Arithmetic reasoning Flashcards
Average
Standard deviation
Arithmetic mean
Take the sum of n numbers and divide by n
1
[Population] Standard deviation = sqrt (— sum(xi - xbar)^2)
N
1 Sample Standard deviation = sqrt(—— sum(xi - xbar)^2)
(N-1)
1 mi = ? ft
1mi = 5,280 ft
Median of odd # of Data
The middle # if the data is listed in increasing order.
Median of an even # of data
Arithmetic mean (average) of the two middle numbers when the data is listed in increasing order.
Mode
Most frequently occurring # in the list. There may be more than one mode for a list of data
Range
Greatest # in the list - least #
Interquartile range
3rd quartile - 1st quartile To divide data into quartiles: 1) arrange in increasing order 2)find median M. M = Q2 3) divide data into 2 equal groups, find their median, it is Q1 and Q3
x + y = -1
Quantity A QuantityB
x y
One equation, 2 unknowns,
D) the relationship cannot be determined from the information given.
Can also plug in
x = 0
Then
y = 0
A certain store sells two types of pens: one type for $2 per pen and the other type for $3 per pen. If a customer can spend up to $25 to buy pens at the store and there is no sales tax, what is the greatest # of pens the customer can buy?
Start by looking at how many of the lesser value, $2, pens the customer can buy if the customer doesn’t buy any $3 pens.
25/2 = 12.5 .
So 12 pens for $2, 0 pens for $3.
Next consider $3 pens
11 $2 pens ($22) and 1 $3 pen = 12 pens total
10 $2 pens ($20) and 1 $3 pen = 11 pens total
9 $2 pens ($18) and 2 $3 pen= 11 pens total.
As the # of $2 pens decreases the total # of pens decreases.
Therefore max amount = 12 pens
A list of numbers has a mean of 8 and standard deviation of 2.5. If x is a # in the list that is 2 standard deviations above the mean, what is the value of x?
x = 2(2.5) + 8 x = 13
The following table shows the distribution of 200,000 physicians by specialty. Which of the following sectors represent more than 40,000 physicians?
A. Pediatrics - 21% B. Internal Medicine - 25% C. Surgery - 24% D. Anesthesiology - 3% E. Psychiatry - 6%
Find what percent of 200,000 is 40,000
40,000/200,000 = 0.2 = 20%
choices over 20% are A, B, C.
Profit
Profit, or gross profit = sales revenue - cost of production
If n is the integer, what is the sum of first two consecutive integers greater than n + 6?
(n+7)+(n+8)
In 2009 the property tax on each home in Town X was p percent of the assessed value of the home, where p is a constant. The property tax in 2009 on a home in Town X that had an assessed value of $125, 000 was $2, 500
Quantity A QuantityB
The property tax in 2009 on $3,000
a home in Town X that had
an assessed value of
$160, 000
$125,000p = $2,500 P = 125000/2500 P = 0.02 or 2%
2% of $160,000 is
160,000 * 0.02 = 3,200
3,200 > 3000
A>B
Congruent angles
Angles that have equal measures. Opposite angles
Acute angle
Less than 90 degrees
Obtuse angle
Angle b/w 90 and 180 degrees
Convex polygon
A polygon in which the measure of each interior angle is less than 180 degrees
Sum of interior angles of a polygon
(n-2)*180
Sum of hexagon (6-2)180=720
Regular polygon
A polygon in which all sides are congruent and all interior angles are congruent.
Quadrilateral
4 sided polygon
Equilateral triangle
Triangle with three congruent sides
Isosceles triangle
A triangle with at least two congruent sides
Congruent triangles
Two triangles that have the same shape and size
Similar triangles
Two triangles that have the same shape but not the same size
Area of a parallelogram
Same as area of a square or rectangle or any quadrilateral.
Area = base* height
Area of trapezoid
Area of the trapezoid equals alf the product of the sum of the lengths of the two parallel sides b1 and b2 and the corresponding height h; that is,
A=1/2(b1+b2)*h
Congruent circles
Circles with equal radii
Chord
Any line segment joining two points on the circle
Pi
Ratio of circumference of a circle to it’s diameter
C 22
– = pi = 3.14 = —–
d 7
Circumference of a circle
C= 2pi*r
Arc length of a circle
Ratio of length of an arc to the circumference
Is equal to the
Ratio of the degree measure of the arc to 360.
Area of a circle
A = pi*r^2
Area of a sector S of a circle
The ratio of the area of a sector of a circle to the area of the entire circle
Is equal to
The ratio of the degree measure of it’s arc to 360.
Concentric circles
Two or more circles with the same center.
Surface area of a right circular cylinder
Acyl = 2pir^2 + 2pir*h
Frequency
Count # of times the category or value appears
Relative frequency
The associated frequency divided by total # of data.
Example:
Number of children Frequency Relative frequency
0 3 3/25 = 12%
1 5 3/25 = 20%
2 7 7/25 = 28%
3 6 6/25 = 24%
4 3 3/25 = 12%
5 1 1/25 = 4%
——————————————————————————–
Total: 25
Histogram
Histograms are graphs of frequency distributions that are similar to bar graphs, but they have a number line for the horizontal axis. Any spaces between bars on a histogram indicate that there are no data in the intervals represented by the spaces
Weighted mean
Ex: 2,4,4,5,7,7,7,7,7,7,8,8,9,9,9,9
2(1) + 4(2) + 5(1) + 7(6) + 8(2) + 9(4) 109
————————————————— = —— = 6.8125
1 + 2 + 1 + 6 + 2 + 4 16
Standardization
Subtract the mean from each value, then divide result by standard deviation
Intersection of S and T (sets)
S () T is the set of all elements that are in both S and T
Union of S and T
The set of all elements that are that are in S or T or both, denoted by S U T
Disjoint or mutually exclusive elements
Sets S and T that have no elements in common
Venn diagram
A useful way to represent two or three sets and their possible intersections and unions.
Inclusion-exclusion principle for two sets
This principle relates the numbers of elements in the union and intersection of two finite sets:
1) number of elements in the union of two sets =
Sum of individual numbers of elements - # of elements in the intersection.
|A U B| = |A| + |B| - |A ^ B|
2) For the union of B and C that are mutually exclusive and have no intersection
|B U C| = |B| + |C|
Multiplication principle
Two choices to be made sequentially one after another and independent of one another. If # of possibilities for first choice is ‘k’ and # of possibilities for the 2 nd choice is ‘m’, then there are ______ different possibilities for the pair of choices.
Km
A meal consists of 1 entree and 1 dessert. If there are 5 entrees and 3 desserts on the menu, how many different meals can be ordered from the menu?
(5)(3) = 15 different meals can be ordered from the menu
A computer password consists of four characters such that the first character is one of the 10 digits from 0 to 9 and each of the next 3 characters is any one of the uppercase letters from the 26 of English alphabet. How many different passwords are possible?
(10)(26)(26)(26) = 175, 760
A computer password consists of four characters such that the first character is one of the digits from 0 to 9 and each of the next 3 characters is any of the non repeating uppercase letters from English alphabet. How many different passwords are possible?
— — — —
(10)(26)(25)(24) = 156,000
If a coin is tossed 8 times, how many outcomes are there?
— — — — — — ——
(2)(2)(2)(2)(2)(2)(2)(2) = (2)^8 = 256 possible outcomes
Permutations and factorials
To order 4 letters in order we use the formula
n(n-1)(n-2)(n-3) Or n! So, to order 4 letters, (4)(3)(2)(1) or 4! = 24 Four letters can be arranged 24 different ways.
10 students are going on a bus trip, and each of the students will be assigned to one of the 10 available seats. What’s the number of different possible seating arrangements?
10! = 3, 628, 800
Determine the # of ways in which you can select 3 of the 5 letters A, B, C, D, E and place them in order from 1st to 3rd.
— — —
(5)(4)(3)= 60
There are 60 ways to select 3 of 5 letters
Permutations of ‘n’ objects taken ‘k’ at a time
Or
The # of ways to select and order ‘k’ objects out of ‘n’ objects.
n
= ——
(n-k)!
How many different five-digit positive integers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if none of the digits can occur more than once in the integer?
This question asks how many ways there are to order 5 integers chosen from a set of 7 integers.
According to the counting principle,
n= set, k= how mamy objects are chosen
n! 7! 7!
——— = ——— = —— = 2,520
(n-k)! (7-5)! (2)!
There are 2,520 different five-digit positive integers that can be formed using 1,2,3,4,5,6,7 and none of the digits repeat in the integer.
Probability
For a random experiment with a finite # of possible outcomes, if each outcome is equally likely to occur, then the probability that an event E occurs is defined by the ratio
The # of outcomes in the event E P(E) = ——————————————————————
The # of all possible outcomes in the experiment
Probability that event E will not occur
1-P(E)
For events E and F,
What is the probability of E and F occuring (rolling a die to get a 4 or an even # or both)
P(E or F or both) = P(E) + P(F) - P(E and F)
If E and F are mutually exclusive events, what is the probability of both of them occuring?
Rolling a die to get a 4 and an odd #?
0
If E and F are mutually exclusive, what is the probability to get either E or F?
Probability of rolling a 2 or an odd # or both?
P(E or F or both) = P(E) + P(F)
If two events are independent, what is the probability of both E and F occuring?
If a 6-sided die is rolled twice, what is the probability of pf rolling a 3 on the first roll and 3 on the second roll?
P(E and F) = P(E) * P(F)
P(rolling a 3 and then rolling a 3) = (1/6)(1/6) = 1/36
P(E or F)
P(E or F) = P(E) + P(F) - P(E and F)
P(E or F) if they are mutually exclusive
P(E or F mut. Excl.) = P(E) + P(F)
P(E and F) independent
P(E and F) = P(E)*P(F)
If a fair 6-sided die is rolled once, what is the probability of getting a 3 and an odd #?
1) independent or dependent events?
Dependent. Rolling a 3 makes certain that the event of rolling an odd # occurs. Therefore
P(E and F) = P(E) = 1/6
Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, but the probability of rolling 6 is twice the probability of rolling a 1. When you roll the die once, the outcomes are NOT equally likely. What are the probabilities of the 6 outcomes?
p = P(1) = P(2) = P(3) = P(4) = P(5) 2p = P(6)
1 = P(1) + P(2) + P(3) + P(4) + P(5) + P(6) 1 = p + p + p + p + p + 2p = 7p
P(1) = 1/7 P(6) = 2/7
