Units, Money and Time

Measurement is where arithmetic meets the world, and it is the branch where a correct calculation most often produces a wrong answer — because the number was right and the unit was not. A length of 350 is nothing until you know whether it is centimetres or metres.

Two of the three systems on this page are decimal and one is not. Metric units and money run on tens, so ordinary place value carries them. Time runs on sixties and twenty-fours, and treating it as though it ran on tens is the source of nearly every time error there is.

Where measuring starts

It starts with comparison, before any number is involved. Longer and Shorter puts two objects side by side, and the whole content is that a comparison needs a common starting line — the ends have to be lined up, or the answer is about position rather than length.

Reading a Ruler is the first measurement with a number attached, and it carries the same idea forward: you measure from zero, not from the end of the plastic, and the reading is the distance covered rather than the marks crossed.

How do the metric units fit together?

By tens, which is the entire point of the system. The prefix tells you the multiplier and it means the same thing whatever it is attached to: kilo is a thousand for metres and for grams alike.

Meters, Centimeters and Kilometers establishes the length family and, just as importantly, a sense of which unit a thing is measured in — a room in metres, a pencil in centimetres, a journey in kilometres. Grams and Kilograms and Milliliters and Liters do the same for mass and capacity, and by the third family the prefixes are doing the teaching.

Converting Lengths is the arithmetic: multiply going to a smaller unit, divide going to a larger one. 350 cm = 3.5 m. 2.4 km = 2,400 m.

Why area units are different

Converting Units of Area is on this page because it is the conversion that catches nearly everybody. 1 m² = 10,000 cm², not 100 — a square metre is a square 100 cm on each side, and 100 × 100 = 10,000.

The rule generalises: area conversions square the length factor, volume conversions cube it. 1 m³ = 1,000,000 cm³. The same fact appears again as the scale-factor rule in ratio and proportion, where enlarging a shape by k multiplies its area by k².

Reading Scales covers the instrument side — rulers, jugs, kitchen scales, dials — and the method is always the same two steps: find the value of one interval by dividing the labelled gap by the number of spaces, then count intervals from the last label.

Telling the time and the 24-hour clock

Telling the Time handles the analogue face, where two hands move at different speeds around the same numbers — which is genuinely harder than it looks, because the numeral 3 means "three o'clock" for one hand and "fifteen minutes" for the other.

The 24-Hour Clock removes the ambiguity by numbering the hours 00 to 23, so every moment of the day has exactly one name. 3:45 pm is 15:45; add twelve to any afternoon time. Midnight is 00:00 and midday is 12:00, which is the pair that trips people up, since 12 am and 12 pm are the two labels the 12-hour clock handles worst.

Working out a duration

Time Durations is the calculation that ordinary subtraction cannot do, because an hour is 60 minutes and not 100. Counting up in stages is both faster and safer:

14:35 to 17:10 → 25 min to 15:00, then 2 hours to 17:00, then 10 min → 2 hours 35 minutes.

The other trap is a duration crossing midnight. Split it at 00:00 and add the two pieces: 22:40 to 06:15 is 1 hour 20 minutes plus 6 hours 15 minutes, so 7 hours 35 minutes.

How does money work?

Money is a decimal with the number of places fixed at two, which is what makes it a good first decimal and an occasionally confusing one. Counting Coins starts before the point exists at all, counting in the values that actually exist rather than in ones, and Dollars and Cents introduces the two-column split.

Making Change is counting up again, in its oldest form: from 6.35 to 10.00 is 65 cents to 7.00 and 3 more, so 3.65. Shopkeepers did this long before tills, and it is still the fastest mental route.

Money as Decimals pins down the notation. 4.05 is four units and five cents, not four and fifty — the second place is the cents column, and a missing digit there is a factor of ten. Money also keeps its trailing zero, so 3.5 is written 3.50, which is a convention rather than a mathematical requirement but a load-bearing one. The decimals guide covers the general place-value rules these follow.

How do you estimate with units?

By carrying a small set of reference objects in your head and comparing against them. Estimating with Units builds exactly that: a door is about 2 m tall, a bag of sugar is 1 kg, a bottle of water is 500 ml, a step is roughly 70 cm.

These matter more than they look, because they are what catch an answer in the wrong unit. A calculation that says a person is 1.7 cm tall or a cat weighs 45 kg has gone wrong by a factor of a hundred, and a reference object catches that in less than a second — no arithmetic, just a comparison against something you have held.

One question, worked end to end

A parcel service charges by weight and refuses anything over 0.03 m³ in volume. A box measures 40 cm by 25 cm by 30 cm and holds 12 bags of 250 g each, plus 400 g of packaging. Does it qualify, and what does it weigh?

The arithmetic is trivial; the units are the question. Convert everything into one system before calculating anything.

• Volume. 40 × 25 × 30 = 30,000 cm³. A cubic metre is 1,000,000 cm³, so this is 0.03 m³ — exactly on the limit, and the answer turns entirely on whether the rule says "over" or "at least".
• Weight. 12 × 250 g = 3,000 g = 3 kg, plus 0.4 kg of packaging, so 3.4 kg.
• Sense check. Three and a half kilograms is about a large bag of potatoes, and a box 40 cm across is about a crate of books. Both are plausible; a result of 3,400 kg would not have been.

The volume step is the one that goes wrong. Dividing 30,000 by 100 rather than by 1,000,000 gives 300 m³, which is a box the size of a house — and the reference-object check is what catches it in a second.

The four mistakes worth naming

• Converting an area with the length factor. 1 m² is 10,000 cm². Square the factor for areas, cube it for volumes.
• Subtracting times in columns. Minutes borrow sixty, not a hundred. Count up in stages instead.
• Reading 4.05 as four and fifty. The second decimal place is the cents column, and it is not optional.
• Counting the marks on a scale. One interval is the labelled gap divided by the number of spaces, which is always one fewer than the marks.

Where this leads next

Metric conversions are place value applied to the world, which is covered from the ground up in whole numbers and place value, and the money arithmetic is decimal arithmetic — see decimals, percentages and interest. Speed, density and every other quantity written as one unit per another belong to ratio, rates and proportion.

Practise it in the game

Math Challenge teaches each step above as an illustrated lesson — the ruler with its intervals marked, the duration counted up in visible stages, the square metre drawn as a hundred-by-hundred grid — inside a catalog of 800+ lessons spanning the whole ladder.

For drill, the Money topic runs change and totals, and the Time topic runs clock reading and durations.

Practise money problems free →