Distance =
Rate*time
Average rate?
Total distance/ total time
or
Total earnings/ Total time
Work =
Rate * time Work= individual rate • number of workers • time
Difference of squares
X2 -Y2 = (x+y)(x-y)
(X+Y)2
X2 +2xy+y2
45-45-90 triangle
1:1: root 2
30-60-90 triangle
1:square root of 3: 2
Diagonal of a rectangular box
D squared = L squared + W squared + H squared
Height of equilateral triangle
H=S * sq. root of 3/ 2
SA of cube
6(side)2
Volume of cube
(Side)3
Volume of rectangular box
L*W*H
Arc length =
Sec angle/360 * 2(pi)(R)
Volume of circular cylinder
(Pi)(r2)(H)
Path. Triangle shortcut
5-12-13 3-4-5 8-15-17 7-24-25 9-12-15
Surface area of box
2(L*w)+2(h*L)+2(h*w)
Area of a trapezoid
A=(b1+b2)/(2)
Area of a polygon
A=B(H)
Sum of angles
180(N-2)
Is it possible for 2 events to happen at the same time? Yes.
P(a) or p(b) = p(a)+p(b) - p(a and b)
Is it possible for two events to happen at the same time? No.
P(a) or p(b) = p(a)+p(b)
When one event happens does this influence/change the outcome of the other? Yes.
P(a and b) = p(a) • p(b|a)
When one event happens does this influence/change the outcome of the other No.
P(a and b) = p(a) • p(b)
Area of polygon with diagonals
A= D1+D2/2
Solve a part:part:whole
4:1:50 5x=50 X=10
Chase problems
Subtract 2 cars rate together. 80-50=30 miles per hour. Distance of 50 miles north D=RT 50=30t 5/3=t
Collision problems
Add 2 cars rate together. 40 + 56 = 96 D=RT d=1200 1200=96t T=12.5
Regular pentagon
All sides and all angles are equal
nCr
1st r values of n!/R!
Find the Nth number in a sequence
1st term + pattern(n-1)
Find the sum of a sequence
1st term + 2nd term/ 2= average Average • # of terms = sum
Students at a school 2/5 - take german 1/7 - take French 1/3 - take Spanish 25 - take Portuguese How many students are there?
7•5•3 = 105 42/105 + 15/105+ 35/105 = 92/105 26 represents 105/105 - 92/105 = 13/105 26=13/105m M=210
1/4 of juniors and 2/3 of seniors are going. If there 2/3 as many juniors as seniors. What fraction of students are not going?
2/3 as many juniors as seniors 20 - juniors 30 - seniors Juniors on trip 1/4(20)=5 Seniors on trip 2/3(30) = 20 50 total student. 25 going. 25 are not going.
√2
1.4142
√3
1.732
Percent change
Difference/original * 100
75 is reduced by x% is 54 X=??
75 minus x percent 75 is 54. 75 - 75/100x = 54 75-54 = 3/4x 21(4/3)=x 28=x
Root 2
1.4142
Root 3
1.732
Percent change
Difference/original * 100
Average =?
sum of n numbers/n
Range
greatest value - least value
Pens cost £0.70 each, and pencisl cost £0.40 each. If Jakob spent £5.20 on 10 pens and pencils, how many pencils did he purchase?
Number of pencils = x
number of pens = 10- x
(Cost per pen x number of pens) + (cost per pencil x number of pencils) = total cost
70(10-x) + 40x = 520
700 -70x +40x = 520
700-30x=520
180 = 30x
x=6
Marco is twice as old as Vladimir. Four years ago, Marco was 6 years younger than 3 times Vladimir’s age at that time. How old will Marco be in 2 years?
M = 2V
M-4 = 3(V-4) - 6
M-4 = 3V-12-6
2V-4 = 3V-18
2V+14=3V
14=V
2(14) = 28 +2 = 30
Movie Theater charges £6 per ticket and each movie showing cost the theatre £1,750. How many people need to see a movie so that the theater makes £1 of profit per customer?
c=6c-1750
-5c=-1750
c=350
Mr Choudury’s class consists of 20 students: 12 boys and 8 girls. If the boys weigh an average of 80 pounds each, and the girls weigh an average of 70 pounds each, what is the average weight in pounds of all 20 students
12 boys x 80 pounds per boy = 960 pounds
8 girls x 70 pounds per girl = 560 pounds
total = 1,520 pounds
1520/20 = 76 pounds
What percent of y percent of 50 is 40 percent of y?
(X/100)(Y/100)50 = (40/100)y
(X/100)(Y/2)= (2/5)y
x= 2y(100)(2)/5y
x=80
A chemist is mixing a solution of acetone and water. She currently has 30 ounces mixed, 10 of which are aretone. How many ounces of acetone should she add to her current mixture to attain a 50/50 mixture of acetone and water if no additional water is added?
50/100 = 10+x/30+x
1/2= 10+x/30+x
Cross multiply: 30+x=20+2x
10=x
Jane scored 15% higher on her secont test than she did on her first test. Jane’s score on her third testwas a 25% decrease from the score on her second test. If Jane got a 69 on her third test, what was her score on her first test?
x(1.15)(.75) = 69
x=80
A hunting lodge has enough fuel to keep 20 rooms heated for 14 days. If the lodge decides to save fuel by turning off the heat in 5 unoccupied rooms, and each room requires the same amount of fuel to heat it, how many FULL days will the fuel supply last?
20(14) = 280
280/15 = 18.67
18 FULL days
Car A driving north frompoix X, traveling at a constant rate of 40 miles per hour. One hour later car B started driving north from point X at a constatn rate of 30 miles per hour. Neither car changed direction of travel. If each car stated with 8 gallons of fuel, which is consumed at a rate of 30 miles per gallon, how many miles apart were the two cars when car A ran out of fuel?
Car A’s distance - Car B’s distance = distance between cars
Car A: 30 miles per gallon x 8 gallons = 240 miles
240/40= 6 hours
Car B: (30 miles per gallon)(6 hours - 1 hour) = 150 miles
240-180= 90 miles apart
One robot, working independently at a constatn rate, can assemble a doghouse in 12 minutes. what is the maximum number of complete doghouse that can be assembled by 10 such robots each working on separate doghouses at the same rate for 2.5 hours?
Individual hourly rate is 60/12 = 5 doghouse/ hour
each robot produces 5 x 2.5 = 12.5 doghouses in 2.5 hours
total of 12 x 10 = 120
A rectagle’s width w is twice is length. Express the rectangle’s area in terms of w?
w=2L
in terms of w solve for L
L = w/2
A=L x W
A = w x w/2
= W^2 / 2
2:3 ratio of boys to girls. 4:3 ratio of students from northside to southside.
number of students?
2x+3x = 5x
3y + 4y = 7y
5 x 7 = 35
Arjen’s tennis record was 3 matches won for every 2 matches lost. If he played 30 games last season how many did he win?
part: part: whole
3: 2:30
3x+2x = 30
5x=30
x=6
6(3) = 18
At an animal shelter, the ratio of cats to dogs is 4 to 7. If there are 27 more dogs than cats. how many cats are at the shelter?
“there are 27 more dogs than cats” becomes Dogs- cats or
7x-4x = 27
3x=27
x=9
4(9) = 36 cats
On Monday, a class has 8 girls and 20 boys. On Tuesday, a certain number of girls joined and twice that number of boys left, changing the ratio of girls to boys to 7:4. How many boys left?
Girls / Boys = 8+x/20-2x = 7/4
4(8+x) = 7(20-2x)
32 + 4x = 140 -14x
x=6
2(6) = 12 boys
Cranberry juice is 3 parts cranberry and 1 part seltzer. Lemonade is 1 part lemon juice and 2 parts seltzer. one glass of cranberry is mixed with an equally sized glass of Lemonade.
Cranberry:Seltzer: whole = 3:1:4
Lemon: Seltzer:whole = 1:2:3
Since the two glasses are the same size, choose a smart number.
Multiply Cranberry ratio by 3 and Lemonade by 4.
Cranberry:seltzer:whole = 9:3:12
Lemon:seltzer:whole = 4:8:12
Total of 24 ounces 11 seltzer water
Oil, vinegar, and water are mixed 3:2:1 to make dressing. If Jozef has 8 cups of oil, 7 cups of vinegar, what is the maximum number of salad dressing he can make?
3:2:1 = 6 cups of dressing
8/3 x 6 = 18 cups of dressing
5/8 of weekly salary on rent. 1/3 of remaining on food. £40 available for other expenses.
weekly salary?
5/8 of salary is spent on rent. 1-5/8 = 3/8 of salary remaining. Of remaining spent 1/3 on food.
(2/3)(3/8) = 2/8 = 1/4
1/4x = 40
x = 160
Mixture of acid and water in a ratio of 1:2. After 200 mL of water is added, the ratio of water to acetone is 2:3. The original volume of the mixture?
water/ acetone = x+200/2x = 2/3
3x +600 = 4x
x=600
original volume of water is 600 mL while the orginal volume of acid is 2x = 2(600) = 1200. Total is 600+1200= 1800
Ratio of Noah’s time to Matthieu’s to paint a house is 3:5. If Noah and Matthieu work together at their respective rates, they can paint a house in 10 hours. How long does it take Noah to paint a house alone?
1/3x+1/5x = 1/10
5/15x+3/15x = 1/10
8/15x = 1/10
15x = 80
x = 16/3
3(16/3) = 16 hours
In a certain town, 2/5 of the population is employed. Among the unemployed population, the ratio of men:women 5:7. if there are 40,000 employed people in the town, how many females are unemployed?
Employed/Total pop. = 40,000/x=2/5
2x=200,000
x=100,000
Unemployed population = 100,000-40,000 = 60,000
Unemployed females/Total unemployed = y/60,000 = 7/12
= 35,000 unemployed females.
A zoo has twice as many zebras as lions and four times as many monkeys as zebras. Which of the following could be the total number of zebras, lions, and monkeys at the zoo?
- 14
- 22
- 28
- 55
- 121
Lions: Zebras: Monkeys = 1:2:8
(1+2+8) = 11
Must be a multiple of 11
22,55,121
Average population in town x was recorded at 22,455 during to years 2000-2010, inclusive . An error was later uncovered: the figure was erroneously recorded at 22,478 in 2009, but should have been 22,500. What’s the average?
There are 22 people not counted.
11 years
divide 22/11 = 2
the average should be 2 more
New average is 22,457
For any evenly spaced set?
the median and mean are equal
Standard Deviation
A measure of how spread out the numbers in a set are - the more spread out he numbers, the larger the standard deviation.
Standard Deviation is the average distance the data values are away from the mean.
What is a quartile?
defined as teh median of half of a set of data
Data Set: 1, 3, 4, 6,6
What is the standard deviation
Mean = 20/5 = 4
Distance from mean
4-1 = 3
4-3 = 1
4-4=0
6-4=2
6-4=2
–> 3+1+0+2+2 = 8/5 = 1.6
Standard Deviation of (22,22,22,22)
0
Variance?
(Standard Deviation)2
Standard Deviation is 3, what is the variance?
Var = 32
= 9
Set A (5,7,8,9,11,14,15,15,18,18)
If set A has a mean of 12 and a standard deviation of 4.4. How many numbers in set A are within in 1 unit of Standard Deviation from the mean?
12 + 4.4 = 16.4
16.4+4.4 = 20.8
12-4.4 = 7.6
7.6-4.4 = 3.2
Numbers between 7.6 - 16.4
6 numbers!
Within 1 Standard deviation of the mean
34% on both sides = 68%
Within 2 standard deviation of the mean
13.5 on both sides plus the exisiting 65% = 95%
Within 3 standard deviations of the mean
2.5% on both sides = 5%
+ 95% = 99%
Set A ( 2,4,7,9,4,5,9,4,9,2,11,2,3,4,3,4)
Find Q1, Q2, Q3
First reorder set
(2,2,2,3,3,4,4,4,4,4,5,7,9,9,9,11)
Find Median = 4+4 = 8/2 = 4
Median = Q2 = 4
Q1 = median of lesser numbers
Middle term = N+1 / 2
8+1 = 9/2 = 4.5 between 4th and 5th term.
Q1 = 3+3 = 6/2 = 3
Q3 = median of the greater numbers
16/2 = 8
Q3 = 8
For finding quartiles if the amount of numbers in the data set is odd?
The median is excluded from both lesser and greater numbers
If a score is in the 40th percentile in a large distubution….
the score is larger than 40% of the distrubution
Percentile of the lowest score is…
the 0th percentile
Percentile with the highest score…
the 99th percentile
Percentiles for normal distrubtions
M - 2d = 2.5th
M - d = 16th
m = 50th
M + d = 84th
M+2d = 99th
what is the probabilty that a month has an R in it?
4 months do not have an R - May, June, July, August.
12-4 = 8
8/12 = 2/3 = 66.67%
Approximate rules
“Or” means add
“and” means multiply
Complement of A
ex: Probablity that Denmark does not win the World Cup.
P(not A) = 1 - P(A)
With Replacement
Whatever selection is drawn put it back into deck and make the next choice from a full/ new shuffled deck.
Each new choice is indepedent of previous
Without replacement
pick a card, place it aside. each new choice is from a smaller deck.
Each new choice is made under a different condition. And the probability for all successive choices is NOT independent.
What is the prob of tossing 3 coins, and getting 3 heads?
Each flip is independent from each other.
1/2 x 1/2 x 1/2 = 1/8
What is the probabilty of rolling 2 six sided dies and getting snake eyes?
The rolls are indepedent of each other.
1/6 x 1/6 = 1/36
3 cards are selected with replacement. what is the probabilty of selecting 3 spades in a row.
With replacement, the selections are independent of each other.
1/4 x 1/4 x 1/4 = 1/64
Events A and Events B are independent. (Not Mutally exclusive) If P(A) = 0.6
P(B) = 0.8
What is the probabilty of A or B happening?
P(A or B) = P(A) + P(B) - P(A and B)
- 6 + .8 - (0.8 x 0.6)
- 4 - .48 = .92
A box has 5 green balls, 7 red balls. Picked with out replacement, what is the probabilty that the 1st 2 balls are both green.
total of 12 balls.
P(1=g) = 5/12
P(2=g/ 1=g) = 4/11
5/12 x 4/11 = 5/33
52 cards, probabilty of picking 3 hearts without replacement.
P(1) = 1/4
.25 x 52 = 13
P(2=H | 2=H) = 12/51 = 7/17
P(3=H | 2=H, 1=H) = 11/50
1/4 x 7/17 x 11/50 = 11/850
Formula for binomials
P= (nCr) x (Pr) x [(1-P)n-r]
P= Probabilty of successes
R = Number of successes
N = Number of trials
10 dice are rolled, what is the probability of rolling 2, 5s.
10c2 = (1/6)2 x (1-1/6)10-2
(1/6)2 x (5/6)8
When you see the words “at least” in a probability question…
use complementary rule.
P(not A) = 1 - P(A)
Roll one die, 8 times. What is the probabilty that we will roll at least one 6.
The complement of at least one 6 is zero 6s.
Probabilty of not getting a six in one roll = 5/6
Probablity of not getting a 6, 8 times = (5/6)8
P= 1-(5/6)8
Cars can be made with gas, hybrid, or electric power. They can be red, black, blue. and the transmission can be either automatic or standard.
how many different cars are possible?
3 types of power x 3 colours x 2 transmission
= 18 types of cars
There are 2 girls and 3 boys.
How many different ways can we seat kids in a row if the girls must sit on the end.
2 x 3 x 2 x 1 x 1
= 12
Lukas, Jakob, Misha, Aleksandr, and Uli are racing. If each person finishes a race and no one ties. How many different possibilities are there?
5 x 4 x 3 x 2 x 1 = 120 or 5!
How many ways can we rearrange 9 letters in the word WONDERFUL
9!
9 x 8 x 7 x6 x 5 x 4 x 3 x 2 x 1 = 362,880 ways
How many ways can 5 letters JKLMN be arranged so that L is not in the middle.
2 ways to solve this problem.
1) Adhere to restriction: 4 x 3 x 4 x 2 x 1 = 96
or
Ignore the restriction : 5 x 4 x 3 x 2 x 1 = 120
Break the restriction: 4 x 3 x 1 x 2 x 1 = 24
120-24 = 96
How many different ways can the word STUDY be rearranged
5 x 4 x 3 x 2 x 1 = 5! = 120
How many different ways can the word NANNY be rearranged.
Total number of letters! / number of letters that repeat !
5! / 3!
5 x 4 x 3 x 2 x 1 / 3 x 2 x 1 = 20
How many ways can the word MISSISSIPPI be rearranged.
M - 1
i - 4
S - 4
P - 2
11! / 4! 4! 2! = 34,650
Comination Formula
nCr = n! / r! (n-r)!
From your group of 5 friends, select 3 to join on a camping trip. In how many different ways can you select 3 friends.
5c3 = 5! / 3! (5-3)!
5! / 3! 2!
20/2 = 10
When to use cominations.. does the order matter?
No? then use combinations!
Yes? then use Fundamental counting principle
12 toppings, how many 3 toppings pizzas can be ordered?
Order does not matter.
12C3 = 12! / 3! (12-3)!
= 12! / 3! 9!
= 200
There are 8 people in a room, everyone shakes hands. How many handshakes.
Does the order matter? No.
8C2 = 8! / 2! (8-2)!
= 8! / 2! 6!
From a group of 10 members, 3 are randomly selected, one as chair, one as treasurer, and one as secretary. These 3 will form a “board” of committee. How many possible boards can be formed?
Does order matter? Yes!
10 x 8 x 9 = 720 ways
In a set of 10 million numbers, one percentile would represent what percent of the total number of terms?
A percentile ALWAYS represents one percent of a set of data.
= 1
Percentiles from mean and standard deviation
M-2d = 2.5th
M-1d = 16th
M = 50th
M+d = 84th
M+2d = 97.5th
150 Students, 75 take latin, 110 take spanish, 11 take neither. Number of students that take only latin?
Total = Group 1 + Group 2 - Both + Neither
150 = 75+110-B +11
B = 46
75-46 = 29
There is an 80% chance David will eat a healthy breakfast and a 25% chance that it will rain. If these events are independent, what is the probablity that David will eat a healthy breakfast OR it will rain.
P(A)+P(B) - P(A and B)
.8 x .25 = 0.2
1.05-.2 = .85
