Math 101!
Arithmetic! Prerequisite: None
Course Description
In this micro-course, we will review the basic concepts of arithmetic most commonly found on the SAT, exponent rules, and a brief ISME review of function problems, ratios, and percent change. We will then review properties of prime numbers, ratios, functions (including weird symbol functions), and percent change.!
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Basic Arithmetic Terms and Operations TERM!
DEFINITION!
Integer!
All positive whole numbers (1, 2, 3, . . . ), negative whole numbers (–1, –2, –3, . . . ), and zero (0)!
Factor!
An integer that divides evenly into another integer!
Multiple!
An integer that can be divided by a smaller integer with no remainder!
Even/Odd!
Even = Divisible by 2! Odd = NOT Divisible by 2 !
Distinct!
Different (e.g. “distinct factors”)!
Inclusive/Exclusive!
The set of integers 1–5 inclusive = {1, 2, 3, 4, 5}! The set of integers 1–5 exclusive = {2, 3, 4}!
Remainder!
The value left over when a number is not evenly divisible by another!
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Basic Concepts
The Rules of Zero!
The Number! Zero (“0”)!
! 0 is… !
positive?
negative?
neither?!
! 0 is… !
even?
odd?
neither?!
! 0 is… !
an integer?
not an integer?!
! Anything divided by 0 is ________________________________! !
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Basic Concepts
The Rules of Prime Numbers! ! 1 is… !
Prime Numbers!
a prime number?
not a prime number?!
! 2 is the ___________________________ prime number! ! ! 2 is the only _______________________ prime number! ! ! List (and memorize) the first 6 primes:! !
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Basic Concepts
The Rules of Exponents! Ø Positive powers—Multiply the base by itself as many times as the exponent tells you to.!
35 =
Ø Negative powers—A negative exponent means that you put a 1 over the base and multiply the base by itself as many times as the exponent tells you to.!
3–5 =
Ø 0 as power—Any number raised to the power of 0 equals 1.!
30 =
Ø 1 as power—Any number raised to the power of 1 equals itself.!
31 =
Ø 1 and 0 with powers—No matter what power 1 is raised to, it remains 1. No matter what power 0 is raised to, it remains 0.!
16 =
Ø Powers on fractions—Multiply the fraction by itself as many times as the exponent tells you to.!
⎛ 3 ⎞ ⎜ ⎟ = ⎝ 4 ⎠
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3
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Basic Concepts
The Rules of Exponents! Ø Negative numbers with powers—If a negative number is raised to an EVEN power, it becomes POSITIVE. If a negative number is raised to an ODD power, it stays NEGATIVE.!
(–3)2 =
Ø Operations on powers—These operations work only when the bases are the same.!
35 x 34 =
Ø Raising a power to a power!
(35)3 =
Ø Distributing an exponent!
(3x)5 =
Ø Fractional powers (aka “Roots”)—The numerator represents the power and the denominator represents the root.!
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1 3
3 =
(–3)3 =
35 = 34
5 2
3 =
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Function Problems
Function Problems – Plug and Chug! ! Identify:! Any problem that includes function notation with numbers: f(2) or g(–15) or !h 3 ! Set Up:! PLUG: The number in the parentheses is the number to plug in to the function wherever you see a variable. ! ! Make Sure:! Don’t be intimidated. Be sure you plug in the number for EVERY instance of the variable. ! ! Execute:! CHUG: Work out the arithmetic, using the number in the!parentheses. !
( )
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Function Problems
Example:! If the function f is defined by f(x) = 3x + 20, what is the value of f(4)? (A) (B) (C) (D) (E)
8 12 27 32 34
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Weird Symbol Notation
Weird Symbol Notation – Plug and Chug! ! Identify:! A problem that involves a weird symbol, and asks you to solve for a number value, such as:! ! 7, or Ω6 ! 3 ,5 ! Set Up:! PLUG: The number in the question is the number to plug in to the operation defined in the question. ! ! Make Sure:! Don’t be intimidated. Be sure you plug in the number for EVERY instance of the variable from the operation. ! ! Execute:! CHUG: Solve the operation by solving the math. !
!
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Weird Symbol Notation
Example:! For all positive values of p, let pJ be defined by p pJ = 2 . What is the value of 2J? p +1 (A) (B) (C) (D) (E)
8 12 27 32 34
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Ratios/Proportions
Ratios/Proportions! ! Identify:! Questions that define a proportional relationship between two or more things.! ! Set Up:! Two fractions equaling one another. The RATIO of one thing to another thing is a fraction:! ! One Thing !Ratio = Another Thing ! ! Make Sure:! Keep your units straight, and answer the question they’re asking.! ! Execute:! Cross-multiply. !
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Ratios/Proportions
Example:! The scale on a wilderness map is such that 1 2
1 4
inch represents
mile. If a trail measures 2 inches on the map, how many feet
long is the actual trail? (1 mile = 5280 feet) (A) (B) (C) (D) (E)
330 2,640 10,560 21,120 43,240
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Percent Change
Percent Change! ! Identify:! Questions that involve percentages, or give “before” and “after” numbers.! ! Set Up:! List all the elements you know, and put them in the percent change formula:! ! Difference Percent C hange = ×100% ! Original ! Make Sure:! Be careful of the denominator: Divide by the Original amount, NOT by the Final amount.! ! Execute:! Just do the math.!
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Ratios/Proportions
Example:! Janet goes to the store to buy herself a new dress, and she finds that there is a 20% off sale going on. Additionally, Janet has a 20% discount coupon. If Janet buys the dress that is on sale and also uses her coupon, what will her total discount be? (A) (B) (C) (D) (E)
20% 30% 34% 36% 40%
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Review Questions Which real number satisfies (2n)(8) = 163 ? (A) 3 (B) 4 (C) 6 (D) 9 (E) 12
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Review Questions If f(x) = –3x2 – 8, then f(–4) = (A) –56 (B) –40 (C) 8 (D) 24 (E) 40
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Review Questions Mars has lower gravity than Earth, so objects on Mars weigh 38% of what they would on Earth. If Jason weighs 175 pounds on Earth, how many pounds would he weigh on Mars? (A) (B) (C) (D) (E)
66.5 89 108.5 241 460.5
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Review Questions Let x ⊕ y = (x – 2y)2 for all integers x and y. What is the value of 5 ⊕ –3 ?
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Review Questions If k is the largest prime number less than 50 and j is the smallest prime number, what is (A) (B) (C) (D) (E)
j ? k
1 47 1 49 2 47 2 49 3 49
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Review Questions Let x∆y be defined by the positive difference between the square root of x and the square root of y. What is the value of 144∆81? (A) 3 (B) 9 (C) 12 (D) 3 7 (E) 63
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Review Questions If the ratio of 2p to 11r is 1 to 6, what is the ratio of 6p to 11r ? (A) (B) (C) (D) (E)
1 to 18 1 to 2 1 to 11 1 to 6 3 to 22
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Review Questions A positive integer is said to be a “perfect cube” if it is equal to another integer raised to the third power. How many positive integers less than 1,000 are perfect cubes?
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Review Questions A clothing store sells a certain type of jacket in two colors: brown and red. If the ratio of the brown jackets to red jackets in stock are 3:4, each of the following could be the total number of jackets at the store EXCEPT? (A) (B) (C) (D) (E)
7 14 21 28 34
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Review Questions What is the largest prime factor of 636 ? (A) (B) (C) (D) (E)
53 54 55 57 61
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Review Questions The integer n is a multiple of both 5 and 3. Which of the following MUST be true? I. n is odd II. n = 30 III. n is a multiple of 15 (A) (B) (C) (D) (E)
III only I and II only I and III only II and III only I, II, and III
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Review Questions For all integers n, let n be defined by follows: n = n2 if n is odd n = 0.5n if n is even. If 3 + 2 = m, what is the value of m3 ?
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Review Questions If the assets of a stock drop by $4.03 billion to $74.02 billion, what was the percent decrease, to the nearest tenth, in the stock’s assets? (A) (B) (C) (D) (E)
0.4% 4.1% 5.2% 25.3% 40.0%
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Review Questions If x is an integer, then the units (ones) digit of x squared CANNOT be which of the following values? (A) (B) (C) (D) (E)
0 1 2 5 6
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Review Questions The ratio of two quantities is 4 to 5. If each of the quantities is increased by 3, what is the ratio of these two new quantities? 1 2 4 (B) 5 (C) 7 8 (D) 17 8
(A)
(E) It cannot be determined from the information given.
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Review Questions If 72 = a b , where a and b are positive integers with a > b, which of the following could be the value of ab? (A) (B) (C) (D) (E)
3 6 12 24 36
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Review Questions What is the least positive integer x for which 12x is the cube of an integer?
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Review Questions Frank buys a 64-ounce can of paint to use for two home improvement projects. He only needs first project, but then uses
3 4
1 8
of the paint for the
of the remaining paint on the
second project. If Frank can return the remaining unused paint to the store for 20 cents per ounce, how much money, in dollars, did Frank receive for returning his unused paint? (A) (B) (C) (D) (E)
$1.20 $2.80 $3.80 $4.00 $8.40
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