# 1.5 Measurement uncertainty, accuracy, and precision  (Page 4/11)

 Page 4 / 11

## Multiplication and division with significant figures

Rule: When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant figures (the least precise value in terms of multiplication and division).

(a) Multiply 0.6238 cm by 6.6 cm.

(b) Divide 421.23 g by 486 mL.

## Solution

(a) $\begin{array}{l}\begin{array}{l}\text{0.6238 cm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}6.6\phantom{\rule{0.2em}{0ex}}\text{cm}=4.11708\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{result is}\phantom{\rule{0.2em}{0ex}}4.1\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}\phantom{\rule{0.2em}{0ex}}\left(\text{round to two significant figures}\right)\hfill \\ \text{four significant figures}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{two significant figures}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{two significant figures answer}\hfill \end{array}\hfill \end{array}$

(b) $\begin{array}{l}\frac{\text{421.23 g}}{\text{486 mL}}\phantom{\rule{0.2em}{0ex}}=\text{0.86728... g/mL}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{result is 0.867 g/mL}\phantom{\rule{0.2em}{0ex}}\left(\text{round to three significant figures}\right)\\ \frac{\text{five significant figures}}{\text{three significant figures}}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{three significant figures answer}\end{array}$

(a) Multiply 2.334 cm and 0.320 cm.

(b) Divide 55.8752 m by 56.53 s.

(a) 0.747 cm 2 (b) 0.9884 m/s

In the midst of all these technicalities, it is important to keep in mind the reason why we use significant figures and rounding rules—to correctly represent the certainty of the values we report and to ensure that a calculated result is not represented as being more certain than the least certain value used in the calculation.

## Calculation with significant figures

One common bathtub is 13.44 dm long, 5.920 dm wide, and 2.54 dm deep. Assume that the tub is rectangular and calculate its approximate volume in liters.

## Solution

$\begin{array}{lll}V\hfill & =\hfill & l\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}w\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}d\hfill \\ & =\hfill & \text{13.44 dm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{5.920 dm}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{2.54 dm}\hfill \\ & =\hfill & \text{202.09459}...\phantom{\rule{0.2em}{0ex}}{\text{dm}}^{3}\left(\text{value from calculator}\right)\hfill \\ & =\hfill & {\text{202 dm}}^{3}\text{, or 202 L}\phantom{\rule{0.2em}{0ex}}\left(\text{answer rounded to three significant figures}\right)\hfill \end{array}$

What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm 3 ?

1.034 g/mL

## Experimental determination of density using water displacement

A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.

(a) Use these values to determine the density of this piece of rebar.

(b) Rebar is mostly iron. Does your result in (a) support this statement? How?

## Solution

The volume of the piece of rebar is equal to the volume of the water displaced:

$\text{volume}=\text{22.4 mL}-\text{13.5 mL}=\text{8.9 mL}={\text{8.9 cm}}^{3}$

(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)

The density is the mass-to-volume ratio:

$\text{density}=\phantom{\rule{0.2em}{0ex}}\frac{\text{mass}}{\text{volume}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{69.658 g}}{{\text{8.9 cm}}^{3}}={\text{7.8 g/cm}}^{3}$

(rounded to two significant figures, per the rule for multiplication and division)

From [link] , the density of iron is 7.9 g/cm 3 , very close to that of rebar, which lends some support to the fact that rebar is mostly iron.

An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder, with results as shown.

(a) Use these values to determine the density of this material.

(b) Do you have any reasonable guesses as to the identity of this material? Explain your reasoning.

(a) 19 g/cm 3 ; (b) It is likely gold; the right appearance for gold and very close to the density given for gold in [link] .

## Accuracy and precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to know both the precision    and the accuracy    of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value. These characterizations can be extended to other contexts, such as the results of an archery competition ( [link] ).

forms of biotechnology
What is a mole?
pls give me 3 type of transition metal
Copper, Scandium, Vanadium, Iron, Chromium, Cobalt
Jumaane-Kalif
ion zinc hydrogen
Abdul
that is not true
Jumaane-Kalif
stop posting foolishness
Jumaane-Kalif
how do I name compounds
depends on the compound. as you may know there's transition metal compounds and there's organic compounds and so on and so forth.
Jumaane-Kalif
what is electricity
electricity refers to the flow electrons
John
Hi. please can you tell me more on chemical equation
When 1 or 2 chemicals react, they rearrange their atomic composition forming new compounds. The total mass before and after is the same/ constant. Chemical equations of same reactants react in same ratios e.g. 1 Na ion reacts with 1 water molecule or a multipier like 1 mole of Na ions react with 1
Abdelkarim
Mole of water molecules. In that example we multipied by 6.02*10^23 or avogadro constant (L). Or 2 Na+ ions with two water molecules. The arrow means '' to form '' Some times conditions or reactants are written above the arrow like H+ or enzyme or temper, sometimes physical states are written
Abdelkarim
Beside the chemical Aqueous (aq) which means solute dissolved on water. Solid (s) etc Some chemical equations are written next to it ΔH= # which means enthalpy change which describes if the reaction is endothermic (+) or exothermic (-).
Abdelkarim
Some are reversible and have half double arrow sign.
Abdelkarim
what is the meaning of atoma
what is theory
what is chemical compound
Lorrita
A compound is the result of chemical bonding between 2 or more different elements.
Abdelkarim
Why is an atom electrically neutral?
Clara
the same number of electron , proton present in an atom.thats why it is electrically neutral
debibandita
difference between Amine and amide
what is the difference between alkanal and alkanone
difference between alkanol and alkanal
Michael
whaatt
Saturday
you are not well at all
Ibrahim
is organic compounds used for drying agent
Sulfuric acid is used as a drying agent.
Abdelkarim
what is an atom
An atom is the smallest part of an element, for instance gold atoms are the smallest part of gold that can enter a reaction. An atom must consist protons and electrons of equal number. You can think of those subatomic particles as spheres, but not orbiting randomly they move in specific way in
Abdelkarim
That was partially described mathematically. As a muslim, we believe that god created all matter from nothing. He is the Able, and only who can create matter in the begging in the big bang that was described in the Quran in chapter 17 verse 30, 1400 years ago, you can read it from internet.
Abdelkarim
what is electron affinity
John
do you have a private jet
Ibrahim
what is acid
Proton donor (H+). Like in lemons, oranges and some medicines.
Abdelkarim
what is titration? and how can i get my readings?
what is electrolysis?
what is the formula you use when calculating for gas law?
favor
PV=nrt
Farid
electro-means electricity while lysis-means splitting up so electrolysis simply means splitting up by means of electric current
John
good day. how may I see previous question asked in this chat, please?
Asha
Asha, A person named Favor asked what is meant by the term electrolysis
John