Basic Math Refresher
A tutorial and assessment of basic math skills for students in PUBP704.
The purpose of this Basic Math Refresher is to review basic math concepts so that students
enrolled in PUBP704: Statistical Analysis for Public Policy will be better prepared for the course.
For some students, this Basic Math Refresher will be rudimentary. Other students may need to
spend more time reviewing the material and working through problems. Students may take the
test as many times as they need to. Each student must score better than 85% on each
assessment.
The Basic Math Refresher covers the following topics: I) Fractions and Decimals, II) Percents;
III) Exponents and Radicals; IV) Order of Arithmetic Operations; V) Basic Algebra and VI) Basic
Coordinate Geometry.
It is recommended that students print off the Basic Math Refresher Tutorial to review before
taking the assessment.
I. Fractions and Decimals
Fractions, decimals and percents are all numbers that represent a part of the whole. We often
need to convert from fractions to decimals or percents or from decimals to percents or fractions.
Example:
Fraction
Decimal
Percent
.20
20%
Converting a fraction to a decimal
Divide the numerator (the top number) by the denominator (the bottom number) in your
calculator.
Example:
2
7
0.2857142 or 0.286
Note that the final decimal in this example is expressed to three decimal places. When you
round a decimal, you round up if the number following is 5, 6, 7, 8, or 9 (as we did here) and you
round down if the number following is 0, 1, 2, 3, or 4 (as in
2 9 0.22222 0.22.)
Converting a decimal to a percent
Move the decimal point two places to the right. What you are doing is multiplying the decimal by
100 and attaching a “%” sign.
Example: 0.286 = 28.6%
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Converting a percent to a decimal
Move the decimal point two places to the left. What you are doing is dividing the percentage by
100 and removing the “%” sign.
Example: 10.8% = .108
Converting a percent to a fraction
Put the number over 100 and reduce (i.e. simplify the fraction so that the top number and
bottom number cannot be divided by the same number.)
Example: 56% =
Convert a decimal to a fraction
Determine how many numbers follow the decimal place (i.e. the number of “decimal places.) If
there is one decimal place, place the number over 10 and reduce. If there are two decimal
places, place the number over 100 and reduce. If there are three decimal places, place the
number over 1000 and reduce. And so on.
Example: 0.06 =
0.8775 =
Adding and subtracting fractions
To add or subtract fractions, each fraction must have the same denominator (i.e. the
same number in the bottom of the fraction.) If the denominators are different, you must
find the lowest common denominator (LCD.) The LCD is the smallest number that is
divisible by the denominator of each fraction. Once you determine the LCD, you
multiply the top and the bottom by the number that will make the denominator of the
fraction equal to the LCD. Then add the two numerators (i.e. numbers in the top) and
put that sum over the LCD.
Example:
=?
These fractions have different denominators (3 and 4.) The LCD is 12 because it
is the smallest number into which both 3 and 4 are evenly divided. To make the
denominator of the first fraction equal to 12, multiply the top and the bottom of
the fraction by 4. To make the denominator of the second fraction equal to 12,
multiply the top and the bottom of the fraction by 3:
So,
is the same as
2
Example:
=?
The LCD is 30.
So,
is the same as
Multiplying and dividing fractions
To multiply fractions, simply multiply across the top and across the bottom of the fractions.
Example:
?
Since 2 x 1 = 2 and 5 x 3 = 15, the answer is
To divide fractions, take the reciprocal of the second fraction (i.e. “flip” the second fraction) and
multiply it by the first.
?
Example:
Flip the second fraction and change the divide symbol ÷ to a multiplication symbol x.
3
2
?
1
5
Then multiply the two fractions
3
6
2
1
5
5
II. Percents
Find one number as a percent of another
Divide the first number by the second number. Move the decimal two places to the right and
attach a “%” sign.
Example: The number 15 is what percent of 48?
Divide 15 by 48: 15 ÷ 48 = .3125
Move the decimal over two places to the right and add % sign: 31.25%
So, 15 is 31.25% of 48 homes.
Find a given percent of a number
Convert the given percent to a decimal. Multiply the number by that decimal.
Example: What is 62.1% of 205?
Convert 62.1% into a decimal: 0.621
Multiply 205 by 0.621: 205 * 0.621 = 127.305
So, 127.305 is 62.1% of 205.
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Find the percent change
Calculate the difference between the first and second numbers. Divide that difference by the
first number. Convert the resulting decimal to a percent.
Example: Calculate the percent increase from 150 to 205.
Find the difference: 205 – 150 = 55
Divide difference by starting value: 55 ÷ 150 = 0.3666
Convert decimal to percentage: 36.7%
So, the percent increase is 36.7%.
III. Exponents and Radicals
Positive Exponents
Exponents (or powers) indicate that a number is multiplied by itself a certain number of times.
In other words, exponents are just a short hand way of writing out a number multiplied by itself.
For example, if we see 52 (said “five squared” and sometimes written as 5^2), it is the same
thing as 5 * 5. The number that is to be multiplied by itself is called the base. How many times
the number is to be multiplied by itself is called the exponent or the power.
Be very careful when a negative number is raised to a power. Be sure to keep track of the sign.
A negative number multiplied by itself an even number of times will give a positive number
result. A negative number multiplied by itself an odd number of times will give a negative
number.
For example, (-2)2 is equal to (-2)*(-2) = 4. (try it!) Note that the exponent is even and the
resulting answer is positive. However, (-2)3 is equal to (-2)*(-2)*(-2) = -8. (try it!) Note that the
exponent in this case is odd and the resulting answer is negative.
Some calculators have a “^” key that allows you to enter in an expression with an exponent
directly. In other words, you can key in 3^2= to get 9. See below.
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If your calculator does not have a “^” key, then you need to type in the multiplication
expression—i.e. for 32 you need to type 3 * 3.
Example: Find 7.8652
7.8652 = 7.865 * 7.865 = 61.858
Example: Find (-3.5)5
(-3.5)5 = (-3.5)*(-3.5)*(-3.5)*(-3.5)*(-3.5) = -525.22
Negative Exponents
Exponents can also be negative. Negative exponents are treated differently. We will not be
using negative exponents in the class.
Radicals
Radicals (or roots) are sort of the opposite of positive exponents. The radical is the symbol
placed over a number. The most common radical is the square root, √ . The square root is
the number you need to multiply by itself twice to get the number that is under the radical. For
example, the square root of 4, or √4, is equal to 2 (or -2) because when you multiply 2 (or -2) by
itself two times, you get 4.
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There are other higher order roots, such as the cube root, √ , or the fourth root, √ . In these
cases, the root is the number that is multiplied by itself 3 or 4 times to get the number under the
radical. For example, the cube root of 8, √8, is 2 because 2*2*2 or 23 equals 8.
Your calculator should have a square root button. On some calculators, you need to hit the
square root button first and then the number. On other calculators, you hit the buttons in
reverse order. Note: Your calculator will only show you the positive square root. Graphing
calculators will have options for calculating higher order roots.
Example:
Find √25
Since 5*5 = 25, 5 is a square root of 25. (-5 is also a square root of 25.)
IV. Order of Arithmetic Operations
When performing a series of arithmetic operations (i.e. addition, subtraction, division,
multiplication, exponents), you must perform those operations in a particular order. There is a
mnemonic to help you remember the order - PEMDAS:
P
E
M
D
A
S
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
If you have a series of operations, do what’s in parentheses first, then apply exponents, then do
any multiplication or division, and finally do any adding or subtracting.
Example: 4
3 2
?
2
Do what’s in parentheses first (find LCD): 2
So now we have 4
3
8
Now apply exponents: 2
So now we have 4 3
2 =?
8=?
6
Now do multiplication: 3
So now we have 4
8
Now do addition and subtraction (find LCD): 4
8 is the same as
So our answer is .
V. Basic Algebra
An algebraic expression is a combination of numbers and variables connected by some
mathematical operation, like addition, subtraction, multiplication or division. A variable is a letter
(we often use x and y) that represents or is a holding place for a number. In statistical analysis,
we will define variables and sometimes will use algebraic equations to relate two (or more)
variables.
Examples of algebraic expressions: 2x + y, 5d, 10-r
Evaluating an Algebraic Expression
To evaluate an algebraic expression, replace the variable with the given number.
Example: Evaluate 2x + 5 when x = 10.
Substitute 10 in for x: 2(10) + 5
Solve: 2*10 + 5 = 25
(Note: 2 * 10 is the same as 2(10).)
Equations
An equation is two expressions set equal to each other. For example, 2 + 2 = 4 is an equation.
Equations can include variables (such as x and y). So, for another example, 3x – 5 = 10 is an
equation (though it is only “true” for one value of x.)
The solution to an equation is the number, such that when you replace the variable, makes the
equation true (i.e. the left side equals the right side.)
Example: Determine if any of these values of x is a solution to the following equation:
3x – 5 = 10
a) x = 5
b) x = -5
Substitute in the value for x (x = 5) and solve: 3(5) – 5 = ?
15 – 5 = 10 √
So, x = 5 is a solution because the left hand side equals 10 and the right hand side
equals 10.
Now try x = -5: 3(-5) – 5 = ?
-15 – 5 = -20 χ
x = -5 is not a solution because the left hand side equals -20 and the right hand side
equals 10.
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General Solutions to a Linear Equation
Sometimes you will be given a linear equation and will need to solve for x (i.e. find a value for x
that makes the equation true.) To solve linear equations, remember the following properties:
-
If you add or subtract a value from both sides of an equation, then the equation is still true.
For example: If you have the equation 2x = 4 and add 5 to both sides (like this: 2x + 5 = 4 +
5) then the equation is still true.
If you multiply or divide on both sides of an equation by the same number (except 0), then
the equation remains true. For example: If you have the equation 2x = 4 and divide both
sides by 2 (like this: 2x ÷ 2 = 4 ÷ 2 or x = 2) then the equation will still be true.
You use these properties to solve a linear equation.
Example: Solve the following equation for x:
x + 5 = 12
Subtract 5 from both sides: x + 5 – 5 = 12 – 5
x=7
So, x = 7 is the solution to this linear equation (i.e. 7 is the value for x that makes the
equation true.)
Example: Solve the following equation for x: 3x – 4 = 8
Add 4 to both sides: 3x – 4 + 4 = 8 + 4
3x = 12
Divide both sides by 3 (to get x alone): 3x ÷ 3 = 12 ÷ 3
x=4
Remember: if you divide by a fraction (e.g. ), it is the same thing as multiplying by the
reciprocal or inverse of the fraction (e.g., or 2)
VI. Basic Coordinate Geometry
Points on a coordinate plane
A coordinate plane is often very useful for writing linear equations with two variables. The
coordinate plane is formed by a horizontal axis (x-axis) and a vertical axis (y-axis). The two
axes intersect at a point called the origin. Points are plotted on a coordinate plane using a set
of ordered pairs (x,y.) The first number in the ordered pair indicates how many spaces to move
along the x-axis and the second number in the ordered pair indicates how many spaces to move
along the y-axis.
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y-axis
origin
(1,2)
1
1
Example:
x-axis
Write the coordinates of the point shown on the coordinate plane.
1
1
Because the point is located 4 units right along the x-axis, the x-value of the point is 4. Because
the point is located 1 unit up along the y-axis, the y-value of the point is 1. Therefore, the x, y
coordinates of the point are (4,1.) Note: points along the x-axis to the left of the origin are
negative and points along the y-axis below the origin are negative.
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