Quant Flashcards
A number is divisible by 3 if __________________
A number is divisible by 3 if the sum of all the digits is divisible by 3
ex) 531
A number is divisible by 6 if __________________
A number is divisible by 6 if the number is divisible by both 2 and 3
cf. a number is divisible by 2 if the ones digit is 0, 2, 4, 6 or 8
ex) 174
A number is divisible by 9 if __________________
A number is divisible by 9 if the sum of all the digits is divisible by 9
ex) 333
The number of factors of 240
240 = 2^4 x 3^1 x 5^1
-> (4+1) x (1+1) x (1+1) = 20
240 has a total of 20 factors
1, 2, 3, 4 …….. 60, 80, 120, 240
x, y 의 최대공약수 x 최소공배수 = ?
The greatest common factor (GCF)
The lowest common multiple (LCM)
xy
√5 x √7
√ 35
√ 33 = √ a * √ b
√ 11 * √ 3
√0.0004
√ 2^2 / √ 10^4 = 2 / 100 = 0.02
|16x+14|=|8x+6|
x = ?
1) 16x + 14 = 8x +6 —–> x = -1
2) 16x + 14 = -(8x+6) —–> x = -5/6
x is 2% less than y
x = k y
k=?
x = 0.98 y
x is 60% less than y
x = k y
k=?
x = 0.4 y
the average of 7, 8, 9, 10, 11, 12
7 + 12 / 2 = 19 / 2 = 9.5
average in a set of consecutive integers =
highest # + lowest # / 2
1, 2 ,3 . . . 101
Median = ?
101+1 / 2 = 51
When n is odd # (101)
1, 2 ,3 . . . 100
Median = ?
100 / 2 = 50 and 51
50+51 / 2 = 50.5
A straight pipe 1 yard in length was marked off in fourths and also in thirds. If the pipe was then cut into separate pieces at each of these markings, which of the following gives all the different lengths of the pieces, in fractions of a yard?
Take the length of the pipe to be 12-meter long (the least common multiple of 3 and 4.)
In this case the branch would be cut at 3, 4, 6, 8, and 9 meters.
3/12=1/4
1/12
2/12=1/6
The ratio of the number of teachers to the number of students is the same in School District M and School District P.
mT:mS - pT:pS
A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?
4C2/2^4 = 6/16 = 3/8
ㅁㅁㅁㅁ
2 x 2 x 2 x2 = 16 total (boy or girl)
4C2 = 6
If 0 < r < 1 < s < 2.
rs must be less than 1?
No.
ex) r=0.99, s=1.99
숫자를 대입해서 풀 때, 범위 안에서 최대한 극단적인 숫자를 선택.
Ex) 0<x<1, 1/2를 넣어볼 수도 있지만, 0.99 or 0.01 을 넣어서 확인 사살.
5 < A < 10 and 2R < A. R의 범위는?
2R < A < 10
=> 2R < 10
=> R < 5
a+30/b+30 is greater than a/b?
Depends… (not sufficient)
If a=1 and b=2, 31/32 > 1/2
But if a=2 and b= 1, 32/31 < 2
(1) apple price is $10 less than orange price
(2) apple discount % is 2% greater than the orange discount %
which one is discounted by the greater dollar amount?
We don’t know
if apple is $10 and orange is $20
vs if apple is $990 and orange is 1000
10 * 12% < 20 * 10%
990 * 12% > 1000 * 10%
In planning for a car trip, Joan estimated both the distance of the trip, in miles, and her average speed, in miles per hour. She accurately divided her estimated distance by her estimated average speed to obtain an estimate for the time, in hours, that the trip would take. Was her estimate within 0.5 hours of the actual time that the trip took?
(1) Joan’s estimate for the distance was within 5 miles of the actual distance.
(2) Joan’s estimate for her average speed was within 10 miles per hour of her actual average speed.
The only thing we need to prove is ?
|ET-AT| < 0.5 hours?
if ET = AT, |ET-AT| = 0.
Therefore, only thing we need to prove is |ET-AT| > 0.5 hours.
T = D / S
ET = 20 / 30 = 1.5
AT = 15 / 40 = 0.375 (to maximize, AT should be minimized as possible)
ET - AT > 1
If x^4 + y^4 = 100, then the greatest possible value of x is between
y=0
x^4 = 100
x^2 = 10
√x = 3.xx
between 3 and 4
There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?
8C2 = 8 * 7 / 2 = 28
What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?
(A) 420
(B) 840
(C) 1,260
(D) 2,520
(E) 5,040
The integer should be divisible
by: 2, 3, 4(=2^2), 5, 6(=2 * 3), and 7.
LCM= 2^2 * 3 * 5 * 7 = 420
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
30/3 = 10
30/3^2 = 3
30/3^3 = 1
10 + 3 + 1 = 14
A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?
(A) $ 1
(B) $ 2
(C) $ 3
(D) $ 4
(E) $12
P * N = 120
(P+1) * (N-10) = 120
“When substituting answer choices, it’s good to start with the middle value”
(C) $ 3 - P
what equation has 1 + √2 as one of its roots?
x^2 -2x -1 = 0
x = a and b
-> x^2 - (a+b) x + ab = 0
The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x,y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w ?
1 + 3C1 + 3C2 + 1 = 8
The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?
5!/2! - 4! = 36
If n = 3^8 - 2^8, which of the following is NOT a factor of n?
(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
apply a^2 - b^2 = (a-b)(a+b)
If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A. 6
B. 12
C. 24
D. 36
E. 48
n^2 = 72 * 몫
= 6^2 * 2 * 몫
n = 6 * √ 2 * 몫
n의 최소값 = 12
n의 값은 무수히 커질 수 있다. n의 최소값은 12. Must는 최소값을 나타낸다.
Explain steps on using a calculator to solve the following:
2570 * 12 - 1125 * 4
2570 * 12 = value M+
1125 * 4 = value M-
And then press MRC once.
(The display will show 26,340, the final result.)
Press MRC twice to clear any previous memory.
홀 + 홀 =
홀 + 짝 =
짝 + 짝 =
홀 x 홀 =
홀 x 짝 =
짝 x 짝 =
홀 + 홀 = 짝
홀 + 짝 = 홀
짝 + 짝 = 짝
홀 x 홀 = 홀
홀 x 짝 = 짝
짝 x 짝 = 짝
Did most respondents who indicated they somewhat agreed that class sizes should be reduced also indicate that they somewhat agreed that music classes should be added?
What does “also” indicate?
교집합
A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?
1) 홀 + 홀 + 홀 -> (1/2)^3
2) 홀 + 짝 + 짝 -> (1/2)^3 * 3
1/8 + 3/8 = 4/8 = 1/2
A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer?
First, draw the box!
ㅁ ㅁ ㅁ
P S T
Total: 10 * 9 * 8
Harry=S: 9 * 1 * 8
Harry=T: 9 * 8 * 1
72 * 2 / 10 * 9 * 8 =1 / 5
15 is what percent greater than 3?
= percentage increase from 3 to 15
= 15 - 3 / 3 * 100%
= 12 / 3 * 100%
= 400%
A car travels from Mayville to Rome at an average speed of 30 miles per hour and returns immediately along the same route at an average speed of 40 miles per hour. Of the following, which is closest to the average speed, in miles per hour, for the round trip?
Set the distance = 120 miles
If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people who are married to each other, how many such committees are possible?
5C3 * 2^3 = 10 * 8 = 80
Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years?
J = B + 14
J + 10 = (B + 24) x 2 (X)
J +10 = (B + 10) x 2 (O)
2^10 = ?
1024
is the following correct?
2^12x4 = 2^12 x 12^4
= 2^12 x 12^4 = 2^16 (2^a x 2^b = 2^a+b)
= 2^48 = 2 ^12 * 2^36 (O)
$100, r% compounded annually.
What’s 원금 + 이자 in 2 years after investment?
$100 ( 1 + r/100 )^2
3/18 = ?
1/6
If x is an integer and y = 3x + 2, which of the following CANNOT be a divisor of y?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
(C) 6
y = 3x + 2 = not a multiple of 3 and therefore not a multiple of 6
Tom, Jane, and Sue each purchased a new house. The average (arithmetic mean) price of the three houses was $120,000. What was the median price of the three houses?
(A) The price of Jane’s house was $120,000.
Is (A) sufficient to know the median price?
Is (A) sufficient to know the median price? YES
Tom + Sue = 24
ex) 10, 12, 14
ex) 12, 12, 12
If it is true that x > -2 and x < 7, which of the following must be true?
A. x > 2
B. x > -7
C. x< 2
D. -7 < x < 2
E. none of the above.
B. x > -7
Is the median of set S even?
(1) Set S is composed of consecutive odd integers
(2) The mean of set S is even.
Is the median of set S even?
(1) Set S is composed of consecutive odd integers
{1, 3, 5, 7} –> median = 3+5/2 = 4 = even.
or {1, 3, 5} –> median = 3 = odd.
(2) The mean of set S is even.
{1, 1, 4}
(1) + (2) : from (1) median = mean and from (2) mean = even. Therefore, median = mean = even
How much greater is the average (arithmetic mean) of the integers in S than the average of the integers in T?
T = 2, 4, 6, 8, 10
S = 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
For any evenly spaced set, median = mean = the average of the first and the last terms
T = 2, 4, 6, 8, 10
S = 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
9+27/2 = 36/2 = 18
For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab ?
I. 24
II. 30
III. 36
III only
The only way the remainder when a is divided by b is equal to the remainder when b is divided by a is a = b since then the remainder in both cases is 0.
There are 10 solid-colored balls in a box, including 1 green ball and 1 yellow ball. If 3 of the balls in the box are to be chosen at random, without replacement, what is the probability that the 3 balls chosen will include the green ball but not the yellow ball?
without replacement
- 8C2 / 10C3
- 8x7x3 /10x9x8
For any numbers a and b if a + b - ab =0,
what CANNOT be a value of b?
a - ab = - b
a ( 1 - b) = - b
a = -b / (1-b)
therefore, “1” can not be a value of b
original value: x
what’s the value when x is increased by 50% / 100% / 200% / 250% ?
increased by 50% : 1.5x
increased by 100% : x + x = 2x
increased by 200% : x + 2x = 3x
increased by 250% : x + 2.5x = 3.5x
original value: x
What’s the valuse when x is decreased by 50% / 100% / 200% / 250% ?
decreased by 50% : 0.5x
decreased by 100% : x - x = 0
decreased by 200% : x - 2x = - x
decreased by 250% : x - 2.5x = 1.5x
2 x 4 x 6 …. X 100 = (2x1) x (2x2) …. x (2x50) = ?
= 2^50 x 50! (O)
= 2 x 50! (X)
Any two consecutive numbers will only share the factor of ____
“1”
ex) 99, 100
11, 12
T = { 1/2, -1/4, 1/8, -1/16 … }
What is the sum of T?
The sum of T = 1/2 ÷ 3/2 = 1/3
first term: 1/2
common ratio: - 1/2
The sum of “infinite geometric progression” with common ratio |r|<1,
b ÷ 1 - r, where b is first term.
Two thirds of the roads from A to B are at least 5 miles long, and 1/4 of the roads from B to C are at least 5 miles long. If you randomly pick a road from A to B and then randomly pick a road from B to C, what is the probability that at least one of the roads you pick is at least 5 miles long?
1) 4 + 4 + 1 / 12 = 9 / 12
2) 1 - none of the roads you pick will be at least 5 miles long = 1 - 1/3 * 3/4 = 1 - 3 / 12 = 9 / 12
What is the largest value of non negative integer N for which 10^N is a factor of 50!?
50/(5^1)+50/(5^2) =10+2 =12
Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address?
Probability that first letter in the right envelope= 1/4
Probability that second letter in wrong envelope = 2/3
Probability that third letter in wrong envelope = 1/2
Probability that forth letter in wrong envelope = 1
1/4 * 2/3 * 1/2 = 1/12
1/12 * 4 = 1/3
is 0 divisible by 5?
YES, because 0 = 5 * K, when K = 0
0 is divisible by EVERY integer.
미지수 2개, 식 1개
두 미지수 값을 구할 수 있음?
It Depends
미지수가 2개여도 식 하나로 미지수 값을 구할 수 있음.
Ex) 23x + 21y = 130
x = 2, y = 6
미지수 2개, 식 2개
두 미지수 값을 구할 수 있음?
It Depends
미지수 2개, 식 2개여도 각 미지수의 값을 못 구하는 경우가 있음.
ex) 식 2개가 결국 같은 정보를 줄 때. 가능하면 확인 사살
Six students are equally divided into 3 groups, then, the three groups were assigned to three different topics. How many different arrangements are possible?
Six students are equally divided into 3 groups: (6c2 * 4c2 * 2c2)/3!
reason we are dividing by 3! is, to eliminate this {AB, CD, EF} {AB, EF,CD}{CD,AB,EF}{CD,EF,AB}{EF,AB,CD}{EF,CD,AB}
The three groups were assigned to three different topics: 3!
** = 6c2 * 4c2 * 2c2 = 90**
I x I > a, 절대값을 풀면?
x > a or x < -a
The probability that a visitor at the mall buys a pack of candy is 30%. If three visitors come to the mall today, what is the probability that exactly two visitors will buy a pack of candy?
3C2 * 0.3 * 0.3 * 0.7
6 hours 40 mins 을 분수로 표현하시오
= 6 * 2/3 (X)
= 6 + 2/3 (O)
