Decimal Places in Addition and Subtraction

Introduction

You have learned that when measured values are multiplied or divided, the answer keeps the same number of significant figures as the least precise value. It would be neat if the same rule worked for adding and subtracting, but it does not. When you add a mass read from a rough balance to a mass read from a precise balance, what matters is not how many digits each value has, but which decimal place each one is reliable to. This page explains the rule for addition and subtraction and, just as importantly, why it is different.

Core explanation

The rule. When measured values are added or subtracted, the answer should be rounded to the same number of decimal places as the value with the fewest decimal places.

For example, a beaker has a mass of 52.3 g on a top-pan balance, and a sample of 1.456 g is added (weighed on an analytical balance). The total is:

52.3 g + 1.456 g = 53.756 g → rounded to 53.8 g

The first value is known only to the nearest 0.1 g, so the total cannot be known more finely than that.

Why decimal places, not significant figures? In addition and subtraction, uncertainties behave in absolute terms — in grams, centimetres or cubic centimetres. The value 52.3 g is uncertain in the tenths place, perhaps by about ±0.1 g. Adding 1.456 g, however precise, cannot remove that ±0.1 g. The hundredths and thousandths digits of the total are therefore meaningless. The column where the uncertainty appears is what limits the answer, and that column is identified by decimal places.

In multiplication and division, by contrast, uncertainties behave in relative (percentage) terms. A 1% uncertainty in one factor becomes roughly a 1% uncertainty in the product, whatever the size of the numbers. The number of significant figures is a rough measure of relative precision, so it is the right thing to count there.

Same units first. Decimal places only make sense when all values are in the same unit. Convert 0.250 kg to 250 g (or the other way round) before adding it to a value in grams, and then apply the rule.

Subtraction can lose significant figures. Consider finding the mass of a solid by difference: a container plus solid is 25.47 g and the empty container is 25.12 g.

25.47 g − 25.12 g = 0.35 g

Both starting values have four significant figures, but the answer has only two. Nothing has gone wrong: the answer keeps two decimal places, as the rule says, but most of the digits cancelled out. This is why chemists try to avoid subtracting two nearly equal numbers when they need a precise result, or they use a more precise instrument.

Addition can gain significant figures. The reverse also happens: 9.7 cm³ + 0.6 cm³ = 10.3 cm³ has three significant figures, although each input had two. Again, decimal places are what count.

Mixed calculations. When a calculation contains both types of step, apply each rule at its own step, but keep extra digits in intermediate results and round only at the end. Mark mentally where the reliable digits end after each step.

Step-by-step reasoning

To add or subtract measured values:

1. Convert all values into the same unit. 2. Note how many decimal places each value has. 3. Identify the value with the fewest decimal places. 4. Carry out the calculation, keeping all digits for now. 5. Round the answer to that smallest number of decimal places.

Visual explanation

Write the numbers one above the other with their decimal points lined up, as in column addition. Shade each column where a digit is reliable. The first column that is empty in any row marks the edge of reliability, so the answer stops at the last column that is filled in every row.

Real-world analogy

Imagine adding a journey measured by a car odometer, to the nearest 0.1 km, to a walk measured with a tape to the nearest centimetre. The total distance is still only known to the nearest 0.1 km, because the car's reading is fuzzy at that level, and precise centimetres cannot fix it.

Real-world example

In titrations, the volume of solution delivered is found by subtracting the initial burette reading from the final one, for example 24.60 cm³ − 1.25 cm³ = 23.35 cm³. Both readings have two decimal places, so the titre is quoted to two decimal places, which matches how burettes are read.

Why?

Why does adding a very precise value to a rough one not help? Because the uncertainty of the rough value is carried unchanged into the sum. Uncertainties in a sum are measured in units, so the largest absolute uncertainty dominates, and it is set by the value with the fewest decimal places.

Common misconception

"When adding, round to the fewest significant figures." This gives wrong answers. For 100.5 g + 0.23 g, the fewest significant figures is two, which would give 100 g, throwing away reliable information. The correct answer, using decimal places, is 100.7 g.

Worked example

Question: A flask has a mass of 105.62 g. After a liquid is added its mass is 128.9 g, measured on a less precise balance. What is the mass of the liquid?

Reasoning: 128.9 − 105.62 = 23.28 g. The value 128.9 g has one decimal place, and 105.62 g has two. The answer must have one decimal place.

Answer: 23.3 g.

Quick check

1. Add 12.52 cm³ and 3.1 cm³ and give the answer to the correct number of decimal places. Answer: 15.62 rounds to 15.6 cm³, because 3.1 cm³ has only one decimal place.

Exam focus

Examiners test whether you can choose the right rule: decimal places for addition and subtraction, significant figures for multiplication and division. Always convert to the same unit first, and show the unrounded result before rounding.

Advanced insight

In more advanced work, uncertainties are combined statistically. For a sum or difference of independent measurements, the absolute uncertainties combine as the square root of the sum of their squares, so two readings each uncertain by ±0.05 cm³ give a difference uncertain by about ±0.07 cm³. The decimal-places rule is a quick approximation to this idea.

Summary

When adding or subtracting measured values, round the answer to the fewest decimal places among the values, after converting to the same unit. This works because uncertainties in sums and differences are absolute. Multiplication and division use significant figures instead, because their uncertainties are relative. Subtracting nearly equal values can leave very few significant figures.

Practice questions

1. Calculate 4.271 g + 0.35 g + 10.1 g to the correct precision. Answer: 14.721 g rounds to 14.7 g, because 10.1 g has one decimal place. 2. A burette reads 0.40 cm³ at the start and 18.95 cm³ at the end. What volume was delivered? Answer: 18.95 − 0.40 = 18.55 cm³, given to two decimal places. 3. Explain why 25.47 g − 25.12 g gives an answer with only two significant figures. Answer: The leading digits cancel in the subtraction, leaving 0.35 g; the answer keeps two decimal places, but only two significant figures remain. 4. Add 1.25 kg and 340 g, giving your answer in grams. Answer: Convert 1.25 kg to 1250 g; 1250 g + 340 g = 1590 g, reliable to the nearest 10 g.