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Welcome to the School of Public Health Online Math Review
Kristin Anderson, who teaches Public Health 6320: Introduction to Epidemiology, thought it would be very useful for people considering enrolling for that class to review math concepts used in the class.

‘Wouldn’t it be great,’ she was saying, ‘if we could post a self-guided math tutorial for people to use before they enroll in PubH 6320 and other courses?’

This site is that tutorial. Since we began building it, we realized that students in other School of Public Health courses might find this useful as well.

You do not have to be registered for any course to use this site. Please tell us if you find it useful; don’t hesitate to suggest what else we might add.

You can do any of the lessons in any order you want. Doing all the lessons back to back should take half an hour to an hour. Click on the links to the left, follow the directions, and have fun brushing up on your math!

Section 1. Ratios & Proportions

  • Ratios
    ratio compares two numbers in order. Ratios are written with a colon, like this:

    4:3

    The order of numbers in a ratio is important. The ratio 4:3 expresses a different relationship than the ratio 3:4 does.

    The units of measurement that each of the two numbers expresses is important too. The two numbers in a ratio do not have to have the same units. One chicken to every pot (1 chicken:1 pot) is a ratio. Saying that one British Pound is worth about 1.5 US dollars (1 GBP:1.5 USD) is another ratio.

    In some cases, it can be convenient to express ratios all in the same units, for example, the ratio of 9 cups to 16 cups may be written in two ways: with a colon 9:16 or as a common fraction 9/16. The fraction 9/16 is another way of saying 9 divided by 16.

    The two numbers in a fraction are referred to so often that they have been given names. The top number in a fraction is the numerator and the bottom number is the denominator:

    numerator / denominator

    A fraction is another way of expressing one number divided by another. What is 24/56? You can take out a calculator and enter

    24 / 56 = 0.42857

    Incidentally, if you wanted to know what percent this fraction represented, simply multiply the number by 100, or move the decimal two places to the right.

  • Proportions
    proportion is similar to a ratio, except that it indicates a part of a whole, and so the numerator arises from the denominator. For instance, a researcher might say that, for every ten students in the residence hall, five were women. Five over ten (5 / 10) is a proportion. Proportions must have all of the numbers in the same units, and are frequently written as fractions.

    What happens when you want to write a proportion, but the numbers are given in different units? Suppose you were asked to write the proportion of 3 cups to 56 ounces (3 cups out of 56 ounces). You must write a proportion of either cups to cups (cups:cups) or ounces to ounces (ounces:ounces), so you will have to convert one of the numbers so that both numbers are expressed in the same unit of measurement. Let’s convert cups to ounces so we can express the ratio as ounces:ounces. Since there are 8 ounces in one cup, 3 cups are equal to 24 ounces:

    3 cups * 8 ounces/cup = 24 ounces

    Now the two numbers 3 cups and 56 ounces can be written as the following ratio:

    24 ounces to 56 ounces,

    24:56,

    or, now that the unit of measurement is the same, 24:56 can also be written as a proportion:

    24/56

    You may need to solve some problems involving ratios.

    If you divided 36 into two parts in the ratio of 1:2 and one part is a and the other is b, you can find the value of a and b:

    You know that

    a+b = 36

    And

    a/b = 1/2

    You can use these equations to solve for a and b, or you can use the following simple method:

    Find out how many units are in 1 part of the ratio. To do this, divide the total by the number of parts.

    Number of parts: 1 + 2 = 3.

    Number of units in each part: 36/3 = 12.

    Then, multiply the number of units in each part by the number of parts in each variable.

    a = 1 * 12 = 12

    b = 2 * 12 = 24

    As an aside . . .

    Percentages are so frequently used that we should spend a little time on them here. The powerful thing about percentages is that they all have the same denominator: 100. When two fractions have the same denominator, comparisions can be made very easily.

    0.21 = 21% = 21 / 100

Section 2: Algebraic Expressions

  • Section 2.1
    You may be asked to solve some simple algebraic equations, mostly with ratios and proportions, and to convert between various units. Recall from Section 1: Ratios and Proportions, that a fraction is one way to write a ratio.

    Let’s say that there is a ratio that is expressed as a fraction, and you would like to know what that same ratio would be with a different denominator. This will be useful for comparing different ratios with one another and for presenting ratios in an easily understood manner.

    Here is an example:

    We’ll start with a ratio of 2.1:10

    This can be written 2.1/10

    Now, for the moment, let’s assume that this represents the number of disease cases in a population of 10 people. It can be awkward to think of fractions of people. You would never walk into a room and see 2.1 people standing there talking about the weather. For this reason, in Epidemiology, it is conventional to avoid fractions of people. To do this, you just increase the population size under consideration (the denominator). This is very easy to do if you increase the denominator size by a factor of ten. In that case, all you need to do is move the decimal one place to the right in both the numerator and the denominator (multiply the fraction by 10/10).

    Like this:

    2.1/10 = 21/100

    You could keep going, if you wanted to,

    2.1/10 = 21/100 = 210/1000 = 2100/10,000 = 21,000/100,000

    Similarly, if you wanted to decrease the size of the denominator by a factor of ten for some reason, you would just move the decimal one place to the left in both the numerator and the denominator (multiply the fraction by 0.1/0.1).

    Like this:

    21/100 = 2.1/10 = 0.21/1 = 0.21

    What if the ratio does not have a factor of ten as its denominator? For example, let’s say we have a ratio of 8:20,000 and you need to know how many that is per 100,000.

    8/20,000 = x/100,000

    Here, we can’t just move the decimal around. Happily, there is a simple method we can use to convert this fraction, and solve for x. Some people call it the ‘flying x’ method, or ‘cross multiplication’, because you multiply across the ‘equals’ sign in an ‘x’ pattern. To do this, first multiply the numerator of one fraction by the denominator of the other. This number goes on one side of the ‘equals’ sign.

    Like this:

    8/20,000 = x/100,000

    8 * 100,000 = ?

    800,000 = ?

    Then, you do the same thing again with the remaining numerator and denominator. This number goes on the other side of the ‘equals’ sign.

    Like this:

    8/20,000 = x/100,000

    8 * 100,000 = 20,000x

    800,000 = 20,000x

    All that is left to do is solve for ‘x’. To do this in our example, we divide both sides by 20,000. So, we get

    8/20,000 = x/100,000

    8 * 100,000 = 20,000x

    800,000 = 20,000x

    x = 800,000/20,000

    x = 40

    so…

    8/20,000 = 40/100,000

  • Section 2.2
    Let’s look at two more examples:

    8/21,463 = x/100,000

    8 * 100,000 = 21,463x

    21,463x = 800,000

    x = 37.27

    so…

    8/21,463 = 37.27/100,000

    Remember, if this were something like case/population size, we would want to avoid fractions of people, so we might write it like this:

    3727/10,000,000

    Here’s the second example:

    0.53/73 = x/100,000

    73x = 0.53 * 100,000

    73x = 53,000

    x = 726

    so…

    0.53/73 = 726/100,000

    This method works just as well if you want a different denominator. Here’s an example:

    1/8 = x/60

    8x = 60

    x = 7.5

    so…

    1/8 = 7.5/60

  • Section 2.3
    Frequently, you will have to consider units of measurement (like centimeters, people, or bushels) in your calculations. Sometimes, it will be important to have all parts of an equation in the same units. To do this, you may have to convert between units. For example, you may need to convert part of your equation from ounces to grams.

    The easiest way to do this is to multiply the part of the equation that you need to convert by a fraction that is equal to one. Recall that a number divided by itself is equal to one, like 7/7, 21/21, or 50,000/50,000. Recall also that multiplying any part of an equation by one does not change its value at all.

    What does all that mean for us here?

    Just this:

    If there are 5 ships in a flotilla, then

    5 ships/1 flotilla = 1,and

    1 flotilla/5 ships = 1

    Here’s a real example:

    There are 0.035 ounces in a gram., so 1 g = 0.035 oz and…

    1 g/0.035 oz = 1 = 0.035 oz/1 g

    You can treat units like numbers. They can be multiplied or divided. This means they can also be cancelled. If you multiply two fractions, and they both contain the same units, except that one fraction has the unit in the denominator and the other in the numerator, they cancel each other out.

    Like this:

    oz/person * g/oz = g/person

    The ounces cancel each other out.

    So, let’s say each person in a group got 8 ounces of steak, and we need to know how many grams each person got.

    8 oz/1 person * 1g/0.035 oz = 8 g/0.035 people

    This is a very awkward fraction, so we change the size of the denominator:

    8/0.035 = x/1

    0.035x = 8

    x = 228.6

    so…each person got 228.6 g of steak.

  • Section 2.4
    Let’s try another example.

    60 inches/1 measuring stick = x cm/measuring stick

    We know from a table that 1 inch = 2.54 cm. So…

    60 in/1 stick * 2.54 cm/1 in = 152.4 cm/1 stick = 152.4 cm/stick

    This works just as well for denominators greater than 1, or for when you need to convert the denominator.

    Here’s an example with a denominator greater than 1:

    1 forest = 40 trees

    20 forest/45 glade = x trees/45 glade

    20 forest/45 glade * 40 trees/1 forest = 800 trees/45 glade, and if you want…

    800 trees/45 glade = 17.8 trees/glade or 178 trees/10 glade

    Remember how to get that?

    An example converting the units of the denominator:

    40 dots/1 inch = x dots/cm

    40 dots/1 inch * 1 inch/2.54 cm = 40 dots/2.54 cm = 15.75 dots/cm

    This method of unit conversion is very helpful when setting up equations. If you do it this way, you can check to see if your equation is set up correctly by looking to see if you end up with the units you want.

  • Section 2.5

    Here is an example using made-up money conversions:

    Let’s say 2 pounds = 1 dollar, and 1 pound = 150 yen, but you need to know how much 20 dollars is in yen.

    You could blithely start out with this equation:

    20 dollars * 1 dollar/2 pounds * 1 pound/150 yen = x

    Is this the right equation?

    Let’s check the units…

    dollars * dollars/pounds * pounds/yen = dollars2/yen

    We get ‘dollars squared over yen’, not plain old yen. So, it can’t be right. Let’s try again using our cancellation method.

    We want to go from dollars to yen via pounds. So, we start with dollars and convert to pounds.

    dollars * pounds/dollars = pounds

    Then we convert pounds to yen.

    pounds * yen/pounds = yen

    When we put this together, we get

    dollars * pounds/dollars * yen/pounds = yen

    This is what we want, so we just plug in the numbers and multiply it out.

    20 dollars * 2 pounds/1 dollars * 150 yen/1 pound = 6000 yen/1 = 6000 yen

Section 3: What is an exponent?

  • Section 3.1

    An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 23) means:

    2 x 2 x 2 = 8.

    23 is not the same as 2 x 3 = 6.

    Remember that a number raised to the power of 1 is itself. For example,

    a1 = a

    51 = 5.

    There are some special cases:

    1. a0 = 1

    When an exponent is zero, as in 60, the expression is always equal to 1.

    a0 = 1

    60 = 1

    14,3560 = 1

    2. a-m = 1 / am

    When an exponent is a negative number, the result is always a fraction. Fractions consist of a numerator over a denominator. In this instance, the numerator is always 1. To find the denominator, pretend that the negative exponent is positive, and raise the number to that power, like this:

    a-m = 1 / am

    6-3 = 1 / 63

    You can have a variable to a given power, such as a3, which would mean a x a x a. You can also have a number to a variable power, such as 2m, which would mean 2 multiplied by itself m times. We will deal with that in a little while.

  • Section 3.2

    First let’s look at how to work with variables to a given power, such as a3.

    There are five rules for working with exponents:

    1. am * an = a(m+n)

    2. (a * b)n = an * bn

    3. (am)n = a(m * n)

    4. am / an = a(m-n)

    5. (a/b)n = an / bn

    Let’s look at each of these in detail.

    1. am * an = a(m+n) says that when you take a number, a, multiplied by itself m times, and multiply that by the same number a multiplied by itself n times, it’s the same as taking that number a and raising it to a power equal to the sum of m + n.

    Here’s an example where

    a = 3
    m = 4
    n = 5

    am * an = a(m+n)

    34 * 35 = 3(4+5) = 39 = 19,683

    2. (a * b)n = an * bn says that when you multiply two numbers, and then multiply that product by itself n times, it’s the same as multiplying the first number by itself n times and multiplying that by the second number multiplied by itself n times.

    Let’s work out an example where

    a = 3
    b = 6
    n = 5

    (a * b)n = an * bn

    (3 * 6)5 = 35 * 65

    185 = 35 * 65 = 243 * 7,776 = 1,889,568

    3. (am)n = a(m * n) says that when you take a number, a , and multiply it by itself m times, then multiply that product by itself n times, it’s the same as multiplying the number a by itself m * n times.

    Let’s work out an example where

    a = 3
    m = 4
    n = 5

    (am)n = a(m * n)

    (34)5 = 3(4 * 5) = 320 = 3,486,784,401

    4. am / an = a(m-n) says that when you take a number, a, and multiply it by itself m times, then divide that product by a multiplied by itself n times, it’s the same as a multiplied by itself m-n times.

    Here’s an example where
    a = 3
    m = 4
    n = 5

    am / an = a(m-n)

    34 / 35 = 3(4-5) = 3-1 (Remember how to raise a number to a negative exponent.)

    34 / 35 = 1 / 31 = 1/3

    5. (a/b)n = an / bn says that when you divide a number, a by another number, b, and then multiply that quotient by itself n times, it is the same as multiplying the number by itself n times and then dividing that product by the number b multiplied by itself n times.

    Let’s work out an example where

    a = 3
    b = 6
    n = 5

    (a/b)n = an / bn

    (3/6)5 = 35 / 65

    Remember 3/6 can be reduced to 1/2. So we have:

    (1/2)5 = 243 / 7,776 = 0.03125

    Understanding exponents will prepare you to use logarithms.

Section 4: What is a Logarithm?

  • Section 4.1
    SECTION 4. WHAT IS A LOGARITHM?

    A logarithm is the power to which a number must be raised in order to get some other number (see Section 3)
    of this Math Review for more about exponents). For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:

    log 100 = 2

    because


    102
    = 100

    This is an example of a base-ten logarithm. We call it a base ten logarithm because ten is the number that is raised to a power. The base unit is the number being raised to a power. There are logarithms using different base units. If you wanted, you could use two as a base unit. For instance, the base two logarithm of eight is three, because two raised to the power of three equals eight:


    log2
    8 = 3

    because


    23
    = 8

    In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written

    log

    and a base ten logarithmic equation is usually written in the form:

    log a = r

    A natural logarithm is written

    ln

    and a natural logarithmic equation is usually written in the form:

    ln a = r

    So, when you see log by itself, it means base ten log. When you see ln, it means natural logarithm (we’ll define natural logarithms below). In this course only base ten and natural logarithms will be used.

  • Section 4.2

    Base Ten Logarithms

    We saw above that base ten logarithms are expressions in which the number being raised to a power is ten. The base ten log of 1000 is three:

    log 1000 = 3

    103 = 1000

    So far, we’ve worked with expressions that have whole numbers as solutions. Here’s one that does not. What is the log of 4?

    log 4 = x

    log 4 ≈ 0.602

    because

    10 0.602 ≈ 4

    Natural Logarithms

    Logarithms with a base of ‘e’ are called natural logarithms. What is ‘e’?

    ‘e’ is a very special number approximately equal to 2.718. ‘e’ is a little bit like pi in that it is the result of an equation and it’s a big long number that never ends. For those of you who have had calculus, you might remember that ex is special because its derivative is itself. If you want to know more about ‘e’, check any trigonometry text, such as page 234 of Ruud, W.L. and T.L. Shell. Prelude to Calculus, 2nd ed. 1993. Boston: PWS Publishing Company.

    Most scientific calculators have an ‘e’ button and an ‘ln’ button, so you don’t need to memorize the value of ‘e’.

  • Section 4.3

    Logarithmic Rules

    Just as exponents have some basic rules that make them easier to manipulate (see Section 3: Exponents), so do logarithms. These rules apply to all logarithms, including base 10 logarithms and natural logarithms. For simplicity’s sake, base ten logs are used in most of these rules:

    1. br = a is the equivalent to logb a=r (This is the definition of a logarithm.)

    2. log 0 is undefined.

    3. log 1 = 0

    4. log (P*Q) = log P + log Q

    5. log (P/Q) = log P – log Q

    6. log (Pt) = t *log P

    7. 10(log a) = a (in the case of natural logarithms, e(ln a) = a)

    8. log (10r) = r (in the case of natural logarithms, ln er = r)

    9. log (1/a) = -log a

    Let’s take a closer look at each of these rules:

    1. br = a is the equivalent of logb a=r. We’ve already looked at how this works, but here’s another example:

    log 14 ≈ 1.146

    is the equivalent of

    10 1.146 ≈ 14

    2. log 0 is undefined. It’s not a real number, because you can never get zero by raising anything to the power of anything else. You can never reach zero, you can only approach it using an infinitely large and negative power.

    3. log 1 = 0 means that the logarithm of 1 is always zero, no matter what the base of the logarithm is. This is because any number raised to 0 equals 1. Therefore, ln 1 = 0 also.

    All the rest of the logarithmic rules are useful for solving complex equations, or equations with unknowns.

  • Section 4.4

    4. log (P*Q) = log P + log Q means that if you take the logarithm of two factors, it is the same as taking the logarithm of each factor, and adding them together. For example:

    log 6 =

    log (2 * 3) =

    log 2 + log 3 ≈

    0.301 + 0.477 = 0.778

    If you were using natural logarithms, it would look like this:

    ln 6 =

    ln (2 * 3) =

    ln 2 + ln 3 ≈

    0.693 + 1.099 = 1.792

    (Note that the numerical value of the natural logarithm is different from that of the base ten logarithm. That’s because, in the second example, 1.792 is the power to which ‘e’ must be raised to get 6, whereas, in the first example, 0.778 is the power to which 10 must be raised to get 6.)

    If you have a variable as one of your factors, it would look like this:

    log 2y = log 2 + log y

    Let’s say log 2y = 36 and solve for y:

    log 2y = 36

    log 2 + log y = 36

    log y = 36 – log 2

    log y = 36 – 0.301

    log y = 35.699

    y = 10 35.699

    which is a really big number.

    5. log (P/Q) = log P – log Q means that if you take a logarithm of one number divided by another, it is the same as taking each logarithm separately, and then subtracting the logarithm of the denominator from the logarithm of the numerator.

    For example:

    log (3 / 2) =

    log 3 – log 2 ≈

    0.477 – 0.301 ≈ 0.176

    If you were using natural logarithms, it would look like this:

    ln (3 / 2) =

    ln 3 – ln 2 ≈

    1.099 – 0.693 ≈ 0.406

    If you have a variable as one of your factors, it would look like this:

    log (y/2) = log y – log 2

    Let’s say log (y/2) = 36 and solve for y:

    log (y/2) = 36

    log y – log 2 = 36

    log y = 36 + log 2

    log y = 36 + 0.301

    log y = 36.301

    y = 10 36.301

    which is an even bigger number.

    6. log (Pt) = t * log P means that the logarithm of a number raised to some power, it is the same as multiplying the logarithm of that number by the value of the power.

    For example:

    log (32) = 2 * log 3

    2 * 0.477 = 0.954

    It looks the same when you use natural logarithms, however, as in example three the numerical value will be different.

    ln (32) = 2 * ln 3

    2 * 1.099 = 2.198

  • Section 4.5

    7. 10(log a) = a (or, in the case of natural logarithms, e(ln a) = a). Logarithms and exponents reverse each other.

    For example:

    10(log 3) = 3

    10(log 8) = 8

    e(ln 3) = 3

    e(ln 8) = 8

    If you raise a number to the power of a logarithm that has that number as its base, it is equal to the number that you used in the logarithm.

    8. log (10r) = r (in the case of natural logarithms, ln er = r)Because logarithms and exponents reverse each other, this rule is similar to rule number seven.

    For example:

    log (102) = 2

    log (103) = 3

    ln (e2) = 2

    ln (e4) = 4

    Any logarithm of its base number raised to some exponent is equal to that exponent.

    9. log (1/a) = -log a means that the logarithm of 1 divided by some number is equal to the negative logarithm of that number. (This is the exactly the opposite of the rule governing exponents where a number raised to a negative number is equal to 1 divided by that number raised to that power.)

    For example:

    log (1/2) = – log 2 = -0.301

    log (1/3) = – log 3 = -0.477

    ln (1/2) = -ln 2 = -0.693

    ln (1/3) = -ln 3 = -1.099

Glossary and Other Useful Links

denominator
the bottom number in a fraction; i.e. in the fraction 19/32, 32 is the denominator.

exponent
the number of times that a given number, such as n, is multiplied by itself; i.e. n to the third power, or n3, means n x n x n

fraction
when the units of a ratio are the same, a ratio may be written as a fraction. Two apples to three oranges (2:3) could not be written as a fraction. The ratio of two apples to three apples may be written in two ways: with a colon 2:3 or as a common fraction 2/3. The fraction 2/3 may also be interpreted as 2 divided by 3. A common fraction has two parts: the numerator and the denominator. In the fraction 2/3, 2 is the numerator and 3 is the denominator.

logarithm
a logarithm is the “power” to which a number must be raised in order to “get” some other number. The two most common logarithms are base 10 logarithms (the base unit is the number being “raised to a power”) and natural logarithms.

logarithm, base ten
a base ten logarithm is the exponent of 10 required for the expression to equal some other number. It is usually written in this form: log a=r, which means that a=10^r.


logarithm, natural

a natural logarithm is the exponent of e ~ 2.71828183 required for the expression to equal some other number. e is a transcendential number, which means it does not end. It is usually written in this form: ln a=r, which means that a=e^r.

numerator
the top number in a fraction; i.e.in the fraction 19/32, 19 is the numerator.

proportion
a proportion is a statement that expresses two equal ratios. A ratio is different from a proportion.

ratio
a ratio is a relationship involving a comparison of two numbers in a definite order, such as “two apples to three oranges.” Ratios are usually written in this form: 2:3. When the units of both numbers in a ratio are the same, the ratio may be written as a fraction.

Other Useful Links

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