Unit conversion, scale and order-of-magnitude estimates

Multiply by a ratio equal to one

Because 1 m = 1000 mm, the ratio (1000 mm)/(1 m) equals one. Multiplying a length by it changes the unit expression, not the length. For example, 2.4 m × (1000 mm)/(1 m) = 2400 mm. The m in the numerator and denominator cancels, leaving mm.

Choose the ratio’s direction by the unit you want to remove. To turn 2400 mm into metres, use (1 m)/(1000 mm). A quick reasonableness check helps: smaller units require more of them to describe the same length. If a metre value becomes a smaller number when expressed in millimetres, inspect the conversion.

Convert the whole unit, including its power

Kilo means a factor of 1000, centi means 1/100 and milli means 1/1000. These are exact factors, not measured approximations. Prefixes attach to the unit: mm means millimetre. A prefix inside a squared or cubed unit is squared or cubed too.

A square 1 cm on each side measures 10 mm on each side. It contains 10 × 10 = 100 squares of area 1 mm². Therefore 1 cm² = 100 mm². The diagram makes visible why multiplying by ten once is insufficient. Similarly, 1 cm³ = 1000 mm³, and 1 m³ = 1 000 000 cm³.

For liquid capacity, 1 litre (L) is exactly 1 cubic decimetre, which is 0.001 m³. Thus 1 m³ = 1000 L. Convert each length before multiplying, or convert the completed volume with a cubic conversion factor; both approaches should agree.

Convert every changing part of a rate

A cart travelling 6 m in 4 s has average speed 1.5 m/s. To express that speed in metres per minute, multiply by 60 s/min: (1.5 m/s) × (60 s/min) = 90 m/min. The cart has not accelerated. The time unit changed along with the number. In a compound unit, check both the numerator and denominator.

Make a rough answer before a detailed one

An order-of-magnitude estimate locates a quantity on a broad scale, often using powers of ten. It can tell us whether a capacity should be nearer a hundred litres than a tenth of a litre. State how rough the estimate is; rounding to the nearest power of ten and simply choosing a convenient power are different conventions.

Estimate the storage box as roughly half a metre in each direction. Its volume is then about 0.5 × 0.5 × 0.5 = 0.125 m³, or roughly 100 L. This is deliberately rough. It will not specify a lid or decide a tight fit, but it will expose a result of 0.072 L as implausible for such a box. Exact conversion factors cannot repair uncertain dimensions or a poor shape assumption.

Check a box capacity two ways

In this constructed case, the rectangular internal dimensions are 0.60 m, 0.40 m and 0.30 m. Their product is 0.072 m³. Multiplying by 1000 L/m³ gives 72 L. Alternatively, use 6 dm × 4 dm × 3 dm = 72 dm³ = 72 L.

Allowing a nominal 80 % filling fraction gives 0.80 × 72 L = 57.6 L. Since our dimensions and fill level are only rough design assumptions, “about 60 L usable capacity” communicates the estimate better than implying a measured 57.600 L. The exact multiplication and the justified precision of a practical claim are different questions.

Try the reasoning

1. A learner writes 2 cm² = 20 mm². What was missed?

   Show the answer and reasoning

   Each of the two length directions changes by a factor of ten, so the area factor is 100. The correct conversion is 200 mm².

2. Why does 1.5 m/s become 90 m/min rather than 0.025 m/min?

   Show the answer and reasoning

   One minute contains 60 seconds, so at that speed the cart travels 60 times as far in a minute. Unit cancellation gives the same conclusion.

3. Would converting an estimated 0.60 m dimension to 600 mm make it a more precise measurement?

   Show the answer and reasoning

   No. An exact unit conversion changes the expression, not the information used to estimate the dimension.

What to carry forward

Cancel units deliberately, apply powers to the entire unit, and compare the result with a rough physical estimate.