Download Basic Mathematics Review (Part 2)

Basic Math Review #2
Solving “Average” Problems
Average =
Total
# of things
To help you organize your information, draw an “Average Pie”:
Total
# of
things
Average
This is how it works:
“Total” is the sum of the numbers being averaged.
“# of things” is the number of different elements that you are averaging.
“Average” is the average (also called the Mean).
Let’s suppose you want to find the average of 5, 8, and 14.
5 + 8 + 14 = 27
27
Mathematically, the “Average Pie” looks like this:
=9
3
27
÷
÷
3
×
9
Think of the horizontal bar as the Division Bar.
If you divide the “Total” by “# of things”, you get the “Average”.
If you divide the “Total” by “Average”, you get the “# of things”.
If you have the “# of things” and the “Average”, multiply them together to get “Total”.
Page 1 of 18 Benefits of using the “Average Pie”:
• It’s an easy way to organize your information.
• One diagram will enable you to solve all average problems – you don’t have to
rewrite formulas depending on which part of the average equation you’re looking
for.
• It makes it clear that if you have two of the three pieces, you can always find the
third. This makes it easier to figure out how to approach the problem. For
example, you’re given the value for “# of things” and it wants to know the average,
the “Average Pie” shows you that the key to unlocking that problem is finding the
total.
Let’s try this out with an example:
The average (arithmetic mean) of a set of 6
numbers is 28. If a certain number, y, is removed
from the set, the average of the remaining numbers in the set is 24.
Column A
y
A.
B.
C.
D.
Column B
48
The quantity in Column A is greater.
The quantity in Column B is greater.
The two quantities are equal.
The relationship cannot be determined
from the information given.
Before removing data value y:
Total
6
# of
things
28
Average
Our “Average Pie” tells us that before the data value y
was removed, “Total” = 6 × 28 = 168
Page 2 of 18 After removing data value y:
Total
6–1=5
24
The “# of things” will be one less after y is removed.
We are given that “Average” is 24 when y is removed.
Our “Average pie” tells us we need to calculate the “Total”:
“Total” = 5 × 24 = 120
Before:
After:
Total
168
Total
120
6
28
# of
Average
things
5
# of
things
24
Average
y = TotalBefore – TotalAfter
=
168
120
y = 48
Now that we know that y = 48, we know the answer is (C) The two quantities are equal.
===================================
Median: The middle value in a sorted list of numbers:
5
9
12
16
19
21
8
12
15
17
20
23
29
← For an odd number of data values
← For an even number of data values
15 + 17
= 16
2
Mode: The data value that occurs the most often
For: 3, 5, 5, 8, 8, 8, 9, 9, 12, 19, 21
Mode = 8
Median =
Page 3 of 18 Range = Largest Data Value - Smallest Data Value
For: 3, 5, 5, 8, 8, 8, 9, 9, 12, 19, 21 Range = 21 – 3 = 18
Standard Deviation: Tells you how much variability there is in the data.
A large standard deviation tells you the data values are spread far away from the Mean.
A small standard deviation tells you the data values are clustered closely around the
Mean.
The Bell Curve (or Normal Distribution):
The Mean is at the line down the center of the curve.
The Standard Deviation is the length of each of the “tick marks”. This length is the
typical (or “standard”) distance between (or “deviation”) a data value and the Mean.
The percentages represent the portion of the data that falls between each line. These
percentages are valid for any question involving a normal distribution.
For example: If Mean = 200 and Standard Deviation = 50, we know:
2%
50
100
14%
150
34%
34%
200
14%
250
2%
300
350
Page 4 of 18 What percentage of the data values are between 200 and 250? 34%
What percentage of the data values are less than 100? 2%
What percentage of the data values are more than 250? 14% + 2% = 16%
The fourth grade at School X is made
up of 300 students who have a total
weight of 21,600 pounds. If the
weight of these fourth graders has a
normal distribution and the standard
deviation is equal to 12 pounds, approximately what percentage of the
fourth graders weights are more than
84 pounds?
A. 12%
B. 16%
C. 36%
D. 48%
E. 60%
First we need to calculate the Mean = 21,600 / 300 = 72 pounds.
Now let’s make a sketch of our normal curve:
2%
14%
34%
34%
14%
2%
36
48
60
72
84
96
108
From our sketch we see that the proportion that is larger than 84 pounds is 14% + 2% =
16%. So our answer is (B).
Page 5 of 18 “Rate” Problems
Rate problems are similar to “Average” Problems. We used “Average Pie” diagrams to
help us with “Average” problems. Now to help us with “Rate” Problems we will use the
“Rate Pie” diagram:
÷
Distance
or
Amount
Time
÷
× Rate
This “Rate Pie” works exactly the way the “Average Pie” does. If you divide the
Distance or Amount by the Rate, you get the Time. If you divide the Distance or Amount
by the Time, you get the Rate. If you multiply the Rate by the Time, you get the Distance
or Amount.
It takes Carla three hours to
drive to her brother’s house at
an average speed of 50 miles
per hour. If she takes the same
route home, but her average
speed is 60 miles per hour, how
long does it take her to get
home?
A. 2 hours
B. 2 hours and 14 minutes
C. 2 hours and 30 minutes
D. 2 hours and 45 minutes
E. 3 hours
Page 6 of 18 Trip to her brother’s house:
Distance
?
Time
3
Rate
50
Distance
150
→
Time
3
Rate
50
Trip from her brother’s house:
Distance
150
Time
?
Rate
60
Our chart tells us that the Time = 150 ÷ 60 = 2.5 hours = 2 hours and 30 minutes.
The answer is choice (C).
Ratios and Proportions
Ratios, proportions, fractions, percentages, and decimals are just different ways of
representing division.
You may see ratios expressed in these ways:
x:y
the ratio of x to y
x is to y
Anything you can do to a fraction you can also do to a ratio:
Cross-multiply,
Find common denominators,
reduce, etc.
Page 7 of 18 Example:
You have 24 coins in your pocket and the ratio of pennies to nickels is 2 : 1. How many
pennies and nickels are there?
“Count the Parts” -- The ratio 2 : 1 contains three parts: there are 2 pennies for every 1
nickel, making a total of 3 parts.
To find out how many of our 24 coins are pennies, we divide 24 by the number of parts
(which is 3), and then multiply the result by each part of the ratio.
24 ÷ 3 = 8 so each of the 3 parts in our ratio consists of 8 coins. Two of the parts are
pennies, and at 8 coins per part, that makes 16 pennies. One of the parts is nickels, so
that makes 8 nickels.
If you want a more “systematic” way to approach this problem, you could use a “Ratio
Box”:
Pennies
Nickels
Total
ratio
2
1
3
multiply by
real
24
The row for “real” represent what we really have, not in the conceptual world of ratios,
but in real life.
In the “Total” column, how do we get from the “3” to the “24”? We multiply by 8. So
our “multiply by” number is 8. So let’s fill that row in:
ratio
multiply by
real
Pennies
2
8
Nickels
1
8
Total
3
8
24
Now just finish filling in the box by multiplying everything else out:
Pennies
Nickels
Total
ratio
2
1
3
multiply by
8
8
8
real
16
8
24
In the “real” world we have 16 pennies and 8 nickels
Page 8 of 18 Geometry
Remember that you cannot necessarily trust the diagrams you are given. Always go by
what you read, not what you see. For example, you have to be told that two lines are
parallel, you can’t assume that they are just because they look like they are. And if they
don’t include a drawing with a geometry problem, it usually means that the drawing
would have made the answer obvious. So you should just draw it and see for yourself if
it becomes obvious as a result.
You need to know the following:
• A line is a 180-degree angle (it can be thought of as a perfectly flat angle).
• When two lines intersect, four angles are formed: the sum of these angles is 360
degrees.
a
x
y
b
• When two lines are perpendicular to each other, their intersection forms four 90degree angles. This is the symbol to indicate a perpendicular angle: ⊥
• Ninety-degree angles are also called right angles. A right angle in a diagram is
identified by a little box at the intersection of the angle’s arms.
• The three angles inside a triangle add up to 180 degrees. This applies to every
triangle, no matter what it looks like.
• An equilateral triangle has all three sides that are equal in length, and because of
this, the angles are equal too. So each angle would have to be 180 ÷ 3 = 60
degrees.
Page 9 of 18 • An isosceles triangle has two of the three sides that are equal in length, so two of
the three angles are equal. So if you know the degree measure of any angle in an
isosceles triangle, you can figure out what the measures of the other two are.
• A right triangle has a right angle (a 90-degree angle). The longest side of a right
triangle (the side opposite the 90-degree angles) is called the hypotenuse. A right
triangle will always have a little box in the 90-degree corner.
Know these relationships between the angles and sides of a triangle:
• The longest side is opposite the largest interior angle. The shortest side is opposite
the smallest interior angle. Equal sides are opposite equal angles.
• The length of any one side of a triangle must be less than the sum of the other two
sides and greater than the difference between the other two sides. So take any two
sides of a triangle, add them together, then subtract one from the other, and the
third side must lie between those two numbers.
Page 10 of 18 The area of any triangle is equal to the height (or “altitude”) multiplied by the base,
divided by 2. The height (or “altitude”) is the perpendicular line drawn from the point of
the triangle to it base.
A = 1 bh
2
6
The Pythagorean Theorem applies ONLY to right triangles. The square of the length of
the hypotenuse (longest side) is equal to the sum of the square of the lengths of the other
two sides.
a
c
a2 + b2 = c2
b
Page 11 of 18 The three angles inside any triangle add up to 180 degrees.
The four angles inside any four-sided figure add up to 360 degrees.
Q
6 miles
P
R
9 miles
In the figure above, driving
directly from point Q to point
R, rather than from point Q to
point P and then from point P
to point R, would save approximately how many miles?
A. 0
B. 1
C. 2
D. 3
E. 4
We have a right triangle, so let’s use Pythagorean’s Theorem to find the length of QR:
Let a = 6, b = 9, and c = length of QR
62 + 92 = c 2
36 + 81 = c 2
117 = c 2
117 = c
10.82 ≈ c
Driving from Q to P (6 miles) and then from P to R (9 miles), we will have traveled 15
miles. Driving directly from Q to R is approximately 11 miles.
That’s a difference of 15 – 11 = 4 miles. So the answer is choice (E).
Page 12 of 18 Circles:
• A circle contains 360 degrees.
• A radius is a line that extends from the center of a circle to the edge of that circle.
Plural is radii.
• A diameter is a line that extends from one edge of a circle to the other edge and
goes through the circle’s center. The diameter is twice as long and the radius.
• The circumference of a circle is the distance around the outside of the circle. It is
equal to 2 times π times the radius, or π times the diameter.
Circumference = 2π r = π d
• For π just remember that it is a little bigger than 3, this will be a close enough
approximation.
• The area of a circle is equal to π times the square of its radius.
Area =
π r2
Vertical angles are the angles that are across from each other when two lines intersect.
They are always equal.
angle a = angle b
a y
angle x = angle y
x b
When two parallel lines are cut by a third, only two types of angles are formed, big
angles and small angles. All of the big angles are equal, and all of the small angles are
equal. The sum of any big and any small angles is always 180 degrees.
Page 13 of 18 The perimeter of a rectangle is the sum of the lengths of its four sides.
The area of a rectangle is equal to its length multiplied by its width.
A square has four equal sides, so the perimeter is 4 times the length of any side. The area
is the length of any side times itself (i.e. the square of any side).
The Coordinate System
The x-axis is the horizontal line, and the vertical line is the y-axis. The four areas formed
by the intersection of these axes are called quadrants. The origin is the point where the
axes intersect.
To find any point in the coordinate system, you first give the horizontal value then the
vertical value: (x,y)
(2,4)
(-6,1)
(-5,-5)
The equation of a line: y = mx + b or y = ax + b
The x and y are the points on the line, b is the y-intercept (where the line crosses the yaxis), and m (or “a”) is the slope of the line.
change in y
vertical change
Slope = rise =
=
run change in x horizontal change
Page 14 of 18 y
P
O
R
x
8
The line y = − x + 1 is graphed
7
on the rectangular coordinate axes.
Column A
OR
Column B
OP
The quantity in column A is greater.
The quantity in column B is greater.
The two quantities are equal.
The relationship cannot be determined
from the information given.
−8
vertical change
OP
=
=
7 horizontal change OR
So the length of segment OP = |-8| = 8
and the length of segment OR = |7| = 7
So the answer is choice (B)
The slope of the line is:
Page 15 of 18 Volume
The volume of a three-dimensional figure is found by multiplying the area of the twodimensional figure by the height (or depth).
Rectangular solid: Volume = (Area of a rectangle) × (depth) = length × width × depth
Circular cylinder: Volume = (Area of a circle) × (height) =
π r 2h
Diagonal inside a three dimensional rectangular box (longest distance between any two
where a, b, and c are the dimensions of the rectangular
corners): a2 + b2 + c2 = d 2
box, and d is the length of the diagonal.
Surface Area
For a rectangular box the surface area is the sum of the areas of all of its sides.
What is the length of the
longest distance between any
two corners in a rectangular box
with dimensions 3 inches by
4 inches by 5 inches?
5
12
5 2
12 2
50
This is a job for our formula
a 2 + b2 + c 2 = d 2
a 2 + b2 + c 2 = d 2
32 + 42 + 52 = d 2
9 + 16 + 25 = d 2
50 = d 2
50 = d 2
25 2 = d
5 2=d
The answer is choice (C).
Page 16 of 18 Probability
0 ≤ Probability ≤ 1
Probability = 0
The event is impossible. It will never happen
Probability = 1
The event is certain to happen.
0 < Probability < 1
It’s possible for the event to happen, but it’s uncertain.
If all of the possible outcomes are equally likely to occur, then:
Number of ways the Event can occur
Probability of an Event =
Number of ways the entire Experiment can occur
When we want to find the probability of a series of events in a row, we multiply the
probabilities of the individual events. Note: you are finding the probability of one event
occurring AND another event occurring. They are BOTH happening.
Probability of events A AND B occurring = (Probability of A) × (Probability of B)
When we want to find the probability of either one event occurring OR another event
occurring, we add the probabilities of the individual events.
Probability of events A OR B occurring = Probability of A + Probability of B
Another important thing to know about probabilities:
(Probability of an event happening) + (Probability of the even NOT happening) = 1
Factorials
5! = 5 × 4 × 3 × 2 × 1 = 120
3! = 3 × 2 × 1 = 6
1! = 1
0! = 1
Permutations
The number of arrangements of things when you have one less to choose from each time,
and order IS important.
To solve, figure out how many “slots” you have, write down the number of options for
each slot, and multiply them.
For example:
Page 17 of 18 How many ways can we choose 3 students from a class of 12 students, where the order
they are selected in IS important (because the first person selected recites poetry, the
second selected does a math problem, and the third has to do an interpretive dance).
Since we are selecting three things, make three “slots”: _____ _____ _____
In the first slot, ask yourself, “How many choices do I have?” Answer: 12
_12__ _____
_____
Go to the next slot, and ask the same question again. Answer: 11
_12__ _11__
_____
Ask the same question at the next slot. Answer: 10
_12__ _11__ __10__
Multiply these numbers together: 12 × 11 × 10 = 1320
There are 1320 ways to choose 3 things from 12 things where order is important (and you
have one less to choose from after each selection).
Combinations
The number of arrangements of things when you have one less to choose from each time,
and order is NOT important.
To solve, figure out how many slots you have, fill in the slots as you would a
permutation, and then divide by the factorial of the number of slots. The denominator of
this fraction will ALWAYS cancel out completely, so you can cancel first before you
multiply.
For example:
How many ways can we choose 3 students from a class of 12 students, where the order
they are selected in is NOT important?
You begin by doing the same steps that you would if it were a Permutation:
Put in 3 “slots” (because 3 things are being selected).
Put the number of choices you have into each slot.
But for a Combination you DIVIDE this result by the FACTORIAL of the number of
things being selected (this is also the number of “slots” you had).
12 ×11×10 4 12 ×11× 510
= 220
=
3 × 2 ×1
3 × 2 ×1
There are 220 ways to choose 3 things from 12 things where order is NOT important (and
you have one less to choose from after each selection).
Page 18 of 18