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Circles
Perfect Squares
Odds and Evens
Least Common Multiples
Common Percent Equivalencies
Distance Problems (Distance, Rate & Time)
Triangles (30-60-90, 45-45-90, 3-4-5, 5-12-13)
Prime Numbers
Work Problems
Percent Problems
Number Properties
Probability
Inequalities
Averages
Divisibility Rules
Ratios
Exponents
Parallelograms
1. Circles
Circumference = 2 * Pi * R where R = Radius or Pi * D where D = Diameter
Diameter = 2 * R
Radius = D/2
Area = Pi * R2
The area of a circle is 36. What is the diameter of the circle?
36 = Pi * R2 so R = 6 / Square root of Pi so D = (12 * Square root of Pi)/Pi
The radius of a circle is 10. What is the circumference of the circle?
Circumference = 2 * Pi * R so circumference = 20 * Pi
What formula do you use to find the ratio of the arcs, sectors, etc. of a circle?
x/360 = arc length/circumference = area of sector/area where x is the degree measure of the angle
2. Perfect Squares
List all the perfect squares from 0 to 100
0 (don't forget 0!), 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
3. Odds and Evens
Odd + Odd = Even
Odd * Even = Even
Even - Even = Even
Even/Odd = Even (or decimal)
Even * Even = Even
Odd + Even = Odd
Odd - Odd = Even
Odd/Odd = Odd (or decimal)
4. Least Common Multiples
What is the least common multiple of 15 and 24?
15 = 3 * 5 and 24 = 2 * 2 * 2 * 3, Eliminate one of each of the common factors (3) and multiply the rest of the prime factors, LCM = 120
What is the least common multiple of 12 and 30?
12 = 2 * 2 * 3 and 30 = 2 * 3 * 5, Follow the same procedure as above, eliminate 2 & 3, LCM = 60
5. Common Percent Equivalencies
Decimals to Fractions
0.4 = 2/5
0.875 = 7/8
0.167 = 1/6
0.85 = 17/20
0.375 = 3/8
Fractions to Decimals
5/8 = 0.625
2/40 = 0.05
5/6 = 0.833
3/5 = 0.6
2/16 = 0.125
6. Distance Problems (Distance, Rate & Time)
Distance = Rate * Time
Rate = Distance/Time
Time = Distance/Rate
7. Triangles (30-60-90, 45-45-90, 3-4-5, 5-12-13)
Draw a 30-60-90 triangle and label its sides and angles
The side opposite the 90 will be 2X, the side opposite the 60 will be X * the square root of 3, the side opposite the 30 will be X
Draw a 45-45-90 triangle and label its sides and angles
The sides opposite the 45 will be Xs, the side opposite the 90 will be X * the square root of 2
(Note: All of the following are right triangles)
If the leg opposite the 30-degree angle is 8, what is the hypotenuse of the triangle?
16
If one of the legs is 6 and the hypotenuse is 10, what is the other leg?
8 (3-4-5 triangle)
If the hypotenuse is 20, what is the length of the leg opposite the 45-degree angle?
10 * the square root of 2
If one of the legs is 10 and the other leg is 24, what is the length of the hypotenuse?
26 (5-12-13 triangle)
8. Prime Numbers
List all the prime numbers less than 22
2 (1 is not a prime number!), 3, 5, 7, 11, 13, 17, 19
9. Work Problems
What is the formula to use for work problems?
(X*Y)/(X+Y) = Combined time of two workers (C) where X equals the time of one worker working alone and Y equals the time of the other worker working alone
If Sue can do a job in 3 hours and Bob can do the same job in 6 hours, how long will it take them working together at their respective rates?
2 hours
If Tom can do a job in 10 hours and Tom and Mary together can do the job in 6 hours working at their respective rates, how long will it take Mary to do the job by herself?
10Y/(10 + Y) = 6 so Y = 15 hours
10. Percent Problems
What number should you always pick for percent problems?
100
What formula will give you the percent increase/decrease?
[(New # - Old #)/Old #] * 100
What formula will give you what percent the new quantity is of the old quantity?
(New #/Old #) * 100
11. Number Properties
What numbers get bigger when you square them?
Negative numbers and numbers greater than 1
What numbers stay the same when you square them?
0 and 1
What numbers get smaller when you square them?
Numbers between 0 and 1
What numbers get bigger when you cube them?
Numbers between -1 and 0 and numbers greater than 1
What numbers stay the same when you cube them?
-1, 0 and 1
What numbers get smaller when you cube them?
Numbers less than -1 and numbers between 0 and 1
12. Probability
What formula will give you probability of an event occuring?
Favorable events divided by total number of events
What method do you use to find the total number of events that could occur?
Multiply each individual event by the number of different things that could happen for that event
What is a popular shortcut to find the probability of "success"?
1 - Probablity of failure
13. Inequalities
An inequality can be treated just like an equal sign with one exception. What is it?
When you are dividing or multiplying by a negative number, you must "flip" the sign
If 6x < -18, what does x have to be?
x < -3
If -10x + 120 > -2(40 + 15x), what does x have to be?
x > -10
14. Averages
What is the formula used to find out the average of a group of numbers?
(Sum of numbers/# of numbers)
If a group contains five numbers and the average of the numbers is 17, what is the sum of the numbers?
85
If the average of a group of numbers is 24 and the sum of the numbers is 192, how many numbers are there?
8
15. Divisibility Rules
How do you know if a number is divisible by 3?
Sum of the digits is divisible by 3
How do you know if a number is divisible by 4?
Last two digits are divisible by 4
How do you know if a number is divisible by 5?
Last digit is either a 0 or a 5
How do you know if a number is divisible by 6?
Number is even and divisible by 3
How do you know if a number is divisible by 7?
Number divides evenly by 7 (there is no shortcut)
How do you know if a number is divisible by 9?
Sum of the digits is divisible by 9
Is 47 a prime number?
Yes
Is 117 a prime number?
No (divisible by 3, 9, etc.)
Is 981,495 a prime number?
No (divisible by 3, 5, etc.)
16. Ratios
If three things are in a ratio of 5:9:11, what does the total number of things have to be a multiple of?
25 (the sum of the numbers)
If the number of things represented by the 5 doubles, can you represent the new ratio as 10:9:11?
Yes, multiplication and division are okay
If you add 6 things to the the number of things represented by the 9, can you represent the new ratio as 5:15:11?
No, not unless you know the absolute number
If the number of things represented by the 9 is reduced by one-third, can you represent the new ratio as 5:7:11?
No, but you could represent it by 5:6:11
17. Exponents
(x2)*(x3) = x5 (Add the exponents)
(x8)/(x2) = x6 (Subtract the exponents)
(x3)2 = x6 (Multiply the exponents)
x-6 = 1/x6
18. Parallelograms
What is the formula for the area of parallelogram?
Base * Height (Not Side * Side)
What things do you know to be true about a parallelogram?
Opposite angles are equal, opposite sides are equal, sides are parallel, interior angles equal 360
What do you know about the area of a parallelogram versus the product of its sides?
The area will always be less than the product of the sides (because the height of of the parallelogram will always be less than the length of the sides).
Is a square always a parallelogram?
Yes
Is a parallelogram always a square?
No
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