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Basic Math Review #2

Solving “Average” Problems TotalAverage = # of things

To help you organize your information, draw an “Average Pie”:

Total

# of Average things

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 93

= Mathematically, the “Average Pie” looks like this:

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”.

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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 num- bers in the set is 24.

Column A Column B y 48

A. The quantity in Column A is greater. B. The quantity in Column B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Before removing data value y: Total 6 28 # of Average things Our “Average Pie” tells us that before the data value y was removed, “Total” = 6×28 = 168

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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 Total 168 120 6 28 5 24 # of Average # of Average things things 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 29 ←For an odd number of data values 8 12 15 17 20 23 ←For an even number of data values

Median = 15 + 17 = 162

Mode: The data value that occurs the most often For: 3, 5, 5, 8, 8, 8, 9, 9, 12, 19, 21 Mode = 8

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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% 14% 34% 34% 14% 2% 50 100 150 200 250 300 350

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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, ap- proximately 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).

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“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

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Trip to her brother’s house: Distance Distance ? 150 Time Rate → Time Rate 3 50 3 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.

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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: Pennies Nickels Total ratio 2 1 3 multiply by 8 8 8 real 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

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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 90-

degree 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.

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• 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.

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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.

12A bh=

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

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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 approxi- mately 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

2 2 2

2

2

6 936 81 117 117

cccc

+ =

+ =

=

= 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).

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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 = 2 rπ = dπ

• 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 = 2rπ 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.

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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 y-axis), and m (or “a”) is the slope of the line.

change in y vertical changeSlope = change in x horizontal changeriserun = =

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y P O R x

The line 8 17

y x= − + is graphed

on the rectangular coordinate axes. Column A Column B OR 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.

The slope of the line is: 8 vertical change7 horizontal change

OPOR

− = =

So the length of segment OP = |-8| = 8 and the length of segment OR = |7| = 7 So the answer is choice (B)

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Volume The volume of a three-dimensional figure is found by multiplying the area of the two-dimensional 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) = 2r hπ Diagonal inside a three dimensional rectangular box (longest distance between any two corners): 2 2 2 2a b c d+ + = where a, b, and c are the dimensions of the rectangular 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 2 2 2 2a b c d+ + =

2 2 2 2

2

2

2

2 2 2 2

3 4 59 16 2550

5025 2

5 2

dd

dd

dd

a b c d+ + =

+ + =

=

=

=

=

+ + =

The answer is choice (C).

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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 occurProbability 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:

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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 12

34

2 1× × =× ×

11 105× ×3 2×

2201

=×

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).