Units and Estimation

Introduction

Numbers need units to mean anything: 5 what? Kilometres, kilograms, dollars? Mixing units is a common source of mistakes. Estimation then gives you a quick check that an answer is in the right range.

Metric conversions

The metric system uses powers of ten.

Quantity	Relationships
Length	1 km = 1,000 m, 1 m = 100 cm, 1 cm = 10 mm
Mass	1 kg = 1,000 g
Volume	1 L = 1,000 mL
Time	1 h = 60 min, 1 min = 60 s

To convert, multiply or divide by the relationship.

2.5 km to metres: 2.5 x 1,000 = 2,500 m.
450 g to kilograms: 450 / 1,000 = 0.45 kg.

Tip: converting to a smaller unit means a bigger number (2.5 km is 2,500 m), and the reverse for a larger unit.

Always write the units in each step and check they cancel correctly.

Quick quiz

How many millilitres are in 1.5 litres?

Speed, distance and time

The relationship: distance = speed x time. Rearranged: speed = distance / time, and time = distance / speed.

A car travels at 60 km/h for 45 minutes. Convert the time to hours: 45 / 60 = 0.75 h. Distance: 60 x 0.75 = 45 km.

A warning: units must match. If speed is in km/h, time must be in hours.

Estimation

Before calculating, round the numbers and estimate.

19.8 x 5.1: about 20 x 5 = 100. The exact answer is 100.98, so it is sensible.
If you typed 19.8 x 5.1 into a calculator and got 10.098, the estimate shows an error, probably a misplaced decimal point.

Estimation helps you:

Catch calculator mistakes.
Make quick decisions, such as whether you have enough money for a shopping basket.
Judge whether claims are plausible.

Rounding to one significant figure is a good quick method. For example, 4,870 / 52 is about 5,000 / 50 = 100. The exact value is about 93.7.

Sensible answers

Ask: Is the size reasonable? A person walking 400 km/h or a recipe needing 600 eggs is a sign of an error. Check the units, and ask if the answer should be larger or smaller than the numbers you began with.

Try it yourself

Convert 3,750 m to km, and 0.6 L to mL. A cyclist rides 18 km in 40 minutes. What is the average speed in km/h? Estimate each answer first.

Summary

Metric conversions multiply or divide by powers of ten, and units must be written and checked.
Distance equals speed times time, as long as the units match.
Estimate first to catch mistakes and judge whether answers are sensible.