Download For number smaller than 1

Estimating
Circle the correct answer
1.
If you have a can of soda which is 12 oz and you want to fill a 20 oz glass, you
will need
a) 1 can
b) less than 1 can
c) more than 1 can
2.
If you have tablets with a strength of 0.25 mg and you must give 0.125 mg, you
will need
a) 1 tablet
b) less than 1 tablet
c) more than 1 tablet
3.
Gas is $5.00 for 2 gallons and you want to fill your almost full gas tank which can
hold an additional 1.5 gallons, you will spend
a) $5.00
b) less than $5.00
c) more than $5.00
4.
You have 2 tablets, one is labeled 0.15 mg and the other is labeled 0.3 mg.
What is the total dosage of these tablets?
a) 0.35 mg
b) less than 0.35 mg
c) more than 0.35 mg
Some Problems
1.
You are making sandwiches for a large crowd. All must be identical and must
contain 2 slices of bread, 2 teaspoons mustard, 3 slices salami, 2 slices of
cheese, 2 lettuce leaves, and 1 slice tomato.
You have on hand
2 loaves of bread, 20 slices each
3 pounds of cheese, each pound gives 16 slices
1 quart of mustard
60 slices of salami
An unlimited supply of tomatoes from your garden
3 head of lettuce, approximately 15 leaves in each
How many sandwiches can you make?___________________________
For which ingredients is (are) there no leftovers? ____________________
In case you’ve forgotten your kitchen conversions: 1 quart = 4 cups;
1 cup = 16 tablespoons; 1 tablespoon = 3 teaspoons
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2.
You like 3 teaspoons of sugar and 1/4 cup of milk in each cup of coffee
(3/4 cup coffee, 1/4 cup milk) you drink. Before leaving for a trip you prepare
a thermos that holds 2 quarts of liquid. What quantities of coffee, sugar and
milk will you use? (Assume the sugar occupies no volume.)
3.
A kilogram (kg) is a little more than 2 pounds (lb).
Which is heavier? 30 kg or 60 lb?
Which is a better buy: bananas for 39 cents / lb or bananas for $ 1.00 / kg?
4.
A kilometer is somewhat more than half a mile in length.
Which is shorter? a mile or a kilometer?
Which is faster? a car going 70 miles per hour or 70 kilometers per hour?
Who is faster: the runner who runs a mile in 8 minutes or the runner who runs a
kilometer in 4 minutes?
5.
A gallon is 4 quarts. The volume of a liter is slightly larger than a quart. If you
put 13.5 gallons of gasoline in your car, about how many liters is this?
Which is cheaper? $ 2.40 for a gallon or 60 cents per liter?
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Exponential Numbers
Exponential numbers are a way of writing very small or very large numbers, a situation
that occurs frequently in scientific measurements.
For number larger than 1
Number form
Exponential form
1
100
10
101
100
= 1 x (10)(10)
102
1000 = 1 x (10)(10)(10)
103
10000 = 1 x (10)(10)(10)(10)
104
Notice that the exponent indicates how many times the number 1 is multiple by 10, and
how many places the decimal point is moved to the right. Example: 1. x 103 = 1000.
Exercise 1. Write the following numbers in exponential form:
10, 000, 000 = _________________
100, 000, 000, 000, 000, 000 = ____________
1 million =
10, 000, 000, 000, 000 = _________________
_________________
Exercise 2. Write the following exponentials in number form
103 = _________________
1019 = _____________________________________
108 = __________________________
1012 = ___________________________
For number smaller than 1
Number form
Exponential form
0.1
= 1  10
10 -1
0.01 = 1  [(10)(10)]
10 -2
0.001 = 1  [(10)(10)(10)]
10 -3
Notice that a negative exponent indicates how many times the number is divided by 10,
and how many places the decimal point is moved to the left. Example: 1. x 10 -3 = 0.001
Exercise 1. Write the following numbers in exponential form:
0.00001 = _________________ 0.0000000000000000000001 = ____________
Exercise 2. Write the following exponentials in number form
10 -5 = _________________
10 -13 = ____________________________________
10 -7= __________________________
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10 -11 = __________________________
Multiplying and Dividing with Exponential Numbers
Multiplication example: 103 x 105 =
In case you’ve forgotten the general rule, remember that exponential can be
expanded:
103 x 105 = (10) (10) (10) x (10) (10) (10) (10) (10) = 108
103 x 105 = 103+5 = 108
General rule: to multiply exponential numbers, ADD the exponents.
10a x 10b = 10 a+b
General rule: to divide exponential numbers, SUBTRACT the exponents.
10a  10b = 10 a-b
Exercise 1: Complete the following
10 5 x 10 6 =
10 8 =
10 5
10 8 x 10 -4 =
10 23 =
10 -9
10 8 x 10 –6 x 10 -5 =
10 4
10 -3 x 10 -9 =
10 -11 =
10 5
10 96 =
10 54
10 5 x 10 9 x 10 -4 =
10 14 x 10 2 x 10 -6
Scientific Notation
Any number can be expressed as a number with an exponent.
For example
Another example
7.5 x 10 2 = 7.5 x 100 = 750.
8.6 x 10 -3 = 8.6 x 0.001 = 0.0086
Notice that a positive exponent moves the decimal place to the right (makes the number
bigger), and a negative exponent moves the decimal place to the left (makes the
number smaller)
When a number is expressed as a number between 1 and 10 times an exponent this is
called scientific notation:
Scientific notation: = number between 1 and 9.999… x 10
example: 6548 = 6.548 x 1000 = 6.548 x 10 3
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a
Exercise 1. Write the following in expanded form (no exponents):
5.62 x 10 5 = _____________________
8.6 x 10 –3 = ____________________
8.923 x 10 11 = _____________________
1.00234 x 10 –4 = ________________
Exercise 2. Write the following in scientific notation:
584, 000, 000 = ____________________
0.00000003456 = ________________
57 = ______________________________
0.258 = ________________________
Exponential and Scientific Notation
More Practice
Exercise 1: Solve the following problems, using exponents and giving your final
answer in scientific notation. No calculators for this.
Example:
(4 x 10 -3 ) (15 x 10 8 ) = 4 x 15 x (10 –3 ) (10 8 ) = 60 x 10 5 = 12 x 10 2 = 1.2 x 10 3
5 x 10 3
5
10 3
5
10 3
(3 x 10 9 ) (8 x 10 –7 ) =
(2 x 10 –2 ) (6 x 10 –5 )
(0.030) (0.60)____ =
(0.00075) (12000)
(4.2 x 10 3 ) (9 x 10 5 ) =
(7 x 10 –4 ) (3 x 10 4 )
(900) (5.4) (8.0 x 10 6 )_ =
(0.00002) (0.027)
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Conversions within the SI system (Metric System)
1.
Important prefixes: The SI system uses a series of prefixes to indicate size.
The prefixes indicated in BOLD below are used frequently in chemistry
pico
nano
micro
milli
centi
deci
kilo
mega
giga
Exercise:
Below each prefix given above, write its numerical value and the
abbreviation. For example: micro means 10-6 and is abbreviated as . (use textbook,
chapter 2)
2.
Using prefixes: It’s important to fully understand how to use prefixes. The
prefixes are added to base units. Some common base units are listed below
gram
Exercise:
meter
liter
second
Below each base unit, write the abbreviation. (use your text, chapter 2)
For example: The abbreviation for megameter is Mm with the M for mega and the m for
meter.
Exercise:
Fill in the following blanks with the correct abbreviations for the units
_____ picogram
_______ milliliter
_______ decimeter
_____ nanosecond
Try to think in concrete terms. For example 1 centimeter = 10 -2 m. This means
100 cm = 1 m or 1 cm = 0.01 m. Which is bigger the meter or the centimeter?
Exercise: Fill in the following blanks with the correct numbers or units
1 pm = _________________ m
1 m = __________________ pm
1 nL = _________________ L
1 L = __________________ nL
1 g = __________________ g
1 g = __________________ g
1 mm = __________________ m
1 m = __________________ mm
1 cm = __________________ m
1 m = __________________ cm
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1 dL = _________________ L
1 L = _________________ dL
1 kg = _________________ g
1 g = _________________ kg
1 ML = _________________ L
1 L = _________________ ML
1 Gg = _________________ g
1 g = _________________ Gg
We can use the equivalences to make unit conversions. This means to change from 1
unit to another. For example: a medium-sized bug is 8.4 mm in length. Express
this length in meters. First, consider if the number should be bigger or smaller than
8.4. Which is the bigger unit? The meter is the bigger unit so the answer should be
smaller than 8.4. The equivalences are used as ratios to make the conversion. The
units in the numerator (on top) will cancel with identical units in the denominator (on
bottom ).There are usually multiply ways to do the same conversion.
8.4 mm x 10-3 m = 8.4 x 10 –3 m
1 mm
or
8.4 mm x 1 m___ = 8.4 x 10 –3 m
10 3 mm
The problem can be done either way. Both are correct.
Exercise: Complete the following. Do each both ways.
A rock is 2.3 cm. Express this length in m.
A person has a mass of 70 kg. Express this mass in g.
A dose of the flu vaccine in 0.0005 L. Express this value in mL.
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Conversions between Metric and English systems
The relationships between metric and English units are not simple numbers and you do
not need to learn or memorize these conversion relationships. They will be given on
exams and quizzes or you can look them up for homework. For example, several tables
are found in chapter 2 and inside the back cover of your book.
Example: 1 kg = 2.20 lb (pounds)
or
0.454 kg = 1 lb
Is 1 kg bigger or smaller than 1 pound? _________________
We can use the equivalences to make unit conversions. This means to change from
one unit to another. For example: a full-term newborn baby weighs 3.26 kg. What
is her weight in pounds? First, consider if the number should be bigger or smaller
than 3.26. Which is the smaller unit? The lb is the smaller unit so the answer should be
larger than 3.26. The equivalences are used as ratios to make the conversion. There
are usually multiply ways to do the same conversion.
3.26 kg x 2.20 lb = 7.17 lb
1 kg
or
3.26 kg x 1 lb___ = 7.17 lb
0.454 kg
The problem can be done either way. Both are correct.
Exercise: Complete the following. Do each both ways.
A piano weighs 952 pounds. Express this weight in kg.
A can of Pepsi is 12 fluid ounces. Express this volume in mL(1 fl oz = 29.6 mL)
(one way only)
A race is 7.5 km. Express this distance in miles. (use table in book)
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Estimating Numbers – Using your Calculator
When using your calculator, there are three things you must do to be sure you have the
right answer:
1.
First estimate the answer. The estimate should be rough. This must be done by
hand without the calculator, to provide an independent check on your
calculator result.
2.
Then, use your calculator to get an answer. If your calculator result is not fairly
close to your estimated answer, go over steps 1 and 2 until the two results agree.
3.
Finally, report your answer with correct number of significant figures (chapter 2 )
Example: (6.293 x 10 8 )(8.31 x 10 –5 ) = ? Estimate: (6 x 10 8 )( 8 x 10 –5 )_ 4 x 10 12
(1546) (7.63 x 10 –12 )
(1.5 x 10 3)(7 x 10 –12)
Calculator result : 4.43 x 1012
For the following, show your estimated and your calculator result.
(87.5)(0.00000046) =
(3.047 x 10 9)
(4.832) (55.04) (439 x 10 -5) =
(3.0062 x 10 15)(4.98 x 10 –3)
(0.00068)3 =
8.74 + 13.6509 – 50.08
Three apples weigh: 49.75 g, 51.93 g, and 50.62 g. Calculate the average mass
(weight) of the apples (show set-up)
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Algebra
The following problems will give you an idea of the sort of algebra you need to know for
this course.
For each of the following, solve for the unknown variable
2x = 34
z + 23 = 204
3x – 21 = 6
0.3y = 0.24
5x + 75 = 45
y2 = 49
3x2 = 48
516 = 6
x
D=m
v
if m = 6 kg
D=m
v
if D = 6 g / mL
Math Review
and v = 9 mL,
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calculate D
and v = 12 mL,
calculate m