Units, Money and Time

Updated 4 October 2026

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 centimeters or meters.

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?

The metric units fit together in tens, hundreds and thousands, and a prefix means the same thing on any unit. Kilo means a thousand, so 1 kilometer is 1,000 meters and 1 kilogram is 1,000 grams. Centi means a hundredth and milli means a thousandth, so a meter is 100 centimeters or 1,000 millimeters, and a liter is 1,000 milliliters. To convert, multiply when changing to a smaller unit and divide when changing to a larger one: 3 km is 3 × 1,000 = 3,000 meters, and 4,000 g is 4,000 ÷ 1,000 = 4 kilograms. This works because the units differ by powers of ten, so converting only moves digits between place-value columns. Quantities in different units must be changed to one unit before they are added: 1 km and 350 m make 1,000 + 350 = 1,350 meters. The size of the number is a quick check: a smaller unit always needs a bigger number, so 3 km cannot be 300 meters.

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 meters, a pencil in centimeters, a journey in kilometers. 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: 350 cm = 3.5 m. 2.4 km = 2,400 m.

Why area units are different

Converting Units of Area is the conversion that catches nearly everybody. $1m^2=10$1 m² = 10,000 cm², not 100 — a square meter is a square 100 cm on each side, and 100 × 100 = 10,000.

The rule generalizes: area conversions square the length factor, volume conversions cube it. $1m^3=1$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 $k^2$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 labeled 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\to 25\min$10 → 25 min to 15:00, then 2 hours to 17:00, then $10\min \to 2$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 written as a decimal with exactly two places after the point: the whole number counts dollars, the two digits after the point count cents, and 100 cents make 1 dollar. Counting Coins starts with the coins themselves, and Dollars and Cents introduces the split between the two parts. The second place after the point is the cents column, so $4.05 is 4 dollars and 5 cents, not 4 dollars and 50 cents. Money also keeps its trailing zero: three and a half dollars is written $3.50, never $3.5. To give change, count up from the price. For a price of $6.35 paid with $10.00, count up 65 cents to $7.00, then 3 dollars to ten, so the change is $3.65. Counting up works because it builds the gap in easy steps, the way a cashier hands over coins and then bills. Adding the change to the price confirms it: 635 + 365 = 1,000 cents, which is the 10 dollars that was paid.

Making Change is counting up in its oldest form. Shopkeepers did this long before tills, and it is still the fastest mental route.

Money as Decimals pins down the notation, and a missing digit in the cents column is a factor of ten. The decimals guide covers the general place-value rules these follow.

How do you estimate with units?

To estimate a measurement, compare it with a reference object whose size you already know. Estimating with Units builds a small set of these: a door is about 2 m tall, a bag of sugar is 1 kg, a bottle of water is 500 ml, and a step is roughly 70 cm. A classroom ceiling is about one and a half doors high, so it is about 3 m. Reference objects work because they turn an unfamiliar size into a comparison you can picture, and a comparison is much easier than guessing a number from nothing. For something too large or too many to compare in one go, estimate one piece and multiply. An apple weighs about 150 g, so a bag of 20 apples weighs about 20 × 150 = 3,000 g, which is 3 kg. Then go back to a reference: 3 kg is three bags of sugar, which is a believable weight for a bag of apples.

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 $0.03m^3$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 meter is 1,000,000 cm³, so this is $0.03m^3$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 $300m^3$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. $1m^2$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 labeled 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.