a number is divisible by 3 if
the sum of its digits is divisible by 3
a number is divisible by 4 if
its last two digits compose a two-digit number that is itself divisible by 4
a number is divisible by 9 if
the sum of its digits is divisible by 9
a number is divisible by 11 if
the difference between the sum of its odd-placed digits and the sum of its even-placed digits is divisible by 11
average value =
sum of values / number of values
sum of values =
average value X number of values
John travels x miles in a hours and y miles in b hours. What is is his average speed?
(x + y) miles / (a + b) hours
How to handle complex rate problems
make a table
row labels = rate, time, and distance
column labels = a to b, b to c, and entire trip
CANNOT add rate
rate X time = distance
combined work total time formula for 2 people/machines
total time = AB / A + B
Bob can paint a room in 3 hours, and George can paint the same room in 2 hours. How many hours does it take Bob and George to paint the room if they work together but independently?
(3 X 2) / ( 3+ 2)
combined total work formula for more than 2 people/machines
1/total time = 1/A + 1/B + 1/C + ... + 1/N
How many minutes is 1/5 of an hour?
formula for overlapping sets
total = group 1 + group 2 - both + neither
overlapping sets approach if there is no one in the neither group
overlapping sets approach for complicated problems
draw a chart
row labels = in group 1, not in group 1, and total
column labels = in group 2, not in group 2, and total
if selection is unordered, then it's a
if the selection is ordered, it is a
the combinations formula is used when
solving for the number of k unordered selections one can make from a group of n items
nCk = n! / k!(n-k)!
A company is selecting 4 members of its board of directions to sit on a committee. If the board has 9 members, any of whom may serve, how many different selection of members could be made?
9C4 = 9! / 4!(9-4)!
County X holds an annual math competitions, where each county high school sends a team of 4 students. If School A has 6 boys and 7 girls whose math grades qualify them to be on their school's team, and competition rules stipulate that the team must consists of 2 boys and 2 girls, how many different teams might school A send to the competition?
6C2 and 7C2
(6! / 2!4!) X (7! / 2!5!)
number of permutations of n items
How many ways are there to rearrange the letters in the word ASCENT?
in combinations, permutations, and probability, AND translates to ________. OR translates to ________.
and = multiply
or = add
How many ways are there to rearrange the letters in the word ASSETS?
6! / 3!
How many ways are there to rearrange the letters in the word REASSESS?
8! / 4!2!
If a fair coin is flipped 3 times, what is the probability of getting at least one tail?
subtract the probability of the undesired outcomes from the total
1 - 1/2 x 1/2 x 1/2
If a fair coin is flipped 5 times, what is the probability of getting exactly 3 heads?
solve for the probability of one possible desired outcome, then multiply by all the permutations of that outcome
HHHTT = 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32
HHHTT = 5! / 3!2! = 10
1/32 x 10
# of desired outcomes / total number of possible outcomes
bisecting an equilateral triangle creates __________.
two 30-60-90 triangles
how to find the longest side of a triangle
longest side is across from largest angle
circumference of circle
pi d or 2 pi r
arc length where angle opens up at center
(n/360) x circumference
arc length where angle opens up at edge of circle
(n/180) x circumference
area of circle
area of sector
(n/360) x area of circle
volume of a rectangular solid
SA of rectangular solid
2lw x 2lh x 2wh
rectangular solids diagonal
diagonal^2 = l^2 + w^2 + h^2
volume of cube
SA of cube
volume of cylinder
pi r^2 h
lateral SA of cylinder
2 pi r h
total SA of cylinder
2 pi r^2 2 pi r h