Term
| The price goes up from $80 to $100. What is the percent increase? |
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Definition
Solution: ($20/$80) X 100% = (0.25 X 100%) = 25%
Concept: Percent Increase/ Percent Decrease
% Increase = (Amt Increase / Initial) X 100%
% Decrease = (Amt Decrease / Initial) X 100% |
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Term
| If m is odd and n is even, will the product (m X n) be odd or even? |
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Definition
Solution: Assume m is 3 and n is 2. Then m X n = 3 X 2 = 6 which is EVEN. Therefore, the m X n is EVEN for any values of m and n.
Strategy: Substitute any odd number for m and any even number for n, then solve. The result (odd vs. even) will be consistent no matter which values you choose for m and n. |
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Term
| What is 12 percent of 25? |
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Definition
Solution: 25 x 0.12 = 3
Strategy: Part = Percent X Whole |
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Term
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Definition
Solution:
45 = (9)(x) <-- Note: in decimal form 100% = 1.0
45/9 = x
x = 5.0
5.0 x 100% = 500%
Strategy:
Part = Percent x Whole |
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Term
| 15 is 3/5 percent of what number? |
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Definition
Solution:
(3/5) percent = 0.6 % = .006
0.6 % = 0.006
(0.006)(x) = 15
15/0.006 = x = 2500
Strategy: Part = Percent x Whole |
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Term
| How can you recognize multiples of 2? |
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Definition
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Term
| How can you recognize multiples of 3? |
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Definition
Sum of digits is a multiple of 3
Example: 432 is a multiple of 3 because 4+3+2 = 9 which is also a multiple of 3 (and actually, because 432 / 3 = 144) |
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Term
| How can you recognize multiples of 4? |
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Definition
Last two digits are a multiple of 4
Example: 2416 is a multiple of 4 because the last two digits are 16, which is a multiple of 4. 2416 / 4 = 602 |
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Term
| How can you recognize multiples of 5? |
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Definition
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Term
| How can you recognize multiples of 6? |
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Definition
Sum of digits is a multiple of 3, and last digit is even
Example: 432 -->
a) 4+3+2 = 9 which is a multiple of 3
b) last digit is 2 which is even |
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Term
| How can you recognize multiples of 9? |
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Definition
Sum of digits is a multiple of 9
Example: 432 -->
4+3+2 = 9 which is a multiple of 9 (9x1) |
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Term
| How can you recognize multiples of 12? |
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Definition
Sum of digits is a multiple of 3, and last two digits are a multiple of 4
Example: 432 -->
a) 4+3+2 = 9 which is a multiple of 3
b) 32/4 = 8, so last two digits are a multiple of 4 |
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Term
| What factors do 135 and 225 have in common? |
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Definition
Solution:
a) 135 = 9 x 15 -->
9 = 3 x 3
15 = 3 x 5
Prime factors of 135 are 3,3,3, and 5
b) 225 = 9 x 25
9 = 3 x 3
25 = 5 x 5
Prime factors of 225 = 3,3,5, and 5
Common prime factors: 3, 3, and 5
Multiply common prime factors in every possible way to get other common factors:
a) 3 x 3 = 9 so 9 is a common factor
b) 3 x 5 = 15 so 15 is a common factor
c) 3 x 3 x 5 = 45 so 45 is a common factor
In conclusion, the common factors of 135 and 225 are: 1, 3, 5, 9, 15, and 45
Strategy:
1. Find prime factors of each number
2. Find prime factors in common
3. Multiply common prime factors in every possible combination to get addition (non-prime) common factors |
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Term
| What is the least common multiple of 28 and 42? |
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Definition
Solution:
a) Determine prime factors of 28:
28 = 14 x 2 --> 2 is a prime factor of 28
14 = 7 x 2 --> 2 and 7 are prime factors of 28
Prime factors of 28: 2, 2, 7
b) Determine prime factors of 42:
42 = 21 x 2 --> 2 is a prime factor of 42
21 = 7 x 3 --> 7 and 3 are prime factors of 42
Prime factors of 42: 2, 3, 7
c) Multiply prime factors by one another to find the LCM. If a prime factor is common to both values, then multiply it the number of times it appears in the value where it appears the greatest number of times:
i) 2 is a common factor of 28 and 42. It is a one-time factor of 42, and a two-time factor of 28, therefore it should be multiplied twice in finding the LCM
ii)7 is a common factor of 42 and 28. It is used once in each case, therefore it should be used once in finding the LCM
iii) LCM = 3 x 2 x 2 x 7 = 84
Therefore, the LCM of 42 and 28 is 84.
Strategy:
First, Note that the LCM of two numbers is NOT necessarily the product of those numbers
Find the LCM by
a) First, find the prime factors of each number
b) Second, determine how many times each prime factor appears as a prime factor
c) Multiply the prime factors by one another the greatest number of time each appears
d) The product will be the LCM |
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Term
| 17.5 is the average (arithmetic mean) of 24 numbers. What is the sum of the 24 numbers? |
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Definition
Solution: 17.5 x 24 = 420
Strategy: Sum of terms = Average x Number of terms |
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Term
| What is the average of all the integers from 13 to 77? |
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Definition
Solution: Average = (13+77)/2 = 90/2 = 45
Strategy: The average of evenly spaced (consecutive) numbers is simply the average of the largest number and the smallest number. |
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Term
| What is the sum of the integers from 10 through 50, inclusive? |
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Definition
Solution:
a) Find average of max and min values:
Average of 10 and 50 = 60/2 = 30
b) Determine number of terms to be summed:
50-10=40
40+1 = 41 terms to be summed
c) Sum = Average x Number of terms
Sum = 30 x 41 = 1230
Strategy:
Sum = Average x Number of Terms |
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Term
| How many integers are there from 73 through 419, inclusive? |
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Definition
Solution:
419 - 73 = 346
346 +1 = 347
Strategy: The number of integers from A to B, inclusive is (B-A)+1 |
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Term
| There are 12 oranges and 20 apples. What is the ratio of oranges to apples? |
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Definition
Solution:
12/20 = 6/10 = 3/5 = 3 :: 5 = Oranges :: Apples
Strategy:
Ratio = of/to |
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Term
| The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there? |
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Definition
Solution:
3/4 = 135/#girls
(4 x 135) = (3 x # girls) = 540
540 / 3 = # girls = 180
There are 180 girls
Strategy: Use cross multiplication |
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Term
| How many 3-digit numbers can be formed with the digits 1, 3, and 5 if each number can only be used once? |
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Definition
Solution:
Given the 3-digit # XYZ:
a) Options for X = 3 options, then one is chosen
b) Options for Y = 2 remaining options
c) Options for Z = 1 remaining option
2. 3 x 2 x 1 = 6 possible ways to arrange the numbers if each number can only be used once (ie. "order matters"
Strategy:
1. Individually look at each digit of the theoretically resultant 3-digit number, "XYZ" to determine how many possibilities there are for each spot.
a) How many options are there for X?
b) After X has been selected, how many options are there remaining for Y?
c) After X and Y have been selected, how many options are there remaining for Z?
2. Multiply the number of options for each spot together to get the number of possible PERMUTATIONS:
a) (options for X) x (options for Y) x (options for Z) = # of Permutations possible |
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Term
| What is the probability of throwing a 5 on a 6-sided die? |
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Definition
Solution: 1/6
Strategy: Probability = desired outcomes / possible outcomes |
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Term
Factor the following polynomials:
1. ab + ac
2. a^2 +2ab + b^2
3. a^2 -2ab + b^2
4. (a^2 - b^2) |
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Definition
1. ab + ac = a(b+c)
2. a^2 +2ab + b^2 = (a + b) x (a + b)
3. a^2 -2ab + b^2 = (a - b) x (a - b)
4. (a^2 - b^2) = (a - b) x (a +b) |
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Term
| Rewrite 7-3x > 2 in its simplest form (meaning, "solve" for x) |
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Definition
Solution:
7-3x > 2
-3x > 2-7
-3x > -5
x < 5/3
Important Note:
Be sure to reverse the direction of the inequality sign if you multiply or divide by a negative number |
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Term
| What is the area of a triangle with height = 5 units and base = 8 units? |
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Definition
Solution:
*** Area of a triangle = (Base x Height) / 2 ***
Area = (8 x 5) / 2 = 40/2 = 20 units^2 |
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Term
| What is an isosceles triangle? |
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Definition
| An isosceles triangle has two equal sides and two equal (corresponding) angles) |
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Term
| What is an equilateral triangle? |
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Definition
| An equilateral triangle has three equal sides (all sides are equal) and 3 angles of 60 degrees each |
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Term
| What does it mean to say that two triangles are "similar"? |
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Definition
For Similar triangles:
a) corresponding angles are equal
b) corresponding sides are proportional |
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Term
| A right triangle has a leg of length 3 and a leg of length 4. What is the length of the third leg and is this leg the hypotenuse? |
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Definition
| The hypotenuse is 5 units long. This is a 3-4-5 triangle. |
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Term
| A right triangle has a leg of length 5 and a leg of length 12. What is the length of the third leg? |
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Definition
| The hypotenuse is 13 units long. This is a 5-12-13 triangle |
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Term
| A right triangle has a leg of length square-root of 3, and a leg of length 1. What is the length of the third leg? |
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Definition
a) The hypotenuse is 2 units long. This is a 1-2-square-root of 3 triangle.
b) The angle between the 1-unit leg and the hypotenuse (2 units) is 60 degrees
c) The angle between the square root of 3 unit leg and the hypotenuse (2 units) is 30 degrees.
d) The angle between the square root of 3 leg and the 1 unit leg is 90 degrees |
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Term
| A right triangle has two legs of equal length x. What is the length of the third leg? What are the angles of the triangle? |
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Definition
a) The third leg is (square-root of 2) x (X) units long.
b) The angle between the two sides of length X is 90 degrees
c) Each angle between a side of length "square-root of two" and a side of length X are 45 degrees. |
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Term
| What is the area of a parallelogram? |
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Definition
Area = (base) x (height)
Height can be determined by drawing a line from the top corner of the parallelogram so that it intersects in a manner perpendicular to the base, forming two 90 degree angles. This will form a right triangle, from which the height can be found if the hypotenuse is known. |
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Term
| How do you determine the circumference of a circle? |
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Definition
| Circumference = (Pi) x (Diameter) |
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Term
| How do you find the area of a circle? |
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Definition
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