### Calculations with Significant Figures

```Sig Figs meet Rounding!
 In Science we take measurements, but those
measurements are sometimes needed to find the
values that we really want.
 For example:
 Volume (Length x Width x Height)
 Volume by Difference using a graduated cylinder
(Height of Water with object – Height of water initially)
 Density (Mass ÷ Volume)
 Calculations cannot be more precise than the
measurements.
 The least precise measurement dictates the precision
 The least precise will be the measurement with the
LEAST decimal places.
 When adding or subtracting – pretend that you are
back in elementary school and line those numbers up!
 Example: Two different pieces of metal are massed
separately using different balances. The first piece has
a mass of 19.473 g. The second piece has a mass of 3.82
g. What is the total mass? Don’t forget to use
significant figures.
 Step 1 – Line them up and do the math.
19.473
+ 3.82
23.293
 Step 2 – Find the LAST significant figure. Start from
the right. Look for the first column that has a
significant figure from both measurements. Underline
the number in the answer that is in that column.
19.473
+ 3.82
23.293
 Step 3 – Round to that place – do not forget to LOOK
right!
19.473
+ 3.82
23.293
Answer : 23.29 g – Don’t forget the Unit!!!
 Subtraction is the SAME!!!
 Example – The mass of a paper cup is 1.284 g. The
mass of the paper cup plus a sample of Manmium is
38.2 g. What is the mass of the Manmium sample?
Step 1 – What is it?
Yep – Line them up!
38.2
- 1.284
36.916
Step 2 – What is it?
Yep – Underline the last significant figure!
38.2
- 1.284
36.916
Step 3 – What is it?
Yep – Look right and ROUND!
38.2
- 1.284
36.916
 489.2 g + 8.03 g = ?
 0.2800 L + 4.8 L =?
 285.0 kg – 3.82 kg = ?
 0.01963 g + 0.290 g =?
 926.028 kg – 30.02 kg = ?
 3902 L + 284.5 L = ?
 489.2 g + 8.03 g = 497.2 g
 0.2800 L + 4.8 L = 5.1 L
 285.0 kg – 3.82 kg = 281.2 kg
 0.01963 g + 0.290 g = 0.310 g
 926.028 kg – 30.02 kg = 896.01 kg
 3,902 L + 284.5 L = 4,186 L
 The least precise measurement dictates the precision
 The least precise will be the measurement with the
LEAST number of significant figures.
 The answer should be rounded to the least number of
significant figures
 Example: Find the area of a rectangle with a length of
24.35 m and a width of 4.09 m.
 Step 1 – Do the math!
 Area = Length x width
 24.35 m x 4.09 m = 99.5915 m2
 Step 2 – Determine the number of significant figures
in the original measurements
 24.35 m (4 sig figs)
 4.09 m (3 sig figs)
 Step 3 – Determine the least number of sig figs
 3 sig figs is less than 4 sig figs
 Step 4 – round the answer to the least number of sig
figs that you determined in Step 3
 99.5915 m2
 Final Answer – 99.6 m2
 Division is the same!
 Example – An object has a mass of 48.309 g and a
volume of 9.28 mL. What is the density of this object?
 What is Step 1?
 Yep – do the math!
 Density = mass/volume
 Density = 48.309 ÷ 9.28 = 5.20571121 g/mL
 What is Step 2?
 Yep – determine the number of significant figures in
each of the original measurements
 48.309 g (5 sig figs)
 9.28 mL ( 3 sig figs)
 What is Step 3?
 Yep – determine the least number of sig figs in the
original measurements
 3 sig figs
 What is Step 4?
 Yep – Round to the least number of sig figs found in
Step 3
 5.20571121 g/mL round to three sig figs
 Final Answer = 5.21 g/mL
 40.38 cm x 3.2903 cm = ?
 0.00382 m x 0.08291 m = ?
 293.0 g ÷ 3.0023 cm3 = ?
 3.0193 x 104 kg ÷ 4.93 L = ?
 293.0 cm x 3.28 cm = ?
 Challenge: 3.927 cm x 12.736 cm x 4.000 cm = ?
 40.38 cm x 3.2903 cm = 132.9 cm2
 0.00382 m x 0.08291 m = 0.000317 m2 or 3.17 x 10-4 m2
 293.0 g ÷ 3.0023 cm3 = 97.59 g/cm3
 3.0193 x 104 kg ÷ 4.93 L = 6,120 kg/L or 6.12 x 103 kg/L
 293.0 cm x 3.28 cm = 89.3 cm2
 Challenge: 3.927 cm x 12.736 cm x 4.000 cm = 200.1 cm3
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