Chapter 2
Review
Math 1030
Section A.1: You Can’t Add Apples and Oranges
Definition of units
The units of a quantity describe what is measured or counted.
We cannot add or subtract numbers with different units, but we can multiply and divide quantities with
different units.
There are four different operations that we are allowed to do with different units:
(1) division
(2) raising to second power
(3) raising to third power
(4) multiplication
Identifying units in a problem
In order to understand what a problem is about, the first thing to do is to look at the units involved in
the problem.
Ex.1
Identify the units in the price for gasoline found by dividing its total cost in euros by the number of liters
of gas.
Ex.2
Identify the units in the volume found by multiplying an area measured in square meters by a depth
measured in centimeters.
Section A.2: Unit Conversions
Unit conversions
It is very important to understand how we can convert one unit to another one.
Ex.3
Convert 5 meters into kilometers.
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Chapter 2
Review
Math 1030
Ex.4
How many seconds are there in one year?
Ex.5
Convert 1 gram2 to hectogram2 .
Ex.6
You want to carpet a room that measures 10 by 12 feet, making an area of 120 square feet. But carpet is
usually sold by the square yard rather than by the square foot. How many square yards of carpet do you
need?
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Chapter 2
Review
Math 1030
Ex.7
You are preparing a vegetable garden that is 20 meters long and 12 meters wide, and you need enough
soil to fill it to a depth of 1 meter. The landscape supply store sells soil by the cubic feet. How much soil
should you order?
Ex.8
At an italian pizzeria, the price for a pizza is 15 euros. What is the price in U.S. dollars? (1 euro =
$1.272).
Section A.3: Problem Solving with Units
Working with units
(1) Identify the units involved in the problem. Use the units to help you decide how to approach the
problem and what units to expect in the answer.
(2) Perform any operations (addition, multiplication,...) on both the numbers and the associated units.
Remember:
• You cannot add or subtract numbers with different units (you cannot add apples and oranges),
but you can combine different units through multiplication, division or raising to powers.
• To make your work with units easier, replace division with multiplication by the reciprocal.
(3) When you complete your calculations, make sure that your answer has the units you expected. If it
doesn’t, then there is a mistake.
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Chapter 2
Review
Math 1030
Ex.9 Gas Mileage.
After a long day of driving, your destination is only 150 miles away. You know that your car gets 30
miles per gallon. How much gas do you need in your tank if you want to reach the destination without
stopping? If you have a 20-gallon tank and the fuel gauge shows it is one-quarter full, will you make
it?
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Chapter 2
Review
Math 1030
Section B.1: Standardized Unit Systems: U.S. and Metric
Different system of standardized units
There are different systems of standardized units: the international metric system, called SI (from the French
Système International d’Unités) and the English system, called the U.S. customary system (USCS).
The U.S. Customary System
• To measure the lengths of objects we have: inches, feet, yards, miles and others.
• To measure weights of objects we have three different ways: Avoirdupois measures, Troy measures
and Apothecary measures. The Avoirdupois measures are commonly used today.
• To measure volumes we have two kinds of measures: Liquid measures and Dry measures. The
liquid measures are used in the case of liquid objects as water, beer and so on. The Dry measures
are used in the case of solid objects such as flour.
Notice that 1 dry pint = 33.60 in3 > 1 liquid pint = 28.88 in3 .
The International Metric System
The basic units of length, mass, time and volume in the metric system are
• the meter for length (m);
• the kilogram for mass (kg);
• the second for time (s);
• the liter for volume (L).
In the international metric system we use powers of 10 to find all the other units ad we write the new units
adding a prefix which indicates multiplication by a power of 10.
Ex.10
At a gas station in Mexico, the price of gasoline is 0.05 pesos per centiliter. What is the price in dollars per
gallons?
We know that 1 peso = $0.09136.
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Chapter 2
Review
Math 1030
Ex.11
How many square meters are in a square mile?
Temperature Units: Fahrenheit, Celsius and Kelvin
Temperature Units
There are three temperature scales commonly used today: Fahrenheit, Celsius and Kelvin.
• The Fahrenheit scale is used in the United States and it is defined so that water freezes at 32◦ F and
boils at 212◦F .
• Internationally, temperature is measured on the Celsius scale, which places the freezing point of
water at 0◦ C and the boiling point at 100◦ C.
• The Kelvin scale is used in science and is the same as the Celsius scale except for its zero point: a
temperature of 0 K is the coldest possible temperature, known as absolute zero and it correspond
to −273.15◦C or −459.67◦F .
The conversions are:
• from Celsius to Fahrenheit:
F = 1.8 C + 32
• from Fahrenheit to Celsius:
F − 32
C=
1.8
• from Celsius to Kelvin:
K = C + 273.15
• from Kelvin to Celsius:
C = K − 273.15.
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Chapter 2
Review
Math 1030
Ex.12
Convert 15◦ F into Celsius.
Ex.13
Convert 4 K into Fahrenheit.
Section B.2: Units of Energy and Power
Definition of energy
Energy is what makes matter move or heat up. The international metric unit of energy is the joule.
Calories
For example, we need energy from food to walk or run. The most familiar energy unit is the food Calorie
used to measure the energy our body can draw from food:
1 Calorie = 4184 joules.
Definition of power
Power is the rate at which energy is used. The international metric unit of energy is the watt:
1 watt = 1
7
joule
.
s
Chapter 2
Review
Math 1030
Ex.14
You burn 1200 Calories playing volleyball for 2 hours. What is the average power during the game in
watts? Are you generating enough power to keep a 100-watt bulb shining? Explain.
Definition of kilowatt-hour
A kilowatt-hour is a unit of energy:
1 kilowatt-hour =
1000 joule
1000 joule
× 1 hr =
× 3, 600 s = 3, 600, 000 joule.
1s
1s
Ex.15
Your electric bill states that you used 2000 kilowatt-hours of energy in November. Determine your total
electrical energy use, in joules and your average power use, in watts.
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Chapter 2
Review
Math 1030
Section B.3: Units of Density and Concentration
Definition of density
Density describes compactness or crowding. There are three different kinds of densities:
g
• Material density is given in units of mass per unit volume, such as grams per cubic centimeter ( cm3 ).
g
g
For example, the density of water is 1 cm3 . Objects with densities less than 1 cm3 float in the water,
g
while objects with densities more than 1 cm3 sink.
• Polpulation density is given by the number of people per unit area.For example, if 500 people live in
a square region that is 2 miles on a side, the population density of the area is
500 people
people
= 125
.
4mi2
mi2
• Information density is used to describe how much memory can be stored by digital media. For example, each square inch surface of a DVD hold about 100 megabytes of information, so we say that a
.
DVD has an information density of 100 MB
in2
Definition of concentration
Concentration describes the amount of one substance mixed with another. There are many types of concentration, for example:
• The concentration of an air pollutant is often measured by the numbers of molecules of the pollutant
per million molecules of air. For example, if there are 20 molecules of carbon monoxide in each 1
million molecules of air, we state the carbon monoxide concentration as 20 parts per million (ppm).
• Blood alcohol content (BAC) describes the concentration of alcohol in a person’s body . It is usually
measured in units of grams of alcohol per 100 milliliter of blood. For example, in most states a
person is considered legally intoxicated f his/her blood alcohol content is at or above 0.08 gram of
0.08 g
)
alcohol per 100 milliliters of blood (
100 mL
Ex.16
A typical glass of red wine contains about 20 grams of alcohol. Consider a 150-pound woman, with
approximately 5 liters (5000 milliliters) of blood, who drinks two glasses of wine. If the alcohol were
immediately absorbed into her bloodstream, what would her blood alcohol content be? Explain why it is
fortunate that, in reality, the alcohol is not absorbed immediately.
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