Significant Figures in Multiplication and Division

Introduction

You measure the mass of a metal block as 52.36 g on a good balance, but its volume as 19 cm³ using a measuring cylinder. The calculator says the density is 2.755 789 474 g/cm³. How many of those digits can you trust? A chain is only as strong as its weakest link, and a calculated result is only as precise as its least precise measurement. For multiplication and division there is a simple rule that captures this idea and stops you reporting digits the data cannot support.

Core explanation

The rule. When measured values are multiplied or divided, the answer should have the same number of significant figures as the value with the fewest significant figures.

In the density example, 52.36 g has four significant figures and 19 cm³ has two. The volume is the least precise value, so the density is given to two significant figures:

density = 52.36 g ÷ 19 cm³ = 2.755 789… g/cm³ → 2.8 g/cm³

Why the fewest figures? The volume 19 cm³ could really be anywhere from about 18.5 to 19.5 cm³. Dividing 52.36 by these limits gives about 2.83 and 2.69 g/cm³. The result is uncertain in the first decimal place, so only two significant figures are meaningful. The extra precision of the balance is wasted because the volume is so much less precise.

Counting, not decimal places. For multiplication and division it is the number of significant figures that matters, not the number of decimal places. A different rule, based on decimal places, applies to addition and subtraction.

Multi-step calculations. When a calculation has several multiplication and division steps, keep all the digits in your calculator throughout. Find the value with the fewest significant figures among all the measured data, and round only the final answer to that number.

Exact numbers do not count. Numbers that are defined or counted, such as the 2 in "twice the mass", the 1000 in 1 kg = 1000 g or the number of samples tested, are exact. They have unlimited significant figures and never limit the answer.

Chemistry examples. The rule applies to density (mass ÷ volume), concentration (amount ÷ volume), moles from mass (mass ÷ molar mass) and many other calculations. In practice, the least precise measurement is often a volume from a measuring cylinder or a time from a hand-held stopwatch, which is why chemists choose burettes, pipettes and balances when precise results are needed.

Formulae

number of significant figures in answer = smallest number of significant figures among the measured values (for × and ÷).

Step-by-step reasoning

1. Count the significant figures in every measured value. 2. Ignore exact numbers such as defined conversion factors and counts. 3. Note the smallest count; this is the precision of the answer. 4. Carry out the calculation, keeping all digits. 5. Round the final answer to that number of significant figures, using scientific notation if needed.

Visual explanation

Imagine each measured value as a coloured bar whose length shows how many significant figures it has. Lined up side by side, the shortest bar sets the length of the answer's bar. However long the other bars are, the result can never be longer than the shortest one.

Real-world analogy

A convoy of lorries can travel only as fast as its slowest vehicle. Adding a faster lorry does not speed up the convoy. In the same way, adding a more precise measurement does not make the result more precise if another measurement is much cruder.

Real-world example

When a laboratory prepares a standard solution, it weighs the solid on an analytical balance to four decimal places and makes it up in a volumetric flask accurate to about ± 0.1 cm³ in 250 cm³. Both are chosen to give four significant figures, so neither measurement wastes the precision of the other.

Why?

Why does the least precise value control the answer? Multiplying or dividing spreads the relative uncertainty of each measurement into the result. The value with the largest relative uncertainty, usually the one with fewest significant figures, dominates, so the result can be no more precise than that value.

Common misconception

"Use the number of decimal places of the least precise value." For multiplication and division, decimal places are irrelevant. For example, 3.456 cm × 12 cm = 41 cm² (two significant figures), not 41.47 cm², even though the first value has three decimal places.

Worked example

Question: A sample of 0.0250 mol of a substance has a mass of 1.46 g. Calculate its molar mass.

Reasoning: Molar mass = mass ÷ amount = 1.46 g ÷ 0.0250 mol = 58.4 g/mol. The mass has three significant figures (1, 4, 6). The amount has three significant figures (2, 5 and the trailing 0; the leading zeros do not count). The least is three.

Answer: 58.4 g/mol.

Quick check

1. To how many significant figures should 6.2 × 3.715 be given? Answer: Two, because 6.2 has only two significant figures; the answer is 23.

Exam focus

State "to the same number of significant figures as the least precise data" when justifying the precision of an answer. Many mark schemes accept answers given to one more figure than the data, but giving a long string of calculator digits usually loses a mark.

Advanced insight

The significant-figure rule is a simplified version of how relative (percentage) uncertainties combine. For multiplication and division, the percentage uncertainties of the measurements add, so the result's percentage uncertainty is at least as large as the largest one. Two values with the same number of significant figures can have quite different percentage uncertainties — 1.1 and 9.9 each have two, but ± 0.05 is about 5% of the first and only 0.5% of the second — so the rule is a guide, not a precise law.

Summary

When measured values are multiplied or divided, give the answer to the same number of significant figures as the least precise measured value. Count significant figures, not decimal places. Keep all digits during the calculation and round only the final answer. Exact numbers, such as counts and defined conversion factors, never limit the precision.

Practice questions

1. State the significant-figure rule for multiplication and division. Answer: The answer should have the same number of significant figures as the measured value with the fewest significant figures. 2. Calculate 4.52 g ÷ 2.1 cm³ to the correct number of significant figures. Answer: 2.2 g/cm³, since 2.1 has two significant figures (the calculator gives 2.152…). 3. A rectangle measures 12.35 cm by 4.0 cm. Calculate its area correctly. Answer: 12.35 × 4.0 = 49.4, which rounds to 49 cm² because 4.0 has two significant figures. 4. Convert 2.35 kg into grams. How many significant figures does the answer have, and why? Answer: 2350 g, best written 2.35 × 10³ g, with three significant figures; the factor 1000 g per kg is exact and does not limit the precision.