Diagnostic test 1
www.mathrevolution.com www.mathrevolution.com
1/ (integer) If a, b, c are positive integers, 33=abc, and a
B. 3
C. 11
D. -3
E. -33
==> From 33=(-33)(-1)(1), the smallest possible value of a is -33. Thus, the correct answer is E.
1 2/ *(inequality) If a is an integer greater than 4 but less than 21 and b is an integer greater than 14 but less than 31, what is the range of a/b? A. 2/3
B. 1/2
C. 5/6
D. 1
E. 7/6
==> Since a and b are integers, 4
3/ (ratio) All data in a certain school are recorded either on computer or paper. Out of inaccurate records on the computer, data on paper make up 80%. And, out of the inaccurate data on paper, records on computer account for 48%. Out of all data, inaccurate data, both on computer and paper, make up 12%. What is the percent of accurate data among all data without regarding the record method-either on computer or paper? A. 45% ==>
B. 56%
80C 48P 12%
2 E 3 C
D. 86%
Computer(inaccurate) Paper(inaccurate) 20C
1 E
C. 72%
52P
E. 90%
Diagnostic test 1
www.mathrevolution.com www.mathrevolution.com
From the above diagram, we can see 80C=48P=12%. Then, 20C=12P=3%, 12P=3%, 4P=1%. Looking back at the diagram: Computer(inaccurate) Paper(inaccurate) 20C
80C
3%
48P 12%
52P =13(4P)=13% Accurate
2
Hence, inaccurate data=3%+12%+13%=28%. Accurate data=100%-28%=72%. Hence, the correct answer is C.
4/ (integer) If n is greater than 20, what number is closest to n 100-n90? A. n90
B. n100
C. n99
D. n190
E. n10
==> n100-n90=n90(n10-1)≒n90(n10)=n90+10=n100. Hence, the correct answer is B.
5/ (inequality) Which of x, x 2, and √ x is the greatest? 1) x>0 2) x<1 ==> For inequality, if x>1, 1/x<1< √ x
4 B 5 C
Diagnostic test 1
www.mathrevolution.com
6/ *(velocity) Tom drove from X to Y and then came back to X from Y afterwards. He took different routes back and forth. It took him 40 miles per hour to travel from X to Y. Meanwhile, it took him 50 miles per hour to travel from Y to X. What was the average speed for his entire travel? 1) The distance of the return trip was 4/5 of the entire distance. 2) The distance from the return trip was 160 miles. 3 ==> This is both an average speed question and common mistake type 4(A) question, which has 4 variables and 2 equations in the original condition. In other words, if a person travels a certain distance in 2 different phases, we have 4 variables (from 40t 1=d1, 50t2=d2, we get d1, d2, t1, t2). Also, there are 2 equations (40t1=d1, 50t2=d2). If the question asks average speed, A or B becomes the answer. However, if one of conditions is number and if the other is ratio, the condition with ratio will be the correct answer. Since the condition 1) is ratio, 4/5, it is the correct answer. In other words, if we treat the entire distance as 5d miles, the distance coming back is 5d(4/5)=4d miles. The distance going to the destination is 5d-4d=d miles. From 40t 1=d, we get t1=d/40. From 50t 2=4d, we get t 2=4d/50=2d/25. Then, the average speed=5d/(d/40+2d/25)=1,000/21. The condition is sufficient and the correct answer is A.
6 A
Diagnostic test 1
www.mathrevolution.com
7/ (set) In school V, all students must take at least one of Mathematics, Physics, or Biology. If none of the students took 3 classes, the number of students who took Mathematics class is 18, the number of students who took Physics class is 23, and the number of students who took Biology class is 20. And also 3 students took both Mathematics class and Physics class, 8 students took both Physics class and Biology class, and 6 students took both Biology class and Mathematics class. What is the total number of students in school V? A. 11
B. 22
C. 33
D. 44
E. 55
==> a d
a+b+c=? f
h
d+e+f=?
g b
c e
g=? h=?
Students who take Mathematics=a+d+g+f=18 Students who take Physics=b+e+g+d=23 Students who take Biology=c+f+g+e=20 If we add three equations above, we get (a+b+c)+2(d+e+f)+3g=61 Students who take both Mathematics and Physics=d+g=3 Students who take both Physics and Biology=e+g=8 Students who take both Biology and Mathematics=f+g=6 If we add three equations above, we get (d+e+f)+3g=17. And, h=0, g=0. If we substitute, from d+e+f=17, (a+b+c)+2(17)+3(0)=61, we get a+b+c=27. The total number of students=(a+b+c)+(d+e+f)+g+h=27+17+0+0=44. Hence, the correct answer is D.
7 D
4
Diagnostic test 1
www.mathrevolution.com
8/ (integer) X, y are positive integers. When x is divided by y, the remainder is 6 and x/y=6.12. What is the value of x? A. 6
B. 50
C. 206
D. 306
E. 336
==> From X=yQ+6, =integer and y>6. Then, 1>6/y=decimal. If we substitute, from x/y=(yQ+6)/y=Q+(6/y)=6+0.12, we get Q=6 and 6/y=0.12, y=6/0.12=50. Then, x=(50)(6)+6=306. Hence, the correct answer is D. 5 9/ (sequence) A sequence Zn=2n-1 for a positive integer n. If S n is the sum of Zm when m is all integers 1 to n, which of the following can be the value of Sn? A. 12
B. 24
C. 36
D. 48
E. 60
==> Since S1=Z1=1=12, S2=Z1+Z2=1+3=4=2 2, S3=Z1+Z2+Z3=1+3+5=9=3 2, we get Sn=n2. Hence, a square is the answer, which is C. Since 36=6 2, C is the correct answer.
8 D 9 C
Diagnostic test 1
www.mathrevolution.com
10/ (integer) If n is a positive integer, how many different factors does n have? 1) N/5 is a prime number. 2) N has only two different prime factors. ==> According to the Variable Approach Method, there is 1 variable (n) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Hence, there is high chance that D is the correct answer. In the case of the condition 1), the number of factors is 4 when n/5=3, n=(3)(5). The number of factors is 3 when n/5=5, n=5 2. The answers are not unique and the condition is not sufficient. In the case of the condition 2), the number of factors is (1+1)(1+1)=4 when n=(31)(51), and (3+1)(1+1)=12 when n=(3 3)(51). The answers are not unique and the condition is not sufficient. Using both the condition 1) and the condition 2), the number of factors is 4 when n=(3)(5). Hence, the answer is unique and the condition is sufficient. Thus, the correct answer is C.
10 C
6
Diagnostic test 1
www.mathrevolution.com
11/ *(sequence) John deposits $10,000 in an account with an annual interest rate of x percent (compound quarterly). If no other activity for this account took place, is John’s account balance over $10,500 after 1 year?
2 ) >1.05 2 2) (1+ ) >1.05 1) (1+
==> We can modify the original condition and the question according to the Variable Approach Method. The question becomes 10,000(1+
4 ) >10,500?,
4 4 ) >10,500/10,000?, (1+ ) >1.05?. There is 1 variable (x). In order to match the number of equations to the number of variables, we need 1 (1+
equation. Hence, there is high chance that D is the correct answer. However, in the case of the condition 2), we only have to square both sides of
4 22 ) =((1+ ) ) >1.052>1.05. The answer is always yes and the condition is sufficient. In the case of the condition 1), we inequality. In other words, (1+
can apply the common mistake type 4(B), which states that if you get the answer A and B too easily, consider the answer choice D. The condition 1)
4 2 ) ≥ (1+ ) >1.05. The answer is also yes and the condition is sufficient (this is because the total amount which includes the also becomes (1+
quarterly interest rate is larger than semi-annually). Hence, the correct answer is D.
11 D
7
Diagnostic test 1
www.mathrevolution.com
12/ (sequence) There is a sequence An for a positive integer n such that when An is divided by An-1 the remainder is An-2. If A2=7, A1=0, which of the following can be the value of A4? A. 48
B. 50
C. 52
D. 56
E. 58
==> If we substitute N=3, we get A 3=A2Q+A1=7Q+0=7Q. If we substitute N=4, we get A4=A3P+A2=7QP+7=7(QP+1)=a multiple of 7. Hence, a multiple of 7 is the answer (P, Q are positive integers), and the correct answer is D.
13/ (integer) What is the remainder when 10 8 is divided by 11? A. 0
B. 1
C. 2
D. 3
E. 4
==> Dividing by 11 is same as treating 10 as -1 (10=11(0)+10=11(1)-1). Hence, the remainder of 10 8 divided by 11 is same as (-1) 8 divided by 11. Since (-1)8=1, the correct answer is B.
12 D 13 B
8
Diagnostic test 1
www.mathrevolution.com
14/ (function) It is well known that the relationship between the revenue from sales of products and the total cost of labor and materials is represented by the function, R(c)=-10c2+kc+m, where R(c) is the revenue and c is the total cost. When the revenue from sales of products is at its greatest, what is the value of the total cost? 1) k=200 2) m=2,000
9
==> We can modify the original condition and the question according to Variable Approach Method.
Revenue R(c)=-10c2+kc+m Greatest revenue
Cost(c)
Generally, when y=ax2+bx+c, and when y=f(x) is at its greatest or at lowest, the value of x is x=-b/2a. Hence, this question can be solved in a similar way. From R(c)=-10c2+kc+m, we get the greatest revenue when c=-k/2(-10)=k/20. Hence, we only have to find k. This means that the condition 1) is sufficient and the correct answer is A.
14 A
Diagnostic test 1
www.mathrevolution.com
15/ (integer) The remainder of n is 1 when divided by 15, 35, and 125. What is the smallest possible integer n? A. 1,926
B. 2,126
C. 2,326
D. 2,426
E. 2,626
==> Since N is divisible by 15, 35, 125, these number become the factors of n. Also, this means n is the multiple of 15, 35, 125. In other words, since n is the multiple of three integers, it is a common multiple. Since the question asks what is the smallest possible integer of n, we have to find the least common multiple. From 15=(3)(5), 35=(5)(7), 125=5 3, the least common multiple is the greatest possible values of indexes. Hence, the least common multiple become 2,625=(3)(5 3)(7). This means, 2625 is divisible by 15, 35 and 125. Since the remainder is 1, we get 2,625+1=2,626, and the correct answer is E. 16/ (integer) What is the value of integer n? 1) The only different prime factor of n is 3 and 5 2) The number of different factors of n is 18. ==> According to Variable Approach Method, there is 1 variable (n) in the original condition. In order to match the number of variable to the number of equations, we need 1 equation. There is high chance that D is the correct answer. In the case of the condition 1), we get n=(3)(5), (3)(5 2)……. The answer is not unique and the condition is not sufficient. In the case of the condition 2), we get n=(3 2)(55), (35)(53),….. the answer is not unique and the condition is not sufficient. Using both the condition 1) and the condition 2), we get n=(3 2)(55), (35)(53), (3)(58)…. The answer is also not unique and the conditions are not sufficient. Hence, the correct answer is E. 15 E 16 E
10
Diagnostic test 1 17/ (exponent)
www.mathrevolution.com
-----|-------|-------|-------|-----217
219
n
If the distances between two consecutive points are the same, what is the value of n? A. 3(218)
B. 3(219)
C. 5(218)
D. 5(219)
E. 7(218)
==> If we treat the distance between two points as d, we get: d
d
11
d
-----|-------|-------|-------|-----217
219
n
d=219-217=(22-1)217=3(217). n=219+2d=(22)(217)+2(3)(217)=(4)(217)+(6)(2 17)=(4+6)(2 17)=10(2 17)=(5)(2)(2 17)= 5(218). Hence, the correct answer is C.
18/ (integer) If n is the remainder when 123,456,789 is divided by 4 and m is the remainder when 987,654,321 is divided by 8, what is the sum of n and m? A. 2
B. 3
C. 4
D. 5
E. 6
==> The remainder when an integer ‘a’ is divided 4 is same as the tens-digit of ‘a’ divided 4. The remainder when an integer ‘b’ is divided by 8 is same as the hundreds-digit of ‘b’ divided by 8. In other words, n is the remainder when 89 divided 4, and m is the remainder when 321 is divided by 8. From 89=4(22)+1, n=1 and from 321=8(40)+1, m=1. Hence, since n+m=1+1=2, the correct answer is A.
17 C 18 A
Diagnostic test 1
19/ (exponent)
()()
www.mathrevolution.com
=?
1) n=5 2) x=2 ==> If we modify the original condition and the question according to Variable Approach Method, we get
= −+−(−)(+) = −+−( −−) = −+− ++) =x4?.
()() only have to know the value of x, the correct answer is B.
12 Since we
20/ (sequence) There is a sequence An and n is a positive integer such that A1=a, A2=b, and An+2=An+1×An. Is A6<0? 1) a<0 2) ab<0 ==> We can modify the original condition and the question according to Variable Approach Method. If we substitute n=1, we get A 3=A2A1=ba. If we substitute n=2, we get A4=A3A2=(ba)b=b2a. If we substitute n=3, we get A5=A4A3=(b2a)(ba)=b 3a2. If we modify n=4, we get A 6=A5A4=(b3a2)(b2a)=b5a3. However, since the question is A6<0?, b5a3<0?--> ba<0? (Since b 4a2 is a positive integer, the sign of inequality does not change even if we both sides by b4a2). Hence, the condition 2) is sufficient and the correct answer is B.
19 B 20 B
Diagnostic test 1
www.mathrevolution.com
21/ (integer) On the 25 th of November in 1990, John was trying to make a new plan for Jan 1991. However, he did not have a new calendar that included Jan 1991. Then, in order to match a new plan for Jan 1991, which of following months of 1990 should he look at? A. March
B. April
C. May
D. June
E. July
==> There are 7 days in 1 week. And if the number is divided by 7, and if the remainder is same, their days will be the same as well. In other words, since May has 31 days, if we divide it by 7, we get 3(31=7(4)+3). Since June has 30 days, if we divide it by 7, we get 2(30=7(4)+2). The remainder of July is 3, of August is 3, of September is 2, of October is 3, of November is 2 and of
December
is
3.
If
we
add
up
all
remainders,
we
get
3+2+3+3+2+3+2+3=21. From 21=7(3), the remainder is 0. Hence, May of one year has the same calendar as January of the next year. So, in order to make a plan for January 1991, we can observe May of 1990. Hence, the correct answer is C.
22/ (probability) N is an integer from 21 to 30. What is the probability that n(n2-1) is divisible by 6? A. 0.3
B. 0.42
C. 0.48
D. 0.72
E. 1.00
==> Probability = Number of one particular case/The total number of cases. The total number of n is 10, so the total number of cases is also 10. Since n(n2-1)=(n-1)n(n+1), it is the multiple of 3 consecutive terms. Hence, it is always a multiple of 6. In any case, n(n 2-1) is always divisible by 6. Hence, the probability is 10/10=1. The correct answer is E.
21 C 22 E
13
Diagnostic test 1
www.mathrevolution.com
23/ (probability) Factory X and factory Y are among the 5 factories in a certain manufacturing business. If the CEO of a business must visit the 5 factories every day, in how many different possible orders can the CEO go to the factories so that factory X is ahead of factory Y? A. 24
B. 30
C. 36
D. 48
E. 60
==> In the case of the permutation that includes the same things, we divide it by their factorials. For example, if we line up c,o,f,f,e,e, there are a total of 6 letters and we get 6!. Since f,f,e,e appears twice each, we can divide 6! by 2! and 2!. Hence, the correct answer is 6!/2!2!. The next logical step is, in case of probability, if things have a certain order, we have to treat them as the same cases. In other words, the CEO visits X,Y,A,B,C. Hence, we should treat X=Y as X is ahead of Y. So, we get X,X,A,B,C. The different possible orders become 5!/2! (since X,X). Hence, 5!/2!=60, and the correct answer is E.
24/ (statistics) We define that the harmonic mean is the reciprocal of the average (arithmetic mean) of the reciprocals of a set of specified numbers. What is the harmonic mean of 2, 3, and 6? A. 3
B. 1/3
C. 2
D. 1/2
E. 4
==> We have to calculate H(2,3,6)=? 1) the reciprocals of a set of specified numbers=1/2, 1/3, 1/6. 2) the average (arithmetic mean)=[(1/2)+(1/3)+(1/6)]/3=1/3. 3) the reciprocal of the average=1/(1/3)=3. Hence, H(2,3,6)=3. Therefore, the correct answer is A.
23 E 24 A
14
Diagnostic test 1
www.mathrevolution.com
25/ (integer) If a, b are integers, and (a-b) 2+8b2=108, what is the number of the ordered pairs (a,b)? A. 2
B. 4
C. 6
D. 8
E. 10
==> From (a-b) 2=100 and b 2=1 or (a-b)2=36 and b 2=9, we get a-b=±10 and b= ± 1 or a-b= ± 6 and b= ± 3. This means we have 8 pairs, making D the correct answer.
15
26/ (sequence) Tom will cultivate bacteria. The number of bacteria he will cultivate after one day is 150 percent of that of the previous day. The total number of bacteria is 4,725 after 2 days. What is the value of the original number of bacteria on the first day? A. 1,200
B. 1,500
C. 1,600
D. 1,800
E. 2,100
==> If we treat the original number of bacteria as a, we know that a(1.5)2=4,725. a=2,100. Hence, the correct answer is E.
27/ (statistics) ) In the exam, 5 people have different scores ranging from 0 to 100, inclusive. The scores are all integers. If the median score is 70, what is the greatest possible average (arithmetic mean) score? A. 80.2
B. 81.2
C. 82.2
D. 83.2
E. 84.4
==> The greatest possible value of the average score is the largest value of the total sum. In other words, ( )+( )+70+( )+( ) needs to the greatest. Hence, we get (68)+(69)+70+(99)+(100)=406. Hence, the greatest possible average is 406/5=81.2. The correct answer is B.
25 D 26 E 27 B
Diagnostic test 1
www.mathrevolution.com
28/ (geometry) A certain right circular cylinder is full of water that fills at a constant rate of 6 cubic feet per minute. As the water level increases, the height of the cylinder goes up at a constant rate of 0.2 feet per minute, what is the radius of the cylinder, approximately, in feet? A. 1
B. 2
C. 3
D. 4
E. 5
==> The rate at which water is being filled up is 6ft 3 /min. Also, the rate at which the water level rises is 0.2ft/min. The volume of a right circular cylinder is v=r2h. Hence, we have 6t= r2(0.2t). t needs to be in minutes. Hence, if we get rid of t from both sides, we get 6= r2(0.2). Then, 6/0.2= r2, 30= r2, 30/ =r2. r2 is approximately 9, which makes r=3. Hence, the correct answer is C.
29/ (integer) Which of the following cannot be the outcome of raising an integer to the 4th power? A. 65,610,000 B. 68,574,961 C. 62,356,782 D. 52,200,625 E. 37,015,056
==> If we square a certain integer, we do not get 2,3,7,8 as units digits. If we raises the integer to the 4 th power, we only get 0,1,5,6 as the units digits. Hence, out of all answer choices, the answer choice without units digits as 0,1,5,6, is C. Hence, since 62,356,782 is the correct answer, C is the final answer choice.
28 C 29 C
16
Diagnostic test 1
www.mathrevolution.com
30/ (absolute value) Is a There are 2 variables in the original condition. We need 2 equations in order to match the number of variables and the number of equations. Hence, there is high chance that C is the correct answer. Using both the condition 1) and the condition 2): In the case of the condition 1), since it is same as |a-b|<|b-0|, this means that the distance between a to b is smaller than the distance between b to a. Hence, we can draw a number line like below: <-------|-------|-------|-----------------|-----------> b (no)
a
b
0
(yes)
It shows that the distance between a to b is smaller than b to 0. However, since we have answers that are yes and no, the condition is not sufficient, and the correct answer is E.
30 E
17
Diagnostic test 1
www.mathrevolution.com
31/ (statistics) The monthly precipitations in the first 5 months in 1999 are 8, 6, 9, 3, and 5, respectively, in inches. If the sixth month ’s precipitation in 1999 is either 2, 6, or 11, in inches, for which of these values does the monthly average (arithmetic mean) precipitation for the first 6 months equal the median? I. 2 A. I only
B. II only
II. 6
III. 11
C. III only
D. I and III only
E. I, II, and III
==> In order for the median and the mean to equal, it should be a consecutive
integers
or
the
sum
of
number
should
be
consistent
symmetrically. In other words, we get (2), 3, 5, 6, 8, 9 3, 5, 6, (6), 8, 9 3, 5, 6, 8, 9, 11 (2), 3, 5, 6, 8, 9 --> 2+9=3+8=5+6 (O) 3, 5, 6, (6), 8, 9 --> 3+9 ≠5+8 3, 5, 6, 8, 9, 11
(X)
--> 3+11=5+9=6+8 (O)
As we can see, only I. 2 and III. 11 have the sum of number that is consistent symmetrically. Hence, the correct answer is D.
31 D
18
Diagnostic test 1
www.mathrevolution.com
32/ *(work rate) There are 4 machines with the same work rate. If it took m hours for 3 machines to finish a task when working together, and took (m-3) hours for 4 machines to finish the task when working together, what is the value of m? A. 8
B. 10
C. 12
D. 14
E. 16
==> If we treat the work rate of one machine as r, then we get rt=w(r:work rate, t:time, w:work done). This is nrt=w (n=number of machines). Hence, 3rm=4r(m-3). If we divide both sides by r, we get =4(m-3)=4m-12, m=12. Hence, the correct answer is C.
33/ (integer) If {x} is the greatest integer less than or equal to x, what is the value of {1/2}+{7/5}+{8/3}? A. 1
B. 2
C. 3
D. 4
E. 5
==> When, {x}=the greatest integer less than or equal to x, we can round down. Hence, since {1/2}+{7/5}+{8/3}={0.5}+{1.4}+{2.6666..}=0+1+2=3, the correct answer is C.
34/ *(integer) If x, y are positive integers, what is the value of xy? 1) yx=9 2) x2y=64 ==> There are 2 variables (x and y) in the original condition. We need 2 equations in order to match the number of variables and the number of equations. Hence, there is high chance that C is the correct answer. Using both the condition 1) and the condition 2), we get x=2 and y=3. The
32 C 33 C 34 B
19
Diagnostic test 1
www.mathrevolution.com
conditions are sufficient and the correct answer is C. However, this is an integer question, one of key questions. Hence, we have to apply the common mistake type 4(A). In the case of the condition 1), we get 9=3 2=91. Since (x,y)=(2,3),(1,9), the answers are not unique and the condition is not sufficient. In the case of the condition 2), from (x y)2=x2y=64=82, we get xy=8, which is unique, making the condition sufficient. By DS question’s definition, when C and B are both answers, B is the correct answer. Hence, the correct answer for this question is B.
35/ (absolute value) If ab-2b=(4-a)b, what is the value of b? 1) |a2-9|≤0 2) a<0 ==> If we modify the original condition and the question according to Variable Approach Method, we get ab-2b=4b-ab, 2ab-6b=0, 2b(a-3)=0. Hence, the number of variables is 2 (a and b) and the number of equations is 1 (2b(a-3)=0). In order for the number of variables and the number of equations to equal, we need 1 equation. Hence, there is high chance that D is the correct answer. In the case of the condition 1), if a=3, the value of b is not unique. Hence, the condition is not sufficient. In the case of the condition 2), if a<0, a-3<0, we always get b=0. The answer is unique and the condition is sufficient. Hence, the correct answer is B.
35 B
20
Diagnostic test 1
www.mathrevolution.com
36/ *(statistics) A certain population has a certain way of distributing its members such that they are symmetric to the mean m 68 percent of the .
distribution lies within one standard deviation d of the mean and 95 percent of the distribution lies within two standard deviations d of the mean. When the average (arithmetic mean) life expectancy of tigers is 15 years and the standard deviation of them is 5 years, what percent of the tigers have the average (arithmetic mean) life expectancy between 20 years and 25 years, approximately? A. 14%
B. 32%
C. 48%
D. 84%
E. 92%
==> m: mean d: standard deviation 0.34
0.34
0.135
0.135
0.025
0.025 m m-d
m-2d
m+d m+2d
From the above diagram, we get m=15, d=5. However, the question asks 20yrs~25yrs, we get (m+d)~(m+2d). Hence, the correct answer is 0.135. This is approximately 14%, and the correct answer is A.
36 A
21
Diagnostic test 1
www.mathrevolution.com
37/ (ratio) Each of 3 cups has the same n number of marbles. When marbles from the 1-st cup are moved to the other 2 cups, the ratio of marbles in each cup was 2:4:3. Then, which of the following represents the marbles that are moved in the 1-st cup, in terms of n? A. n/4
B. n/3
C. n/2
D. n
E. 4n/5
==> Since n:n:n --> 2k:4k:3k, the total number of marbles does not change. Then, from 3n=2k+4k+3k=9k, we get n=3k. In other words, from 3k:3k:3k -> 2k:4k:3k. Hence, the total number of moved marbles is k that became 3k->2k(3k-2k=k). Hence, k=n/3, and the correct answer is B.
37 B
Missing questions’ number
Estimating score
1~5
49~51
5~10
45~48
11~15
41~44
16~20
36~40
22
