Converting with Prefixes

Introduction

You now know what kilo, milli, micro and the other prefixes mean. The practical skill is to move a value from one prefixed unit to another — kJ to J, cm³ to dm³, nm to m — quickly and without mistakes. Errors here are among the most common reasons for lost marks in chemistry calculations, because a slip of a factor of 1000 makes a correct method produce a wildly wrong answer. This page gives two reliable methods and a habit of checking.

Core explanation

Method 1: replace the prefix. Every prefix symbol stands for a power of ten, so you can literally substitute it.

- 4.7 kJ = 4.7 × 10³ J = 4700 J - 35 mg = 35 × 10⁻³ g = 0.035 g - 520 nm = 520 × 10⁻⁹ m = 5.2 × 10⁻⁷ m

This method is best for converting into the base unit .

Converting between two prefixes. Convert to the base unit first, then to the new prefix. For example, 0.25 mm into µm: 0.25 mm = 0.25 × 10⁻³ m. Micro is 10⁻⁶, so divide by 10⁻⁶: 0.25 × 10⁻³ ÷ 10⁻⁶ = 0.25 × 10³ = 250 µm. You can also count the gap in powers: from 10⁻³ to 10⁻⁶ is three powers of ten, so the number grows by 10³.

Method 2: conversion factors. A conversion factor is a fraction whose top and bottom are equal quantities, so its value is exactly one. Multiplying by it changes the unit but not the quantity. To convert 2.5 kg to g, multiply by (1000 g ÷ 1 kg):

2.5 kg × (1000 g ÷ 1 kg) = 2500 g

The "kg" on top and bottom cancel, leaving g. Choose the fraction so that the old unit cancels. This method shines in longer calculations because the cancelling units show you whether you have set it up the right way round.

The decimal shift. Because most prefixes differ by 1000, each step on the prefix ladder moves the decimal point three places . Going to a smaller unit moves it right (bigger number); going to a larger unit moves it left (smaller number). From g to µg is two steps down, so the point moves six places right: 0.004 g = 4000 µg.

Always check direction and size. After every conversion ask: "Did I go to a larger or a smaller unit? Should my number have become smaller or larger?" If 2.5 kg has become 0.0025 g, the direction is wrong.

Step-by-step reasoning

To convert 18 500 mL into dm³:

1. Recognise that 1 mL = 1 cm³, and 1 dm³ = 1000 cm³. 2. So 1 dm³ = 1000 mL — a larger unit. 3. Divide by 1000 to move to the larger unit: 18 500 ÷ 1000 = 18.5. 4. Answer: 18.5 dm³. Check: larger unit, smaller number — correct.

Visual explanation

Draw the prefix ladder vertically: G, M, k, (base), m, µ, n, with "×1000" on every downward arrow and "÷1000" on every upward arrow. Put your finger on the starting unit, count the steps to the target, and apply the operation once per step. Deci and centi sit between the base and milli as special short steps of ×10.

Real-world analogy

Changing prefixes is like changing currency between pounds and pence. £2.50 becomes 250p: you moved to a smaller unit, so the number grew by 100. Nobody would say £2.50 is 0.025p, because it obviously makes no sense — train yourself to have the same instinct for grams and milligrams.

Real-world example

Medication errors caused by confusing mg and µg are a recognised safety problem in hospitals, because the difference is a factor of 1000. For this reason, many hospital guidelines advise writing "micrograms" in full rather than "µg", and double-checking every dose conversion. Careful unit conversion is literally a matter of life and death.

Why?

Why does a conversion factor not change the answer? Because (1000 g ÷ 1 kg) is a fraction whose top and bottom describe exactly the same mass, so its value is one. Multiplying any quantity by one leaves it unchanged; only the way it is expressed changes.

Common misconception

"To convert cm³ to dm³, divide by 10 because deci and centi differ by 10." That is true for lengths (10 cm = 1 dm), but volumes are cubed, so 1 dm³ = 10³ cm³ = 1000 cm³. Prefix rules for squared and cubed units need extra care, which is the subject of the next page.

Worked example

Question: A reaction releases 0.0625 MJ of energy. Express this in kJ and in J.

Reasoning: Mega to kilo is one step down the ladder, so multiply by 1000: 0.0625 × 1000 = 62.5 kJ. Kilo to the base unit is another step down: 62.5 × 1000 = 62 500 J. Check: moving to smaller units gave larger numbers each time.

Answer: 62.5 kJ, which is 62 500 J.

Quick check

1. Convert 0.75 g into mg. Answer: 0.75 × 1000 = 750 mg.

Exam focus

Before substituting into any equation, convert every value into consistent units, usually the base unit or the unit the question asks for. Show the conversion as a separate line so that examiners can award method marks. A final answer out by exactly 1000 is a signal to recheck prefixes.

Advanced insight

The conversion-factor method generalises into dimensional analysis , where a chain of factors converts several units at once — for example, turning a speed in km h⁻¹ into m s⁻¹ by multiplying by (1000 m ÷ 1 km) and (1 h ÷ 3600 s). Keeping the units in every line of a calculation is a powerful way to catch errors before they reach the answer.

Summary

To convert prefixed units, either replace the prefix with its power of ten or multiply by a conversion factor equal to one. Each step of 1000 on the prefix ladder moves the decimal point three places: right when going to a smaller unit, left when going to a larger one. Always check that the size of the number has changed in the sensible direction.

Practice questions

1. Convert 3.4 × 10⁻² m into mm. Answer: 3.4 × 10⁻² m = 0.034 m, and 0.034 × 1000 = 34 mm. 2. Convert 4500 µL into mL and into L. Answer: 4500 ÷ 1000 = 4.5 mL; 4.5 ÷ 1000 = 0.0045 L. 3. Use a conversion factor to change 0.080 kg into g, showing the units cancelling. Answer: 0.080 kg × (1000 g ÷ 1 kg) = 80 g; the kg cancels to leave g. 4. A student converts 250 nm to 0.25 m. Explain the error and give the correct value. Answer: Nano means 10⁻⁹, not 10⁻³; 250 nm = 250 × 10⁻⁹ m = 2.5 × 10⁻⁷ m. 5. Convert 1.5 MPa into kPa. Answer: 1.5 × 1000 = 1500 kPa.