# How do you handle significant figures in multi-step calculations?

Jun 17, 2014

Three simple rules are:
1. Use enough digits to avoid unintended loss of significance.
2. Use as few digits as are reasonably convenient.
3. Too many are better than too few.

Suppose that you can round off the final result of a multi-step calculation can to 2 significant figures.

It may not be safe to round off the intermediate steps in the calculation to 2 significant figures. Round off errors can accumulate.

To avoid round off errors, we carry one or two extra guard digits in the intermediate calculations. Then we round properly when we write the final result.

If you have to write an intermediate answer on paper, include the guard digits.

For the next step, you do not have to re-enter the number in your calculator. You can just use the number that is already in the display or in memory.

Always record all required answers to the appropriate number of significant figures.

Example

A sample of 50.0 g of oxygen occupies a volume of 48 L. The amount of gas changes at constant pressure until the volume is 79 L. What mass of oxygen is now in the container?

1. Calculate the initial moles of oxygen.

${n}_{1}$ = 50.0 g O₂ × $\text{1 mol O₂"/"32.00 g O₂}$ = 1.5625 mol O₂

Five significant figures are OK, because this is not a required answer.

2. Calculate the new number of moles.

${n}_{1} / {V}_{1} = {n}_{2} / {V}_{2}$

n_2 = n_1 × V_2 /V_1 = "1.5625 mol" × "79 L"/"48 L" = 2.57164583 mol (by calculator).

This is not a required answer, so record it as 2.572 mol — two significant figures plus two guard digits.

3. Calculate the mass of oxygen.

The number already in the calculator is 2.57164583. You can use that number or use extra energy to re-enter 2.572. Either way, on paper you write:

Mass of O₂ = $\text{2.572 mol O₂ × 32.00 g O₂"/"1mol O₂}$ = 82 g O₂

The calculator gives 82.29166667, but you must round off to 82 for this required answer.

Here is another example

Hope this helps.