Rates, units and rounding: marks lost before the maths

Write the unit on every number

Every number in this course is a number of something. Write what it is, next to it, every time. Not for neatness - because the unit is what tells you whether the next step is a multiplication or a division, and a page of bare numbers gives you nothing to check against.

The conversions that cost the most are the ones people think they know. Square and cubic units do not convert the way lengths do: a square metre is ten thousand square centimetres, not a hundred, and a cubic metre is a million cubic centimetres. If you convert an area by moving the decimal point the way you would for a length, you will be out by a factor nobody spots afterwards.

The other one is time. Hours to minutes is sixty, not a hundred, and 2.5 hours is two hours thirty minutes, not two hours fifty. A rate per hour turned into a rate per minute by dividing by a hundred is a mistake that produces a believable answer, which is what makes it expensive.

Which way round is the rate

A rate is one quantity per another, and half the marks in rate questions go to people who divided the right two numbers the wrong way round. Litres per hundred kilometres and kilometres per litre are both fuel consumption and they are reciprocals: one goes up when the other goes down.

The way to be sure takes five seconds. Say the rate out loud as a sentence with the word per in it, then check the answer against common sense. A car that uses eight litres per hundred kilometres is ordinary. A car that uses a hundred litres per eight kilometres is not a car. If your answer fails that test, you divided the wrong way.

When a question gives you a rate and asks for a total, you multiply; when it gives you a total and asks for a rate, you divide. Writing the units through the calculation makes which one it is obvious, because the units you do not want have to cancel.

• Say the rate as a sentence with per in it before you use it.
• Check the size of your answer against something you know.
• Carry the units through the working so the wrong ones cancel.
• A rate per hundred kilometres is not a rate per kilometre.

Scale, and what it is a scale of

A scale of one to fifty on a plan means one unit on the paper is fifty of the same unit in the world. It says nothing directly about areas, and that is where the marks go: doubling every length multiplies the area by four and the volume by eight.

So a question that gives you a scale and asks for a floor area wants you to scale the lengths first and find the area afterwards, or to scale the area by the square of the ratio. Scaling the area by the ratio itself is the single most common error in this part of the course, and it is invisible in the answer.

Trapezoidal or offset surveys are the same idea with more arithmetic. Draw the shape, mark the numbers you were given on it, and label the units. A sketch takes thirty seconds and prevents the whole class of errors where two of the given numbers were never in the same units.

Round once, at the end

Carry the full value through the working and round only the thing you are asked to write down. Rounding at every step compounds, and on a multi-stage financial question it can move the final answer by several dollars - enough to be marked wrong even though every method was right.

Then round to what the question asked for, and say what you did. Nearest cent, two decimal places and three significant figures are three different instructions, and answering one when you were asked another loses the mark even when the number is otherwise correct.

Money in a real context rounds up in some places and down in others. If a question asks how many buses are needed for a hundred and thirty students on a fifty-seat bus, the answer is three, not two point six, and writing two point six is answering a different question.

Working somebody else can follow

In this course the working is worth more than the answer, and it is the part people skip when they are short of time. A correct answer with no working gets less than a wrong answer with clear working, because the second one shows a method and the first one shows nothing.

Write the formula before you put numbers in it. Write the substitution. Then the answer with its unit. Three lines, every time, including the questions that feel too easy to bother with - those are the ones where a slip goes unnoticed because there is nothing on the page to check.

And answer the question that was asked. A question asking whether a plan is affordable wants a sentence saying yes or no with the number that decides it, not just the number.

Where it usually goes wrong

None of these five are about being bad at arithmetic, which is what makes them fixable in a week.

• Converting an area or a volume as though it were a length.
• Dividing the two numbers in a rate the wrong way round.
• Rounding at every step instead of once at the end.
• Giving a number when the question asked for a decision.
• Skipping the working on the questions that looked easy.

What to practise next

Take ten questions you have already done and, without redoing them, write the unit beside every number in your working. You will find the ones where two numbers were never in the same units, and that is where your marks went.

Then do five rate questions and say each rate out loud as a sentence before calculating. It feels slow for about two questions and then it stops being something you have to do deliberately.

Check yourself

Three questions on what is above. Have a go before you open them - reading an answer you have not tried to give is the version of this that does nothing.