NUMBERS AND MATH OPERATIONS
Measured numbers or exact numbers.
Scientific notation.
Significance and rounding.
Dimensional analysis.
Problem solving techniques.
The temperature box.
Overview of pH. (In acidbase section)
The pH box. (In acidbase section)
Calculator use with pH box. (In acidbase section)
The pKa box. (In acidbase section)
Percent.
Basic algebra.
Proportionality.
Conversions using DA.
Practice with W5P method.
W5P and DA review problems.
Back to the beginning of Numbers and Math
Operations.
MEASURED vs. EXACT NUMBERS
Exact numbers are numbers that are exact by definition, such as:
1 inch = 2.54 cm or 1 gallon = 231 cubic inches or
1 foot = 12 inches or 1 meter = 100 centimeters
or
numbers that come in integers and are not likely to be available in amounts
smaller than integers.
(When you ask for seating in a restaurant, the number
of people is an integer, an exact number.)
Measured numbers are an estimated amount, measured to a certain
number of significant figures without the benefit of any natural unit
or numbers that come from a mathematical
operation such as averaging. You would get a measured number from using a
meterstick to find the length of a board or using a graduated cylinder to
find the volume of a liquid. Average numbers of people would make a measured
number, even though people naturally come only in integers. The average
family in the U.S. has 2.37 children.
Back to the beginning of Numbers and Math Operations.
SCIENTIFIC NOTATION
There are many very large and very small numbers in
scientific studies. How would you like have to calculate with:
1 mol = 602,200,000,000,000,000,000,000 atoms
or
1 Dalton = 0.000,000,000,000,000,000,000,00165 g
You can streamline large or small numbers with
scientific notation. The standard is that you place the decimal
point after the first significant digit and adjust the exponent
of ten so that there is no change in the value of the number.
Think of the change as creating a new number with two parts, a
digit part and an exponent part, from the old number. To put the
decimal point behind the first digit, you must divide or multiply
the original number by some integer power of ten. Then you must
do the opposite (inverse) to the exponent part of the new
expression so that there is no change in the value of the number.
(0.000,000,000,000,000,000,000,00165
x 10^{24}) x
1/_{10}24 = 1.65 x 10^{24} or 1.65 E24
(602,200,000,000,000,000,000,000/_{10}23) x
10^{23 } = 6.022 x
10^{23} or
6.022 E23
The original numbers have the same value as the exponential
forms, but the exponential forms have the decimal point in
the right place. The 'right place' is to the right of the first
digit. The "E" in the number stands for exponent. Your
scientific
calculator will use the numbers in the shortened form, usually
best represented by the "E" form. Don't get caught
making
too much of this. You have seen it before. The number "five
point two million" is the same as 5.2 E6. The number of the
power of ten only indicates how many places you need to move the
decimal to get the long form of the number back.
The only question you might have trouble with is WHICH WAY to
move the decimal. The easy way to remember that
is: numbers that are less than one have negative exponent numbers
in the scientific notation form, and numbers that are
larger than one have positive exponent numbers. Very often
chemistry professors will tell you they want answers in
scientific notation if the number is larger than one thousand or
smaller than one thousandth. Keep your professor happy. Find out
exactly what is required in your course and follow the instructions to the letter.
The table below shows the fully written out number, the way we say
the number in English, and the way to write the number in
scientific notation. The numbers in scientific notation are to
two significant digits.
Full number
 Words for number
 Sci. notation styles

5,000,000,000  five billion
(USA) 
5.0 x 10^{9
} or 5.0 E9 
500,000,000  five hundred
million 
5.0 x 10^{8} or 5.0 E8 
50,000,000  fifty million
 5.0 x 10^{7} or
5.0 E7 
5,000,000  five
million  5.0 x 10^{6} or 5.0 E6 
500,000  five hundred
thousand  5.0 x 10^{5} or 5.0 E5 
50,000  fifty thousand
 5.0 x 10^{4} or
5.0 E4 
5,000  five thousand  5.0 x 10^{3} or 5.0
E3 
500  five hundred  5.0 x 10^{2} or 5.0 E2

50  fifty  5.0 x 10^{1} or 5.0 E1

5  five  5.0 x 10^{0} or 5.0 E0 
0.5  five tenths  5.0 x 10^{1} or 5.0 E1 
0.05  five hundredths  5.0 x 10^{2 } or 5.0 E2 
0.005  five thousandths  5.0 x 10^{3 } or 5.0 E3 
0.000,5  five tenthousandths  5.0 x 10^{4} or 5.0 E4 
0.000,05  five hundredthousandths  5.0 x 10^{5} or 5.0 E5 
0.000,005  five millionths  5.0 x
10^{6 } or 5.0 E6 
0.000,000,5  five tenmillionths  5.0 x
10^{7} or 5.0 E7 
0.000,000,05  five hundredmillionths  5.0
x 10^{8} or 5.0 E8 
0.000,000,005  five billionths  5.0 x
10^{9 } or 5.0 E9 
Back to the beginning of Numbers and Math Operations.
SIGNIFICANCE AND ROUNDING
All the numbers in the above table have only two significant
digits. Only the five and zero in each number has any
numerical meaning other than placeholding. Let's say that
Delhi, India has five million people. We have expressed that
number in one significant digit. That might be sufficient for
such a number. Are we considering just people within the city
limits? How about people who live outside the city and come in
only for business? What year are we specifying? Let's
say we have enough information to number the population of Delhi
at 5.1 million. That is now more a more accurate
number that claims a two digit significance. What if we were to
go absolutely wild and say that Delhi has 5.1376504 x
10^{6} people? That is the same
number, isn't it? First, it ridiculously claims eight
significant digits. Even worse, that calculates
to 5,137,650.4 people, and people just don't come in
fourtenths of a person. Yes, you can claim to be far too
accurate. With that in mind, here are the rules for considering
the number of significant digits:
1. All nonzero digits are significant.
Every
1, 2, 3, 4, 5, 6, 7, 8, and 9 claims significance. The number, 123,456.789 has nine significant digits.
2. All leading and following zeros that are only placeholders are not significant.
These numbers have some magnitudeindicating zeros. The red (underlined) zeros are not significant in 4,200,000 (2 SF) and 0.000,004,090 (4 SF).
3. All zeros between two other digits are significant.
The number 6.023 has a significant zero for a total of four significant digits.
4. All zeros to the right of the decimal and to the right of other digits are significant.
For instance, the number 43.500 has five significant digits, two of which are zeros.
You may hear the phrases significant digit, significant numeral, or significant figure to describe this idea. Chemtutor
will sometimes shorten it to "sig figs," or just "SF."
We can round numbers to the proper number of significant digits by lopping off all digits past the number needed if less
than five and rounding up the last needed digit if the following digit is five or more. For examples, here rounding to three significant digits:
3.4848 becomes 3.48;
4.1550 becomes 4.16;
5,786,899 becomes 5,790,000 (or 5.79 E6); and
0.000,347,00 becomes 0.000,347 (or 3.47 E4).
How do you know when you need to round? In multiplication and
division, the answer cannot have more significant
digits than the number with the smallest number of significant
digits used to calculate the answer. So a four significant digit
number multiplied or divided by an eight significant digit number
will result in a number that can only claim four significant
digits. Wisconsin has 379 cities with about 5.1 thousand people
in it. How many people live in all these small towns? 379
x 5.1 E3 = 1.9329 E6, but the answer can only claim two
significant digits. The answer must be 1.9 E6 people because
the number 5.1 E3 only has two significant digits.
How do you know where to round in actual measurements? The
last digit is the one we get by estimation. For instance,
if you have a graduated cylinder marked in milliliters and tenths
of a milliliter, you should be able to estimate between the
lines (interpolate) of tenths of milliliters and measure
hundredths of milliliters.
The allowed significance works differently with adding and
subtracting. The addition of a family of four to a city of three
million does not significantly change the population of the city.
If you have 1,578,000 chickens and you add 2,717
chickens to them, you have 1,581,000 chickens. Align these numbers one on
top of the other so you can more easily see the reasoning behind this.
No answer in subtraction or addition can have significant digits in
COLUMNS in which on the right there is not a significant digit in
each participant number. In the example, the columns with the zeros (in red) are not significant. The raw answer of 1,580,717 must be rounded to show that columns E, F, and G are not significant. The answer must then be rounded up because column E in the answer is greater than 5. The answer is 1,581,000 chickens.

A 
B 
C 
D 
E 
F 
G 

1 
5 
7 
8 
0 
0 
0 

______ 
_______ 
+_____ 
2_____ 
7_____ 
1_____ 
7_____ 

1 
5 
8 
0 
7 
1 
7 








It is just as important to know WHEN to round as HOW to round.
In any math problem you should wait until the end to round; Only
the final answer should be rounded. Carry as many significant
digits as you can throughout the problem. On a calculator, the
most efficient way to carry the maximum is to do all the
calculation on the calculator. Arrange the problem so that you do
not have to copy an intermediate answer only to reenter it into
the calculator. If you do find yourself needing to save numbers
outside the calculator, copy several more significant digits than
you think you need.
Back to the beginning of Numbers and Math Operations.
DIMENSIONAL ANALYSIS
Dimensional analysis is a system of using the units of
quantities to guide the mathematical operations. There are other
names for the very same idea, for instance, unit conversion or
factor label or factorunit system. Here we will abbreviate it to
DA. It can be used for conversions, to check work done with
formulas, or, in uncomplicated problems, instead of
formulas. As we will use it, DA can do much of the mathematical
work of chemistry. We will also be doing some more
complex math problems for which the use of formulas is necessary.
The general rule for whether to use DA or W5P is: if
the answer is the same dimension as the given units, you will
likely use DA, but a formula best accounts for any change in
dimension. (W5P method. See below.) To use the DA system for
conversions, you should check to see that the unit
given and the unit into which you want to change are the same
dimension. To someone unfamiliar with that idea, the
prospect of finding out how many feet are in 2.5 acres almost
seems a reasonable one.
Begin with the KNOWN QUANTITY. Place all the known quantity in
the numerator of the beginning fraction if there is
no denominator. To use DA, one must know the dimensions of units
and conversion factors. Definitions, such as those
found in the metric prefix and unit sheet can serve as conversion
factors by dividing one side of an equation by the other,
thus: since 1 mile = 5280 ft.
We can multiply any quantity by anything equal to 1 without
changing the value of the quantity. Therefore, 5280 ft/mile or
1 mile/5280 ft is a conversion factor for length. These
conversion factors can change 6.20 miles to a number of meters
in serial fashion thus:
Notice you can visualize the definitions you know from the
unitsanddefinitions section in each one of the conversion
factors. The definition 1 mile = 5280 ft can convert miles to
feet. We want to cancel out the miles, so we place 1 mile
under the 5280 ft. Now the mile units can cancel with one in the
denominator and the other in the numerator. If we were
to stop at this point, the answer would be in units of feet.
Similarly, use 1 foot = 12 in. to go from feet to inches, placing
the foot unit in the denominator to cancel with the foot unit in
the numerator. The definitions and conversions in the units
and measures and definitions chapter become very convenient for
use as conversion factors. It is necessary to know
them by rote to be able to easily use the system. If you know the
definitions well, not only will you escape having to look
up the right definition, but you will much more easily spot the
best way to convert the numbers. The metric system
definitions come from the powers of ten of the metric prefixes.
The metric "staircase" method allows for changes from
any magnitude to another in one conversion step by "counting
the steps up or down the staircase."
We can use the density of a material as a conversion factor
between mass and volume rather than a formula in this
manner: The density of mercury is 13.6 g/cc. What is the mass of
two liters of mercury?
This math could have been done without stopping to calculate
the intermediate answer of 27200 g, but the math always
works, no matter in what order it is done. In both cases we
carefully accounted for the units by canceling. A stepwise
orderly progression of easily remembered changes can convert
units to anything you need. The metric system is
particularly easy to use with DA. The definitions are multiples
of ten and need scrupulous care to stay untangled. Here is a way
to think of it using a change from Kilometers to millimeters.
Kilometers being the larger unit, begin with one of the larger
unit. The number of smaller units, (mm) is the power of ten that
is the number of steps up the metric staircase. This process
keeps the exponent positive.
1 km = 1x 10^{6} mm or 1 km = E6 mm
Problems that involve many substances or a lot of addition and
subtraction can become difficult with DA, but for
conversions within a dimension or simple multiplication or
division problems, even in serial fashion, it is a powerful tool.
We can use DA when working formula problems by consistently using
and canceling the appropriate units. The answer
must come out in the proper dimension according to the units, or
you should suspect something wrong. You may then
easily convert the answer to units you want by using simple
definitions as conversion factors. As you may have
suspected, DA has many uses in the study of chemistry. Many
common dailyliving problems could be helped by
thinking in DA.
Back to the beginning of Numbers and Math
Operations.
PROBLEM SOLVING TECHNIQUES  THE W5P METHOD
One of the really difficult things to learn or teach in any
science is the way to do word problems. For some students,
there seems to be a mental block about it. Chemtutor teaches a
method that may work for you, but the success of the
method depends upon the background knowledge of the student in
the units, dimensions, and formulas.
W5P METHOD (WILD & WONDERFUL WAY TO WORK WORD PROBLEMS)
GIVEN  List all pertinent information with dimension symbol, number and unit.
FIND  List the dimension of the quantity requested in problem.
FORMULA  With the dimensions in GIVEN
and FIND, list the formula of formulas that fit.
SOLVE  Solve the formula for what you are looking for (FIND),
substitute the number values in GIVEN, and perform the math on both the units and the numbers.
ANSWER  Check the answer for likeliness, make sure the units are appropriate, express the answer in scientific notation and to the accuracy required, and draw a box around it so it is obvious which number your answer is.
Here is a sample problem using W5P method. Lead is 11.3 g/cc. What is the volume of 24.5 kg of lead? If you
recognize the units of density, you can immediately spot that D = 11.3 g/cc. The symbol of the dimension, density, is D.
We show the number and units of the density with the symbol. The mass is 24.5 kg. (m = 24.5 kg. ) Find the volume
(V). The formula that has all the dimensions needed is the density formula from the formula list. We write it here in its
original (memorized!) form. Solve for what you are looking for (volume here) using just a little bit of good old basic
algebra. You would like to get the equation in the condition where you have V on one side all by itself. Do the algebra as
stepwise as you need to in order to show yourself you have actually done it right. Next, substitute the measurements for
the dimension symbols. Note that the kilogram and cc and gram units cancel, along with the number one thousand in the
math. All we have left is to do the division and processing of the answer. The answer calls for three significant digits. The
number does not need scientific notation because the number is less than one thousand and more than one thousandth.
Notice that the "GIVEN" section includes the symbol for the
dimension of the information, the number and unit. "FIND" shows
the dimension of the information requested. The formula is not
labeled as such, but it is the relationship that has the
requested information among the dimensions and all other
dimensions in the formula are known. (Here we are looking for V
and we know D and m.) The next step is to solve the formula for
the dimension we need. Then substitute the given quantities for
the known dimensions and process the answer.
Here is an opportunity to show you a small but useful way to
save yourself some grief. Consider the unit g/cc in the
above math. The unit is a fraction in the denominator. It is
easier to think of as cc/g in the numerator. (The rules of math
permit you to move a fraction from the denominator to the
numerator if you invert the fraction.)
Fraction A comes from the original substitution of the numbers
and units into the solved equation. Fraction B shows the
inversion of the fractional unit into the numerator. This is a
really useful way to simplify fractions that seem complex.
Fraction C shows the same fractional unit integrated into the
whole fraction and cancels the unit of gram. Step C is not needed
if you want to cancel the gram unit in the numerator. Fraction D
shows what happens when you cancel a unit without its metric
prefix. The k is just one thousand now, so k (cc) is the same as one
thousand times cc which is a liter. Fraction E shows the final
unit. Many students make math mistakes by failing to
simplify fractions in the denominator.
Back to the beginning of Numbers and Math
Operations.
THE TEMPERATURE BOX
There are two types of common conversion that are not easily
fit for calculation by DA because they require formulas among
the different units. Temperature conversions are among the units
Celsius, Kelvin, Fahrenheit, and Rankine. Acidity units are pH,
hydrogen ion concentration, hydroxide ion concentration, and pOH.
A conversion box is a fine teaching device for this type of unit
association group.
(For going around the box, use K = C + 273.15 and
R = F + 459.67 to get more accurate results.)
SOME COMMON TEMPERATURES in all four scales to
the nearest whole degree
Material Point
 Kelvin
 Celsius
 Fahrenheit
 Rankine

copper m.p.  1356  1083  1981  2441 
gold melt pt  1336  1063  1945  2405 
aluminum m.p.  933  660  1219  1680 
lead melt pt  601  327  621  1081 
H2O bp1atm  373  100  212  672 
chicken body  313  40  104  564 
human body  310  37  98.6  558 
room temp.  293  20  68  528 
H2O m.p.  273  0  32  492 
zero °F  292  19  0  460 
mercury
m.p.  234  39  38  422 
°C = °F  233  40  40  420 
absolute
zero  0  273  460  0 
There are only three temperature measurement scales commonly
used. The Kelvin, Fahrenheit, and Celsius or
Centigrade scales all have common uses. The Rankine scale is not
commonly used, but it is needed here for the learning
experience. Consider four identical liquidtype thermometers
sidebyside with different markings on them. This is a
good mental image because These four scales measure exactly the
same thing. The small superscript "zero" before the
symbol means "degree." Some people make a big issue of
omitting the degree sign before Kelvin. They say "Kelvins" instead of "degrees Kelvin."
The Kelvin and Rankine scales are both absolute temperature scales. Since temperature is a measure of the average velocity of the atoms or molecules in a material, it makes sense that there is a temperature that represents the condition of no molecular motion. That temperature is called "absolute zero." Both Kelvin and Rankine scales are absolute scales. The zero mark on these scales is at absolute zero. There are no negative temperatures on either scale. Since they have the same zero point, the relationship between them is a simple proportion. You will remember the formula as the Dog Equation   Canine equals Arfive. Yes, K 9 = R 5 or, as you more commonly see it, 9 K = 5 R. No applause, please.
The size of the Celsius degree is the same as the size of the
Kelvin degree. The Celsius scale just has its zero at the
freezing point (or melting point, for they are the same) of
water, 273.15 K. The size of the Fahrenheit degree is the same
as the size of the Rankine degree. To calculate the temperature
in Rankine, add 459.67 to the Fahrenheit temperature.
For instance, the freezing point of water is 32 °F or 491.67 °R.
The easy way to be sure to know the conversion between Fahrenheit and
Celsius is as follows: (a) Only remember one formula. (b)
From the Dog Equation the proportion of Celsius and Fahrenheit is
nine to five. 9/5 C must be the right proportion, and
(c) Fahrenheit zero is 32 degrees below the freezing point of
water. Therefore, the equation is:
KNOW THIS
This is the only equation you will need or want. Some texts will ask you
to memorize two equations, one solved for Fahrenheit
and the other solved for Celsius. To get Celsius temperature from
Fahrenheit, you must solve for 'C,' but, for the
convenience of remembering only one easily related equation, the
price is cheap.
The temperature box is a device that can give you as much
practice as you need. Here is how to use it. Choose a
temperature, for instance, 37 °C, normal human body
temperature. Calculate around the box either clockwise or
counterclockwise. You should return to the same Celsius
temperature. For rough onetime calculations, you can use the
more generally quoted change between Celsius and Fahrenheit and
between Fahrenheit and Rankine. For going around
the temperature box, you should use the more exact numbers in the
parentheses.
Back to the beginning of Numbers and Math
Operations.
PERCENTAGE CALCULATIONS
Converting raw numbers to percentages is easy once the parts
are defined. A percentage is the target over the total
multiplied by one hundred percent.
There are thirty people in the classroom. Of them, seventeen
are male. What percentage of males are in the classroom?
'Seventeen males' is the target we have defined. 'Thirty people'
is the total. Seventeen divided by thirty times one hundred is
56.66667 to seven significant digits or 56.7 to three significant
digits. Males are people, so we cancel the units. The answer is
56.7 percent.
In many cases, the most difficult part of using percentages is
identifying the target and the total. Percentages do not have
any other unit attached to them other than the percent. After
dividing one unit by the same type of unit and cancelling the
units, that makes sense.
Back to the beginning of Numbers and Math Operations.
BASIC ALGEBRA
Algebra operations can be performed on equations with symbols, numbers, or measurements. An equation states that
the right side of the equation is equal to the left side. The rules of manipulation of the two sides of an equation without
changing the equation are: (a) you can multiply or divide both sides by the same thing, (b) you can add or subtract the
same thing from both sides, (c) you can raise both sides to the same power, you can change both sides to or from any
exponent. Also, you can substitute anything in an equation for something of equal value. The rules are easy, but becoming proficient takes practice. If you need the practice, there are many basic algebra textbooks and web sites. Most people need to actually DO the math in the textbook or web site rather than just reading about them.
One of the really big mistakes many novice calculators make is
attempting to do too much between the ears before
enough practice has been done. The best advice is that
calculating with a pencil (or writing implement of any type) is
much more reliable than calculating mentally. You may have heard
of "back of the envelope" calculations by scientists
or engineers. Most people of science or technology understand that
even something scratched on the back of an envelope
is more reliable than mental calculation. Show your work, if not
to your teacher, to yourself. Show the addition of the
same thing to both sides. Show the cancellation of units. Show
the stepwise conversion of what you know to what you
need. Except for log and antilog work for pH calculations, the
use of more complex algibraic operations in basic chemistry
course is rare. The use of quadratics or complex factoring is
rare.
Back to the beginning of Numbers and Math Operations.
PROPORTIONALITY  DIRECT PROPORTION
A boy and a girl are in the back seat of the car on a long
trip through the countryside. Their mother suggests that they
count the cows on their side of the car to keep them occupied.
The boy sees a large number of cows on his side and
begins furiously counting, "1, 2, 3, 4, 5." It is
obvious from his disappointment that he did not get a chance to count all
his cows. Then a much larger herd appears on the girl's side. She looks at the herd and proudly announces, "Six
hundred and seventy seven cows." The boy angrily
says. "Oh, you couldn't have counted that many so
quickly." She retorts, "It's easy. All you have to do is count
the legs and divide by four. Mathematically, of course, she is correct. The number of cows
is proportional to the number of legs; as the number of
cows increases, the number of legs increases and visaversa. This
could be expressed mathematically as:
You would read the proportionality as: "The number of cows is proportional
to the number of cow legs."
An example in chemistry of proportionality is the relationship
of the pressure and temperature of the same gas at a
constant volume. When the temperature increases, the pressure
increases. When the temperature decreases, the
pressure decreases. The formula for that can be expressed:
PROPORTIONALITY  INVERSE PROPORTION
Consider the table of hydrogen ion concentrations versus
hydroxide concentrations. As the [H^{+}] increases, the [OH^{}]
decreases. As the [H^{+}] decreases,
the [OH^{}] increases. This
relationship is inverse proportionality.
It is something of a joke in corporate circles that the importance of a
person is inversely proportional to the number of keys carried. The CEO of
the company (of 'infinite importance') does not have to keep any keys because
all doors are opened for the CEO. The janitor (lowest in importance) has a
large ring of keys. This can be shown in the proportion manner by:
Ths reads, "The importance is inversely proportional to the
number of keys." Using the pressure and volume of the same gas at constant temperature,
inverse proportionality can be expressed mathematically by:
HOW FORMULAS SHOW DIRECT AND INVERSE PROPORTION
It is easy to spot proportionalities in mathematical formulas.
Let's use the universal gas law, PV = nRT, as an example. If
the two variables are both in the numerator (or both in the
denominator) on opposite sides of the equation, as P and T in
the universal gas law, they are proportional. It two variables
are on the same side of an equation, they are inversely
proportional, as in P and V in the same equation.
Back to the beginning of Numbers and Math Operations.
FOR AN EXPLANATION AND DISCUSSION OF EACH PROBLEM, CLICK ON
THE QUESTION MARK FOLLOWING THE QUESTION. A SET OF ANSWERS FOLLOW EACH PROBLEM
SET.
These problems may not have much to do directly with chemistry, but
they are designed to help you understand the methods of thinking about and
solving problems in chemistry.
PROBLEM SET # 1  CONVERSIONS USING DA
DO THESE CONVERSIONS IN THE DIMENSIONAL ANALYSIS STYLE. ANSWER
IN SCIENTIFIC NOTATION IF > E3 or < E3.
1. What is 1.50 mm in km ?
2. How many nanoseconds are in 1.50 days?
3. How many fluid ounces (floz.) of water do you have in exactly half of a ten gallon can?
4. A car is going 60.0 MPH. How fast is that in ft/sec?
5. A car is going 62.0 MPH. How fast is that in KPH?
6. How many mm^{2} are there in one square kilometer?
7. How many in^{2} are there in an acre?
8. How many mm^{2} are there in 0.550 acre?
9. A car goes 0 to 60.0 MPH in 5.00 sec. Write that acceleration in m/_{sec}2?
(Start with (60 mi)/(hr.)(5 sec))
10. Light travels at 3.00 E8 m/sec. How fast is that in MPH?
11. An elephront has a mass of 1.80 tonnes. What is its weight in ounces?
12. Mercury has a density of 13.6 g/cc. What is that in #/gal?
13. A light year is the distance that light goes in a year. Using data from #10, how long is a light year in miles ? (Rate times time = distance)
ANSWERS TO PROBLEM SET #1
1. 1.5 E6 km  2. 1.30 E14
nsec  3. 640 Floz.  4. 88.0
ft/sec  5. 99.8 KPH 
6. E12 mm2  7. 6.27E6
in2  8. 2.23 E9 mm2  9. 5.36
m/sec  10. 6.71E8
MPH 
11. 6.34 E4 Oz.  12. 113 #/gal
 13. 5.88 E12 mi   
Back to the beginning of Numbers and Math Operations.
PROBLEM SET #2  PRACTICE WITH W5P METHOD
1. How deep would the water in a trench three inches wide and sixteen feet long have to be to contain 24.0 cubic feet of water?
2. Cloth is four feet wide as it comes on the bolt. What length of cloth would you need to cover a surface of 58.0 square yards?
3. It is 762 miles from here to Chicago. An obese chemistry teacher jogs at a rate of one mile every twenty minutes. How long would it take him to jog to Chicago if he jogged continuously and REALLY wanted to get there?
4. The density of lead is 11.4 g/cc. What is the mass of a block of lead two cm by ten cm by four cm?
5. How many liters of water could you fit into a container measuring 10.0 ft by 13.0 ft by 8.00 ft?
6. A box of qrtpsdfgh measures 24.0 x 4.00 x 12.0 inches. It has a density of 38.0 pounds per cubic foot. What is the weight of qrtpsdfgh in the box? (Notice the English system density!)
7. My basement was flooded after a big rain. The basement is rectangular 12.0 ft by 17.0 ft. The water was measured as it drained out at 51.0 cubic feet. How deep was the water in the basement?
8. A standard expressway lane is 11.0 ft wide. How long would a four lane expressway be if it had a square mile of pavement?
9. The density of gasoline is 720 g per liter. What is the mass of a jar of gasoline three cm by four cm by five cm?
10. How many liters of water could you fit into a hot tub that measures 1.50m deep and three meters in diameter? How many people? ( 1 cubic meter = 1000 L )
11. Mr. Richland has a swimming pool 10.0 ft by 20.0 ft. He had a wild party and they threw his son's Volkswagen into the pool. The VW has a volume of 10 cubic feet. It was sealed up, but it sank because of all the lead bricks (?!?!?*?*?)
in it. (a) What is the area of the pool? b) How much did the VW raise the water level?
12. How long is a string?
ANSWERS TO PROBLEM SET #2
1. 72 in (6 ft)  2. 43.5
yd  3. 254 hours  4. 912 g
 5. 2.94 E4 L 
6. 25 1/3 #  7. 0.25 ft
 8. 120 miles  9. 43.2
g  10. 1.06 E4 L 
11a. 200 sq.ft.  11b. 0.6 in or
1/20 ft.  12. yes  
\
Back to the beginning of Numbers and Math Operations.
PROBLEM SET #3  W5P AND DA REVIEW PROBLEMS
1. How many ounces is 45.0 tonnes?
2. How many liters are in 8.55 cubic miles of ocean water?
3. Water has a density of one gram (1.00g) per cubic centimeter. What is that in pounds per gallon?
4. A roadsign posts a 50.0 MPH limit. What is that in KPH?
5. How large in square feet is a 525 square meter house?
6. How many square yards of paint are needed on the ceiling of a room that is 24.0 feet wide, 51.0 feet long and ten feet high?
7. Gasoline has a density of 5.83 pounds per gallon. What volume tank would you need to store three tons of gasoline?
8. The bathtub in the residential suite of the White House had to be enlarged for William Howard Taft. He weighed 370 pounds. (Burp.) The bathtub is four feet wide, six feet long and three feet high. How many gallons of water are needed to fill the WHT Memorial Bathtub? (1 gal = 231 cu.in.)
>9. The density of water is 8.3 #/gal. What weight of water is needed to fill the WHT Memorial Bathtub?
10. Mercury is 13.6 kg/liter. What mass of mercury is needed to fill the WHT Memorial Bathtub?
11. The velocity of light is 186,000 mi/sec. It takes radio signals 17.6 minutes to go from Earth to Jupiter at their closest approach. How far apart are the two planets at that time?
12. A car takes on 22.7 L of gasoline. How many Floz is that?
13. Osmium metal is 22.0 g/mL. What's the volume of 2.50 kg of it?
14. Concentrated sulfuric acid is 1.84 kg/L. What is the mass of 500 mL of it?
15. A man is 5 ft. 11 inches tall. How many meters is that?
16. Why is a duck?
ANSWERS TO PROBLEM SET #3
1. 1.59 E6 Oz.  2. 3.56 E13
L  3. 8.30 #/gal  4. 80.5
KPH  5. 5.65 E3 ft^{2} 
6. 136 sq.yd.  7. 1.03 E3 gal
 8. 539 gal  9. 4.49 E3
#  10. 2.76 E4 kg 
11. 1.96 E8 mi  12. 770
Floz  13. 114 mL  14. 0.920 kg
 15. 1.80 m 
16. because     
Back to the beginning of Numbers and Math
Heuristics
Units and Measures
Atomic Structure
Elements
Periodic Table
States of Matter
Compounds
Reactions
Mols, Stoichiometry, and Percents
Oxidation and Reduction Reactions
Gases
Solutions
Acids and bases
Kinetics
Thermochemistry

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