Quant Flashcards
-6*(-3+(-5))
(-6-3)+(-6-5)=18+30=48
Multiply two negatives
Always positive
What is the sum of the greatest negative integer and the smallest positive integer
-1 +1 =0
What is an integer
Whole number. No fractions
If 7 is a factor of 21, then 21 is a __ of 7
Multiple
2^6
(22) * (22) * (22) = 444 = 164 = 64
What is the 3rd root of 27?
3 to the what power = 27? The answer is 3.
What does PEMDAS stand for
Parentheses, Exponents, Multiply or Divide, Addition or Subtraction
“The quantity three plus two” signals
parentheses (3+2)
8ba + ab^2 - 5ab + ab^2 - 2ba^2
3ab + 2ab^2 -2a^2b
Factor a negative x out of -2x^3 + 5x^2 + 3x
-x(2x^2 - 5x - 3)
Factor the expression 4x^2 + 3xy - yx + 6x
2x(2x + y + 3)
Is 6,750 divisible by 18?
Yes. Ends in 0, it is even, so divisible by 2. Digits of 6750 sum to 18, and 18 is divisible by 9. So 6750 is also divisible by 9.
What are positive multiples of 18 that are less than 60?
18, 36, 54
What is a prime number
Divisible by exactly two numbers: themselves and 1:
2, 3, 5, 7, 11, 13, 17, 19….
x = 39. What are prime factors, factors, of x?
Prime = 3, 13; factors = 1, 3, 13, 39
What are two greatest odd factors of 90
15, 45
y^5 * y^3
y^8
-3^a / -3^2
-3^a-2
(5^2)^x
5^2x
Filmmaker Hayao is more famous than any filmmaker in the history of the art.
“any” should be “any other”
It is possible that the earthquake may have caused the building’s collapse.
“possible” and “may” are redundant
5^3
555
10^3
1,000
2^45 positive or negative
Positive because positive times positive is positive no matter how many time syou multiply
7^0
Any number to the 0 power = 1
Divide terms with same base a^5/a^3
SUBTRACT a^2
Multiply terms with the same base a^5*a^3
ADD a^8
d^7/d^8
d^-1
a^-2
1/a^-2
Negative exponent on top of a fraction, move to bottom -OR- negative exponent on bottom to top
Becomes positive!
3^-3 Raise anything to a negative power
1/3^3 = 1/27; get 1 over that same thing but sign switches to positive
When you raise something that already has an exponent to another power,
multiply the two exponents together.
(3/4)^-2
3^-2 / 4^-2 = 4^2 / 3^2 = 16/9
If you apply an exponent to a product
then you apply the exponent to each factor in the product
If you apply an exponent to an entire fraction
then you apply the exponent separately to the top and the bottom
a^3b^3 =
(ab)^3
x^5 + x^3 = x^3*x^2 + x^3 =
x^3(x^2 +1)
Square root of 2
1.4 (2.14 = valentines)
Square root of 3
1.7 (3.17 = st pattys)
Square root of 70 between which two consecutive integers (approximate)
Nearby squares are 64 and 81, so must be between 8 and 9.
Square root of 0.5 is greater/less than
SQuare root of a number between 0 and 1 is greater than the original.
Square root of 1, 0
1, 0
square root of 27 * square root of 27
= sq root of 27 ^ 2 = 27
sq root of 7^22
= (7^22)^(1/2) = 7^(22/2) = 7^11
Simplify 64^(2/3)
Take cube root of 64, squared = 4 squared = 16
Square root of 8 * sq root of 2
sq root of 8*2 = sq root of 16 = 4
Sq root of 27 divided by sq root of 3
sq root of 9 = 3
Sq root of 50
Sq root of (25*2) = sq root of 25 * sq root of 2 = 5 * sq root of 2
Sq root of 360
Sq root of 36 * sq root of 10 = 6*sq root of 10
Sq root of 12
Sq root of (2^2 * 3) = 2 * (sq root of 3)
Sq root of 96
Break down into sq root of factors… 4*sq root of 6
9a + 4b / 3ab
Split into fractions: 9a/3ab + 4b/3ab = 3/b + 4/ca
b+6/6 - 3+b/6
Common denoms of 6, write out with negative sign: b+6-(3+b) / 6 = b+6-3-b / 6 = 3 / 6 = 1/2
Simplify 8 / (2-2/3)
8/ (6/3-2/3) = 8 / (4/3) = 8*(3/4) = 24/4 = 6
20 is 16% of what number?
20=16%x. Turn 16% into fraction = 20(16/100)x. 20(4/25)x. Multiply 4/25 * 25/4(20) = x. 125 = x.
Inclusive of 50 and 60
includes 50 and 60 in the range
Multiplying the numerator of a positive fraction by a number greater than 1…
INCREASES the numerator, which increases the value of the fraction
Dividing a positive number by a positive fraction
INCREASES the value of the number
Simplify 6z + 5z =
11z Combine like terms
Simplify
Find a common denominator
Simplify 2ab + 4b = 2b(a+2)
Pull out a common factor
Simplify 5y^3 / 25y
y^2/5 cancel common factors
Solve one-variable expression 3x+5 = 26
Follow PEMDAS in reverse: subtract 5 from both sides. Divide both sides by 3
x+y = 9
2x=5y+4
Line up: x+y=9
2x-5y=4
Multiply the first equation by -2 = -2x-2y=-18
Add equations together = -7y=14, y=2
What is value of x/y?
(1) x+y/y = 3
(2) y=4
First, notice the question asks for a combo. Manipulate any given info to try to match the combo. Keep the denominator where it is. Split to simplify x/y + y/y = 3. x/y+1 = 3. x/y = 2. Statement 1 sufficient = A or D. Statement 2 provides no information about x, so it is not sufficient = A.
If 2x+y=18 and x+2y=12, what is value of x+y?
Look for the R&R
relationship between variables (is positive in this case) and ratio of coefficients - the values in front of the variables - (is 1:1 in this case, one x and one y). How to combine to give addition relationship and 1:1 ration between variables? 2x+y=18 and x+2y=12 combine to make 3x+3y=30. Divide both sides by 3 = x+y=10.
|x| = 5
Absolute value. Refers to the positive value of the expression within the absolute value brackets. So, x could = +5 or -5. Absolute value generally have two solutions, positive or negative.
Solve for w, given that 12 + |w-4| = 30
First, isolate the absolute value expression by subtracting 12 on both sides |w-4|=18. Solve for two cases - when w is positive, w=12. When w is negative,-(w - 4) = 18, w-4 = -18, w = -14.
How to calculate percentage increase?
First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100. % increase = Increase ÷ Original Number × 100.
What is 10% of 150?
10% * 150 = Y. 0.10 * 150 = Y. 0.10 * 150 = 15
What percent of 60 is 12?
12/60 = P%. 12/60 = 0.20.
25 is 20% of what number?
25/20% = X. 25/0.20 = X. X = 125
Fractions: pizzas/people
As pizzas increases, bigger value. As people increases, smaller value.
207/23 =
230-23 / 23 = 10-9 = 1
96/8 =
80+16 / 8 = 10 + 2 = 12
143/13 =
130+13 / 13 = 10 + 1 = 11
Is 151/7?
Nearest multiple of 7 = 140+7+4/7 = 20+1+4/7 = 21.5ish
13(25) = think quarters
3 dollars 25 cents = 325
675/25 = think quarters
24 + 3 = 27 quarters
What is each form “out of” Fractions, Decimals, Percents, Ratios?
Fractions are out of the denominator
Decimals are out of 1 (the whole)
Percents are out of 100 (per hundred)
Ratios are out of the second term in the ratio.
I ate 5/4 boxes of cereal (I ate more than one box)
= 1.25 boxes
=125% of one box of cereal
=the ratio of what I ate to a whole box of cereal was 5 to 4
To convert a decimal to a percent
1.7 =
.0005 =
Move decimal point to the right two places
170%
.05%
To convert a percent to a decimal
225%
Move two places to the left
2.25%
Simplify .4 or .75
=4/10 = 2/5 or 75/100 = 3/4
Simplify 2.5%
0.025 = 25/1000 = 1/40
Simplify 4%
4/100 = 1/25
Simplify 3.6%
0.036 = 36/1000 = 9/250
Convert 1/10 to a decimal
Memorize 1/10 = 0.1
Convert 1/50 to a decimal
Multiply top and bottom so the denominator becomes a power of 10, but only if the denominator contains only 2’s and 5’s. 1/50=12/502 = 2/100 = 0.2
Convert 7/8 to a decimal
Do long division! 7.000/8
1/10 Fraction Decimal Percent 2/10 = 1/5 3/10 4/10 = 2/5 5/10 = 1/2 6/10 = 3/5 7/10 8/10 = 4/5 9/10 11/10 12/10
- 1 10%
- 2 20%
- 3 30%
- 4 40%
- 5 50%
- 6 60%
- 7 70%
- 8 80%
- 9 90%
- 1 110%
- 2 120%
1/8 Fraction Decimal Percent 2/8 = 1/4 3/8 4/8 = 1/2 5/8 6/8 = 3/4 7/8 10/8 = 5/4 12/8 = 3/2
1/20
0.125 12.5% .25 25% .375 37.5% .5 50% .625 62.5% .75 75% .875 87.5% 1.25 125% 1.5 150%
0.05 5%
One-tenth = 1/10 = what power?
10^-1
You can write any decimal as a fraction with a power of 10 in the denominator, or as a product with a power of 10. For example, what is 0.03 =
0.03 = 3/100 = 3/(10^2)
43.8723 * 10^3 = move decimal…
3 places to the right.
57,234 / 10^4 = move decimal…
4 places to the left.
Solve 7 / 10^-2
= 7 * 10^2 = 700
Solve 6 * 10^-2
= 6 / 10^2 = 0.06
Multiply two decimals
0.6 * 1.1
Ignore decimals, multiply integers, place decimal point by counting the original number of decimals.
Multiply small decimal and big number 50,000 * 0.007
Trade decimal places from teh big number to the small one = 50 * 7 = 350
Divide two decimals by moving the decimal points in the same direction to eliminate decimals as far as you can. 3.39 / 0.003
12.6/0.3
3,390 / 3 = 1,130
12.6/0.3 = 126/3 = 42
What percent of 125 is 25?
= y/100 * 125 = 25
simplify to take 5 out of both
=y/5 * 5 = 25
16 is 2% of what number?
Change 2% to 0.02 16 = 0.02 * y 16/0.02 = y Move decimal = 1600/2 = 800 And that makes sense because 2% is a small percent of a large number.
What percent of =
y/100
30% of =
30%y
21 is 30% of what number?
21 = 30/100y (100/30)*21 = y (10/3)*21 = y (10*21) / 3 = y 10*7 = y
You have $200 in a bank account, and deposit an additional $30, by what % did the value of the account increase?
Percent change (as % of original) = change in value / original value = 30/200 = 15/100 = 15%
You have $200 in a bank account, and deposit an additional $30. What does the new value as a percent of the original value?
New percent (as % of original) = new value / original value =230/200 = get denom to 100 by dividing by 2 = 115/100 = 115%
You have $200, take out 40%. How much money remains?
Original percent (% of original) + change percent (% of original) = new percent (% of original)
100% + -40% = 60% , now find that value
New percent (as % of original) = new value / original value
60% = y/200
60/100 = y/200
60/100(200) = y
60*2 = 120
New percent of original percent
new % = new value / original value
30 is what percent more than 20?
Find the increase from the original number 20 to the new number 30. What % of 20 do you need to take and add to 20 to get to 30? Different of 10. What percent of the original number does 10 represent?
20 = 100%
10 = 50% , so 30 is 50% more than 20.
% change = 10/20p = 50/100 = 50%
Cars:Trucks
Cars:Total vehicles
Trucks: Total vehicles
Part to part
part to whole
part to whole
A bouquet contains 5:3 white to red roses. What percent of all roses are red?
Part to part to whole ratio
Ratio = 5 white + 3 red = 8 total
Red to total = 3/8 = 37.5%
If 20% of animals in a zoo are skunks, what is the ratio of non-skunk animals to skunks in the zoo?
20% skunks + 80% non-skunks = 100%
Ratio = 1 skunks : 4 non-skunks
Ratio of non-skunks to skunks is 4:1
Percent change formula
% change = change / original
56 increases by 14, what is percent change?
=14 / 56 = 1/4 = 25%
If 75 reduced by x percent is 54, what is value of x?
Use logic to estimate %. Reducing by 9% would only reduce by about ~7. Reducing by 72% would reduce by almost all of it. Therefore answer is 28%.
120% of 30% of 400?
120/100 * 30/100 * 400 = there are four zeros on top and four zeros on bottom, cancel everything = 1234 = 144
To simplify for a square root on one side of the equation
Square both sides of an equation
“The cost is marked up by “m” dollars:
c + m
The sum of the three funds combined
a+b+c
The positive difference between q and r, if q > r
q - r
Profit
revenue minus cost = P = R - C
Quotient, Per, Ratio, Proportion
division
“n” persons have “x” beads each. Total beads = ?
total beads = nx
A is 4 times the length of B (show as ratio too)
A = 4B or A : B = 4:1
The ratio of x to y
x/y = x:y
Kelly is three times as old as Bill. In 5 yrs, Kelly will be twice as old as Bill will be. How old is Bill?
K = 3B K+5 = 2(B+5) (3B)+5 = 2(B+%) 3B + 5 = 2B + 10 B= 5 Bill is 5 years old
Average of a and b
(a+b) / 2
Total revenue =
Price * Quantity
Total cost to consumer =
Price * Quantity
Profit =
Profit = revenues - costs
Encounter a money relationship ,”The cost of 8 watches is $1,200…”
Write Total cost = price * quantity, including more items or a fixed cost as necessary.
Add or subtract quantities with units
Ensure the units are the same - convert first if necessary
Multiply quantities with units
Multiply the units, canceling as appropriate. 10 bagels/hr * 3 hr = 30 bagels
Rate =
Distance / Time
Chipo took 4 hours to travel 60 miles
60 miles / 4 hours = 15 miles per hour
Average miles per hour =
Total miles / Total hours
Part + Part = Whole
Miles for first part of a trip + miles for second part = total miles
Hours for first part of a trip + hours for second part = total hours
Average rate for the journey
Total Distance / Total Time
If you see a rate problem
Use Rate * Time = Distance
Rate * Time = Work. Jay builds a chair in 3 hours; Kay builds a chair in 5 hours. How long will it take them together to build 8 chairs?
The work is 8 chairs. THe time is unknown. 8/15 * T = 8. T = 8(15/8). T = 15. It takes Jay and Kay together 15 hours to build 8 chairs.
If a < 0 and b < c which of following is true: ab < c ac>b ab>0 ac<0 ab>ac
ab>ac “must” = test cases on this problem by plugging in real values.
If k is an integer and 0.02468 x 10^k is >10,000, what is the least possible value of k?
6 – multiply shifts the decimal to the right.
What is the “units digit” in 64?
The ones digit. In 64, the units digit = 4
How to get exponent out of denominator?
x^(a+b) / x^b
=x^(a+b)-(b) = x^a
Square root = ? To remove a square root on one side of the equation, ?
x^2
Square the other side of the equation.
Factor x^2 + 11x + 18
Find a factor pair of 18 that sums to 11 = 9 & 2:
x+9)(x+2
Factor x^2 - 8x + 12
Find a factor pair of 12 that sums to 8, then make both numbers negative = (x-6)(x-2)
Factor x^2 + 6x - 16
Find a factor pair of 16 that differs by 6 then make the bigger number positive = (x+8)(x-2)
Factor x^2 - 5x - 14
Find a factor pair of 14 that differs by 5, then make the bigger number negative = (x-7)(x+2)
Factor -2x^2 + 16x - 24
Pull -2 out from all terms first, then factor normally =
-2(x^2 - 8x + 12) = -2(x-6)(x-2)
Solve x^2 + 11x = -18
Rearrange to make one side 0, factor the quadrati side, then set factors equal to 0 =
x^2 + 11x + 18 = 0
(x+9)(x+2) = 0
x = -9 or -2
Solve x^2 = 25
Take the positive and negative square roots of booth sides
x = -5 or 5
Solve (z-7)^2
Take positive and negative square roots of both sides
=square root of (z-7)^2 = z-7
=square root of 225 = 15 or =15
z = 22 or =8
Solve x^3 = x
Set equation equal to 0, factor, and set factors equal to 0 = x^3-x = 0 x(x^2-1) = 0 x(x+1)(x-1) = 0 x = 0 or x+1 = 0 or x-1 = 0 x = 0, -1, 1
Multiply or divide both sides of an inequality by a negative number
Flip the inequality sign:
45w
If you have a variable inside absolute value signs
Drop the absolute value and set up two equations, one positive and one negative
average =
can be shown as
sum divided by # of terms
S/n = A
OR can be shown as
A * n = S
Sam earned a $2,000 commission on a big sale, raising his average commission by $100. If Sam’s new average commission is $900, how many sales has he made?
Keep track of two average formulas in a table. New avg commission is $900, which is $100 higher than old average, so old average was 900-100 = $800.
Old: 800 * n = 800n This sale: 2000*1 = 2000 New total: 900 * (n+1) = 900 (n+1) 800n + 2,000 = 900 (n+1) 1,100 = 100n; 11 = n and n+1 = 12
Standard deviation?
When small?
When large?
how far from the average the data points typically fall.
Small = set is clustered closely around the average
Large = set has some points appearing far from the mean
Standard deviation = 0
All numbers in the set are identifcal
Rate x Time =
Distance or Work
Average speed =
Total Distance / Total Time If an object moves over the same distance but at different rates each time, the slower trip is weighted more heavily .
Work / Time =
Rate
When two machines are both contributing to the same job
ADD the rates
If one machine is taking away from a job,
SUBTRACT the rate of that machine.
If “a” is divided by 7 or 18, an integer results. Is a/42 an integer?
Prime factors include all the factors of those two numbers: 2, 3, 3, 7. Therefore, any integer that can be constructed as a product of any of these prime factors is also a factor of a. 42 = 2 * 3 * 7, all of which are in the prime box, so 42 is also a factor of a.
If 7 is a factor of n and 7 is a factor of p, is n+p divisible by 7?
If two numbers are both multiples of the same number, then their sum is also a multiple of that number. Since n and p share the common factor 7, the sum of n and p must also be divisible by 7.
If 6 is a divisor of r and r is a factor of s, is 6 a factor of s?
Factor foundation rule: if 6 is a factor of r and r is a factor of s, then 6 is a factor of s.
If s is a multiple of 12 and t is a multiple of 12, is 7s + 5t a multiple of 12?
If s is a multiple of 12, then so is 7s. If t is a multiple of 12, then so is 5t. Since 7s and 5t are both multiples of 12, then their sum (7s+5t) is also a multiple of 12.
Inclusive integers formula
How many integers are there from 14 to 765, inclusive?
Last - First + 1
=765-14+1 = 752
How many multiples of 7 are there between 10 and 80?
(last-first)/increment + 1
find the least multiple of 7 and the greatest.
77-14/7 + 1 = 63/7 + 1 = 9 + 1 = 10
The mean and median are equal to eachother when?
Evenly spaced sets. 4, 8, 12, 16, 20 = mean and median is 12.
What is mean of 4,8,12,16,20,24?
Average of the two middle numbers - 12 and 16 - is 14.
For all evenly spaced sets, the average =
First + Last / 2
What is the sum of all integers from 20 to 50, inclusive?
Sum of evenly spaced set = average * number of terms
=(20+50)/2 = 35 Number of terms = 50-20+1 = 31. Sum = 35*31 = 1,085
x^5/x^2 =
x^3
x^3x^5=
x^3+5 = x^8
(xy)^2
x^2y^2
(x^2)^3
X^6
x^0
1
If exponents are the same…
can multiply and divide the bases directly
To add/subtract exponents
pull out common factor
Adding or subtracting from the exponents?
Just multiply or divide the base that many times
Matching up primes to powers can tell us…
the powers even when we have unknowns elsewhere in the equation.
2^x 3^z = 2^ 5 3^ y
x = 5
