# Multiplying & Dividing with Sig Figs

For calculations with sig figs, one performs the arithmetic on the numbers as they normally would
and then
rounds the result to to demonstrate the limited accuracy of the result due
to the limited accuracy of the measurement used in the calculation.
What follows are the rules for rounding the results of multiplication & division.

When multiplying or dividing two measurements, the result should have the same number of
sig figs as the measurement with least sig figs.

As an example of such a calculation, one may want to know the total mass of sugar in a crate
of sugar packets.
The mass of each packet was measured and found to be

1.013 g/packet

(4 sig figs) and
we further know each crate has roughly 1.30e6 packets

(3 sig figs).
Therefore we can calculate the mass of sugar in the crate by the following calculation.
1.30e6 packets sugar | 4.023 g
-------------------------------------
| 1 packet

Entering

1.30e6 × 4.023

into a calculator gives
5229900.0

.
This result has far more sig figs, and is thus more accurate, than either of the input measurements.
To reflect the limited accuracy of the measurements used in the calculation,
we round the result to same number of sig figs as the least accurate measurement;
the measurement with the least sig figs.
This ensures that our results are no more accurate than any of the measurements
used in the calculations.
The least accurate measurement is 1.30e6 packets

,
which has 3 sig figs and rounding our results to 3 sig figs
gives us 5.23e6 g sugar

.
In the following practice examples, we'll give you the arithmetic result of multiplying or dividing
two numbers and you can practice working out the rounded result with the proper number of sig figs.

2.31 * 1.5

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You'll want to determine how many sig figs the answer should have using the rule for
multiplication & division:
the answer should have the same number of sig figs as the multiplied or divided number
with the least sig figs.

So look at the two multiplied numbers and determine the sig figs of each. The one with the least sig figs specifies the number of sig figs in the rounded answer.

Now you can simply round the arithmetic result to the right number of sig figs.

So look at the two multiplied numbers and determine the sig figs of each. The one with the least sig figs specifies the number of sig figs in the rounded answer.

Now you can simply round the arithmetic result to the right number of sig figs.

102.35 / 15.72

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As in the previous example, determine the sig figs of each of two numbers in this division.
Then round the calculator's result to have the same sig figs as the number with the least sig figs.

36.901 * 1.03e5

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For this example you'll want to remember the rule for determining the sig figs of a number
in scientific notation. Additionally, recall that the sig figs of some number can only
be represented with scientific notation & you can enter such an
answer using the e shorthand; i.e.

1.03e5

=
1.03e5

.
When multiplying or dividing a measurements by an exact number, the answer should be
rounded to the same number of sig figs as the measurement.

As a quick example, lets say a certain reaction requires

1.075 g Copper

and we want to know how much copper we'd need if we scale the reaction back by a factor of 6.
Entering 1.075 ÷ 6

into a calculator gives
us 0.1791666667

.
Using this rounding rule, we'll round this result to 0.1792 g Copper

as the original measurement had four sig figs and 6 is an exact number in this example.
In the sig fig calculator and in the rest of this tutorial, exact numbers will be denoted
with the under-bar notation where the number is underlined; i.e.

6x

.
Additionally, you can enter such numbers into the sig fig calculator by adding an
x
suffix to the end of the number.
For instance, the example calculation we just looked at can be entered into the sig fig
calculator as
1.075 / 6x

.
Here are some quick examples to practice multiplying & dividing sig figs with exact numbers.

10x * 27.9

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Using this new rounding rule, we know the answer should have the same number of sig figs
as the measurement (non-exact number).
Simply round the arithmetic result to the right number of sig figs.

4320 / 3.6x

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Exact numbers can be decimals and the sig figs in the rounded answer are still determined
by the sig figs of the other number - the measurement.

The rules for multiplying & dividing with sig figs are…

- When multiplying or dividing two measurements, the result should be rounded to the same number of sig figs as the measurement with the least sig figs.
- When multiplying or dividing a measurement with an exact number, the result should be rounded to the number of sig figs as the measurement.

Next, lets checkout how to
add & subtract with sig figs.
→