Scaling vs. Summing

Multiplication is NOT repeated addition when working with measured quantities.

As a general chemistry student, you have learned the two fundamental rules for significant figures: one for addition/subtraction and one for multiplication/division. A common and confusing paradox arises when a single operation can be viewed in two different ways.

Consider calculating the total mass of three identical samples, each weighing 10.1 g. Should you add (10.1 + 10.1 + 10.1) or multiply (3 × 10.1)? The two rules give different answers.

In this case, the answers happen to be the same, which doesn’t help us understand which approach is correct. Consider a different mass: 50.1 g.

These answers are not the same. Which one is correct? The solution lies in distinguishing between inexact (measured) numbers and exact numbers.

The Core Principle: Inexact vs. Exact Numbers

This concept resolves all significant figure ambiguity.

Inexact Numbers

Inexact numbers have uncertainty and typically come from a measurement. Their significant figures indicate the precision of the value. Examples include a mass from a balance (23.5 g) or a volume from a graduated cylinder (150 mL).

Exact Numbers

Exact numbers have infinite precision and zero uncertainty. They do not limit the precision of a calculation. They arise from two sources:

  1. Counting: 4 hydrogen atoms in CH4; 3 trials in an experiment.
  2. Definitions: 1000 in 1000 g = 1 kg; 12 in 12 inches = 1 foot.
TipThe Golden Rule of Significant Figures

Exact numbers never limit the number of significant figures in a calculation. You can treat them as having an infinite number of significant figures.

The Guiding Question: Are You Scaling or Summing?

With the Golden Rule in mind, you can solve any problem by asking a simple question:

“Am I summing separate, independent inexact numbers, or am I scaling one inexact number with an exact number?”

  • Summing (Use Addition/Subtraction Rule): This applies when combining two or more separate, independent measurements. The result is rounded to the last place value that is significant in all of the numbers being added or subtracted.
  • Scaling (Use Multiplication/Division Rule): This applies when you take a single measurement and multiply or divide it by a count or a defined ratio (an exact number). The precision is limited by the significant figures of your inexact numbers.

Case Studies in Action

Let’s apply this principle to common chemistry scenarios where the choice between summing and scaling is critical.

Case Study 1: Molar Mass of Sucrose (C12H22O11)

Calculating molar mass is a multi-step process. The correct method follows the order of operations, applying the relevant sig fig rule at each step. A common error is to misapply one rule to the entire calculation.

We will use the following atomic masses:

  • C: 12.01 g mol−1
  • H: 1.01 g mol−1
  • O: 16.00 g mol−1
Correct Method: Scaling, then Summing

The subscripts (12, 22, 11) are exact counts. We scale (or multiply) each atomic mass first, then sum the results.

  1. Scale (Multiplication):
    • C: 12 × 12.01 = 144.12
    • H: 22 × 1.01 = 22.22
    • O: 11 × 16.00 = 176.00
  2. Sum (Addition): The least precise place value in all three results is the hundredths place.
    • 144.12 + 22.22 + 176.00 = 342.34
  3. Apply Addition Rule: The answer must be rounded to the hundredths place.
  • Final Correct Answer: 342.34 g/mol
Incorrect Method: “Fewest Sig Figs” Rule

A frequent mistake is to find the input with the fewest sig figs and apply that rule to the final answer, ignoring the order of operations.

  1. Analyze Initial Sig Figs:
    • C (12.01): 4 sig figs
    • H (1.01): 3 sig figs (the fewest)
    • O (16.00): 4 sig figs
  2. Perform Calculation:
    • The raw sum is still 342.34.
  3. Apply Flawed Logic: The student incorrectly rounds the final answer to 3 significant figures because of hydrogen.
  • Final Incorrect Answer: 342 g/mol

This error incorrectly discards known precision by misapplying the rules.


Case Study 2: Enthalpy Changes (Summing vs. Scaling)

This case study provides a direct comparison where adding two numbers versus multiplying by two gives different results, and both are correct in their specific contexts.

Scenario A: Hess’s Law (Summing)

You perform two separate experiments to measure the enthalpy of two different reactions. By chance, they both yield the same value.

  • Rxn 1: ΔH = +50.1 kJ
  • Rxn 2: ΔH = +50.1 kJ

You combine these reactions using Hess’s Law.

Analysis: You are summing two independent, inexact measurements. Apply the addition rule. The least precise place in both numbers is the tenths place.

  • Calculation: 50.1 + 50.1 = 100.2
  • The answer must be rounded to the tenths place.
  • Final Answer: 100.2 kJ
Scenario B: Doubling Stoichiometry (Scaling)

You take one reaction with a measured enthalpy of ΔH = +50.1 kJ and double its coefficients (A → B becomes 2 A → 2 B).

Analysis: You are scaling a single measurement (50.1, 3 sig figs) by an exact count of 2. Apply the multiplication rule.

  • Calculation: 2 × 50.1 = 100.2
  • The answer must be rounded to 3 significant figures.
  • Final Answer: 1.00 × 102 kJ


This direct comparison shows that context is everything. The same numbers can lead to different answers depending on whether you are summing independent measurements or scaling a single measurement by an exact number.


Operations Summary Table

Quick Reference

This summary table can be found on the Scaling vs. Summing quick reference page.


The Take-Home Message

The apparent conflict between significant figure rules is not a conflict at all. It is a call to think like a scientist. Before you apply a rule, look at your numbers. Are they fallible, inexact numbers from a lab instrument, or are they perfect, unchanging counts and definitions?

When you encounter a calculation, always ask yourself the guiding question:

Am I combining separate measurements (summing), or am I scaling one measurement by a perfect count (scaling)?

Answering that question will tell you exactly which rule to use, every time.