001 FOM- Arithmetic Flashcards
Define: integer
A whole number
Express x as a factor of y for some integer n
y=xn
What is another name for factor?
Divisor
What are the quotient and remainder for the following equation?
=28/8
Quotient is 3
Remainder is 4
Express quotient and remainder as a formula. What are the bounds of r?
y = xq+r
0
What are the quotient and remainder when a smaller integer is divided by a larger one?
Q= 0 R = the smaller integer
Y=xq+r
5=7*0+5
Define prime numbers
A positive integer with two different positive divisors, 1 and itself.
What is the smallest prime number?
2
Express consecutive integers as an equation
N, n+1, n+2, n+3…where n is an integer
Express consecutive even integers as an equation
2n, 2n+2, 2n+4…
Define least common multiple
The smallest common multiple of two numbers
How do you divide by a fraction?
Invert the divisor (ie take the reciprocal of the fraction) and multiply
Define a mixed number.
A number consisting of a whole number and a fraction
How to change a mixed number into a fraction?
Multiple the whole number by the denominator and add to the numerator
7 2/3 = 23/3
Express 0.0321 as a fraction
0/10 + 3/100+2/1000+1/10,000
= 321/10,000
lcm is 10,000
How to multiply decimals?
Multiply the numbers as whole numbers. Insert the decimal so that the number of digits to the right is equal to the sum of the numbers of digits to the right of the decimal points in the numbers being multiplied.
2.09 x 1.3 = 627+2090= 2.717
How to divide decimals?
Move the decimal point of the divisor to the right until it’s a whole number. Move the decimal of the dividend the same number. Divide. Put the decimal in the quotient directly above the decimal in the new dividend.
12.4[698.12
=124[6981.2
=56.3
Express xy+xz alternatively
X(y+z)
Define proportion re ratios
Statement that two ratios are equal
How can you find the % increase?
1) find amount of increase
2) divide increase by original amount
3) express as a %
How does the % increase from x to y compare to a % decrease from y to x?
Decrease will be a smaller %, as it is the same amount of change, but the comparable is a larger number
What happens when you square a number between 0 and 1?
It gets smaller
1/3 squared = 1/9
How many square roots does a positive number have?
Two: one positive and the other negative
How many cube roots does a real number have
One
Define mean
The average
Define median
Center for a list of numbers (which are in order from least to greatest). If there are an even amount of numbers, the median is the average of the two middle numbers
What is the relationship between median and mean?
None. Median can be less, equal or more.
Define mode
The number occurring most frequently in a lost. Can be multiple modes for a given list.
What values influence standard deviation most?
Those farthest from the mean. The closer data is grouped around the mean, the smaller the SD
How to calculate standard deviation?
1) calculate mean of x numbers
2) calc each diff from mean (x-mean)
3) square each diff
4) calc mean of squared diff
5) find positive square root of this mean
How to determine mean using frequency distribution?
1) multiply each data value by its frequency
2) sum the results
3) divide by the number of data values
What is union of any two sets A and B?
The set of all elements that are in set A or B or both. represented by a capital U
What is the intersection of sets A and B?
The set of all elements that are both in A and B. Represented by the shaded part of a Venn diagram or an inverted U
Express the Union of two sets mathematically
s U t = set s + set t - Intersection s and t
What is the factorial of n?
n!
Defined as the product of all integers from 1 to n.
4! = 123*4=24
Define permutation and express as a formula
A selection process in which objects are selected one by one in a certain order. Represents all the ways the objects can be ordered.
For a set of n objects:
n(n-1)(n-2) =n!
Formula for a set of n objects from which a complete selection of k objects is to be made without regard to order, where 0
N
K
=
N!/k!(n-k)!
Probabilities are between what range?
0 (impossible) to 1 (certain)
Express probability as a formula
Number of outcomes in E/total number of possible outcomes.
Where E represents a possible event
What is the probability of event E and event F happening, where E and F are independent events?
What is this called?
P(E)*P(F)
Intersection
What is the probability of E or F happening, where E and F are independent? What is this called?
= P(E)+P(F)-P(E and F)
Called the union
What are alternative ways of saying 15 is divisible by 3?
1) 15 is a multiple of 3
2) 3 is a factor of 15
3) 3 goes into 15
4) 3 divides 15
Is zero even or odd? Why?
Even, because it is divisible by 2 (0/2 = 0)
What is the number underneath an exponent?
The base
What does a root do to an exponent?
Undoes it
What is a perfect square?
The square of an integer (the square root of a perfect square therefore is always an integer).
What is the order of arithmetic?
PEMDAS (Parantheses, exponents, multiply, divide, add, subtract).
How do you state (3+2)*2?
The quantity 3 plus 2, all times 2. Quantity refers to parantheses.
Name the first 8 prime numbers.
2,3,5,7,11,13,17,19
If a/b and b/c, then…?
a/c
If d is divisible by two primes e and f, then what is true?
d is also divisible by ef. ex: 20 is divisible by 52=10
Define a prime factor
The unique set of prime numbers multiplied to arrive at a given integer.
If x is divisible by a and b, what is true about the LCM of a and b?
x is also divisible by that
What is true re LCM when two numbers share no prime factors?
The LCM is the product of the two numbers.
What is true about LCM when numbers x and y share prime factors?
The LCM of x and y is less than x*y.
What is 11^2?
121
What is 12^2?
144
What is 13^2?
169
What is 14^2?
196
What is 15^2?
225
What is 4^3?
64
What is 5^3?
125
What is 2^5?
32
What is 2^4?
16
What is 2^6?
64
What is 2^7?
128
What is 2^8?
256
What is 2^9?
512
What is 2^10?
1024
What is 3^3?
27
What is 3^4?
81
What is true about even exponents?
They can hide the sign of the base. (-2)^2 = (2)^2
x^3 * x*4 =?
x^7
x^4 / x^3?
x^1
What is 0^0?
Undefined
4^0 = ?
1
a^-2 =?
1/a^2
What is (a^3)^2?
a^6
What is true about the square root symbol?
It only gives a positive result.
Simplify sq root{3^2 + 4^2}
sq root {25} = 5
Simplify sq root{3^2 * 4^2}
sq root {3^2} * sq root {4^2} = 3*4 = 12
What is an improper fraction?
Where the numerator exceeds the denominator (ie the number is more than 1).
What is the easiest way to see which of two fractions is larger?
Cross multiply - the side with the bigger number is the larger one (generates the numerators of the fractions were you to re-express them using the common denominator)
What is a shortcut way to convert an improper fraction to a mixed number?
Express the number as a sum, then split the fraction and simplify.
ie 13/4 = (12+1)/4 = 12/4 +1/4 = 3 1/4
How do you convert a decimal to a fraction?
Put a power of 10 in the denominator of the fraction. The power depends how many places to the right of the decimal there are.
ie 0.007 = 7/1000
When converting a fraction to a decimal, how do you know when the decimal will end?
If the denominator contains only 2’s and/or 5’s as factors.
If you know a fraction converted to a decimal will end, what’s a shortcut way to convert it?
Make the denominator a power of 10 by multiplication.
ie 1/4 * 25/25 = 25/100 = 0.25
How do you convert a fraction to a percent?
Convert is to a decimal, then convert to a percentage.
Convert to a percent: 1/8
12.5%
Convert to a percent: 3/8
37.5%
Convert to a percent: 2/5
40%
Convert to a percent: 3/5
60%
Convert to a percent: 5/8
62.5%
Convert to a percent: 7/8
87.5%
Convert to a percent: 5/4
125%
Translate “what percent”
= x/100
How do you find the percent change?
Change in value / original value
What is another way to state “percent increase”?
Percent more than
What’s the difference between an expression and an equation?
Expressions represent a number, and does not contain an equals sign. You may simplify an expression, but you never change that expression’s value. You determine the expression’s value by evaluating it.
An equation is a complete sentence: subject equals object.
What is true re simplification of expressions and equations?
You can simplify one side of an expression, but you must do the same thing to both sides of an equation.
For an equation involving exponents, what is true about the bases?
Once the bases are the same, the exponents must be the same (except for bases 1, 0 and -1).
What are two strategies to solve a system of equations?
1) Isolate, then substitute
2) Combine equations to eliminate a variable
In solving a system of equations, which variable should you isolate?
The variable you ultimately don’t want (ie if the problem asks x=? the isolate and sub the variable y).
How many solutions does a quadratic equation usually have?
2
What does x^3 in an equation indicate?
There could be 3 solutions.
What is the square of a sum?
(x+y)^2 = x^+2xy+y^2
What is the square of a difference?
(x-y)^2 = x^2-2xy+y^2
What is a difference of squares?
(x+y)(x-y) = x^2-y^2
middle terms cancel
What should you do if you see a special product or something close to it?
Convert a special product to its other form, especially for a difference of squares. If it’s close to a special product, rearrange the equation to get there.
What is true about solutions to inequality expressions?
They have a range of solutions.
What happens when you multiply or divide both sides of an inequality by a negative number?
You must flip the inequality sign.
What is true about a variable inside of an absolute value expression?
It generally has two values.
What is the formula for Total Cost?
Fixed Cost + Price x Additional Quantity
What is the process to solve a “work” word problem?
Use Rate x Time = Work, choosing work units and often adding rates. Invent widgets (small units of work) to avoid dealing with fractions (i.e. half a 100 box warehouse a day is 50 boxes per day).
