|
| Guess whether to multiply or divide | Arrange the units and read the answer off |
| Take three fifths for a 3 to 5 ratio | Add the parts: it is three eighths |
| Use a direct proportion for workers and days | Multiply for the fixed total, then divide |
| Say 1 square metre is 100 square centimetres | Square the factor: it is 10,000 |
| Convert in stages with rounding between | Write the whole chain, then compute once |
| Compare two totals directly | Reduce both to a common unit first |
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