Math Revolution Approach (PS)

[GMAT math practice question]

(1/26)+(1/27)+……+(1/49)+(1/50) is included in which of the following ranges?

A. (1/6)~(1/5)
B. (1/5)~91/4)
C. (1/4)~(1/3)
D. (1/3)~(1/2)
E. (1/2)~1

=>

1/26 > 1/50, 1/27 > 1/50, …, 1/49 > 1/50
(1/26)+(1/27)+……+(1/49)+(1/50) > 1/50 + 1/50 + … + 1/50 = (1/50)*25 = 1/2.
1/26 < 1/25, 1/27 < 1/25, … , 1/49 < 1/25, 1/50 < 1/25.
(1/26)+(1/27)+……+(1/49)+(1/50) < 1/25 + 1/25 + … + 1/25 = (1/25)*25 = 1.

Therefore, the answer is E.
Answer : E

[GMAT math practice question]

Tom traveled the entire 90 km trip. If he travelled the first 18 km of the trip at a constant rate of 36 km per hour and the remaining trip at a constant rate of 72 km per hour, what is his average speed?

A. 30 km/h
B. 36 km/h
C. 45 km/h
D. 48 km/h
E. 60 km/h

=>

90 / (18/36 + 72/72) = 90 / 1.5 = 60

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

If f(x+y) = f(x)f(y), which of followings can be the value of f(0)?

I. 0
II. 1
III. 2

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III



=>

When we plug in x = 0 and y = 0, we have f(0) = f(0)f(0) or f(0)f(0) – f(0) = 0.
Then f(0){ f(0) – 1 } = 0, so we have f(0) = 0 or f(0) = 1.

Therefore, D is the answer.

Answer: D

Can you explain this one ?

[GMAT math practice question]

If x=-1 and n is the sum of all prime numbers less than 100, what is the value of x^n+x^{n+1}+x^{n+2}+x^{n+3}+x^{n+4}+x^{n+5}?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

Whatever the value of n is, (-1)^n+(-1)^{n+1}+(-1)^{n+2}+(-1)^{n+3}+(-1)^{n+4}+(-1)^{n+5} = 0, since two consecutive integer n and n+1 always have an odd integer and an even integer, and so (-1)^n+(-1)^{n+1} = 0.

Thus, the answer is C.

Answer: C

The edge of the larger cube is 3 and the area of one face is 9.
And the cube has 6 faces.

The edge of the smaller cube is 1 and each face has the area 1.

6*27 means the sum of 6 faces area of the smaller cube times 27 cubes.
6*9 is the number of faces 6 times the area of each face 9.

6*27 - 6*9 is the answer.

Happy Studying !!!

[GMAT math practice question]

When x/y = 1.2, (x^2-y^2)/(x^2+y^2) =?

A. 10/51
B. 11/51
C. 10/61
D. 11/61
E. 11/71

=>

x/y = 6/5.
(x^2-y^2)/(x^2+y^2) = ((x/y)^2-1)/((x/y)^2+1) = ((6/5)^2 – 1 ) / ((6/5)^2 + 1 )
= (6^2-5^2)/(6^2+5^2) = (36-25)/(36+25) = 11/61

Therefore, the answer is D.

Answer: E

[GMAT math practice question]

(3^3+3^4+3^5+3^6)^2=?

A. 2^43^65^2
B. 2^63^45^2
C. 2^83^65^2
D. 2^63^85^2
E. 2^63^65^2

=>
(3^3+3^4+3^5+3^6)^2= (3^3(3^0+3^1+3^2+3^3))^2=3^6(1+3+9+27)^2=3^640^2=3^62^65^2

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

Which of the following equations has a unique integer solution?

A. y=3x
B. y=(x/3)+4
C. y=√2x
D. y=(1/2)x
E. y=(-1/x)+2


=>

A. x = 0, y = 0 / x = 1, y = 3
B. x = 0, y = 4 / x = 3, y = 5
C. y=√2x has a unique integer solution, that is, x = 0 and y = 0. If x is not 0, y is an irrational number, which is not an integer.
D. x = 0, y = 0 / x = 2, y = 1
E. x = 1, y = 1 / x = -1, y = 3

Therefore, the answer is C.
Answer : C

[GMAT math practice question]

What of the followings is the range of X = 1 + (1/2^2) + (1/3^2) + ...... + (1/100^2),

A. 0 < X < 1
B. 1 < X < 2
C. 2 < X < 3
D. 3 < X < 4
E. 4 < X < 5

=>

1/A^2 < 1/{(A-1}A} = 1/(A-1) – 1/A

It is clear that 1 + (1/2^2) + (1/3^2) + ...... + (1/100^2) > 1.

1 + (1/2^2) + (1/3^2) + ...... + (1/100^2)
< 1 + 1/(1∙2) + 1/(2∙3) +1/(3∙4) + … + 1/(99∙100)
= 1 + ( 1/1 – 1/2 ) + ( 1/2 – 1/3 ) + … + (1/99 – 1/100)
= 1 + 1/1 – 1/100 = 2 – 1/100 < 2

Thus 1 < X < 2.
Therefore, B is the answer.

Answer: B

[GMAT math practice question]

The probability that an event A occurs is 1/2 and the probability that an even B occurs is 1/3. If event A and event B are independent, what is the probability that event A occurs but event B does not occur?

A. 1/6
B. 1/3
C. 1/2
D. 2/3
E. 5/6


=>

Since events A and B are independent, P(A∩B) = P(A) ∙P(B).
The probability P(A-B) = P(A) - P(A∩B) = P(A) - P(A) ∙P(B) = 1/2 – (1/2)(1/3) = 1/2 – 1/6 = 2/6 = 1/3.

Thus, the answer is B.
Answer: B

[GMAT math practice question]

What is the sum of the remainders when the first 50 positive integers are divided by 5?

A. 60
B. 70
C. 80
D. 90
E. 100

=>

1, 2, 3, 4, 5 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.
6, 7, 8, 9, 10 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.
…
46, 47, 48, 49, 50 have remainders 1, 2, 3, 4, 0 and the sum of their remainders is 1 + 2 + 3 + 4 + 0 = 10.

10 * 10 = 100

Therefore, the answer is E.

Answer: E

[GMAT math practice question]

In a pin factory, it needs to produce 100,000 pins every day, and 5 machines take x hours to produce 100,000 pins. One day, one of the machines is out of order, so only 4 machines worked. These four machines took x+2 hours to produce 100,000 pins, and each machine’s work rates are the same. What is the value of x?

A. 5
B. 6
C. 7
D. 8
E. 9

=>

Assume T is the time that one machine takes to produce 100,000 pins.

1/x = 5/T and 1/(x+2) = 4/T
5x = T and T = 4(x+2)
5x = 4(x+2)
5x = 4x + 8
x = 8.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

In the normal x-y coordinate plane there are 4 points A(-1,3), B(3,3), C(-1,7), and D(3,7). If a line passing through the origin bisects the area of the rectangle ABCD, what is the slope of the line?

A.5
B. 6
C. 7
D. 8
E. 9

=>

The line should pass through the center of the rectangle ABCD.
Then the center is ((-1+3)/2, (3+7)/2) or (1,5)
The slope of the line passing though (0,0) and (1,5) is 5-0/1-0=5.

Therefore, the answer is A.
Answer : A

[GMAT math practice question]

I. 1/2,2/3,3/4,4/5,5/6
II. 1/2,1/3,1/4,1/5,1/6
III. 1/6,2/6,3/6,4/6,5/6

For which of the above lists is the average of the numbers less than the median of numbers?

A. I only
B. II only
C. III only
D. II and III
E. I, II and III




=>
I. The differences between the consecutive terms are 1/6 (= 2/3 – 1/2 ), 1/12 (= 3/4 – 2/3 ), 1/20 (= 4/5 – 3/4 ), and 1/30 (= 5/6 – 4/5). As these are decreasing, the average is smaller than the median.

II. The differences between consecutive terms are - 1/6(= 1/3 – 1/2),-1/12(= 1/4 – 1/3 ), - 1/20(= 1/5 – 1/4), and -1/30(= 1/6 – 1/5). As these are increasing, the average is larger than the median.

III. As the data are symmetric, the average and the median are equal.

Therefore, the answer is A

Answer: A

[GMAT math practice question]

Which of the following is true if -3<x<9?

A. |x-3|<4
B. |x-3|<5
C. |x-3|<6
D. |x-3|>4
E. |x-3|>5

=>

|x-a|<b
⟺ -b < x – a < b
⟺ a - b < x < a + b
So, a is the midpoint of the end points a – b and a + b, and
b is half the distance between the end points a – b and a + b.

The midpoint of -3 and 9 is [(-3)+9]/2 = 6/2 = 3. So, a = 3.
Since the distance between -3 and 9 is 12, b = 12/2= 6.
Thus, |x-3| < 6.

Therefore, the answer is C.

Answer: C

[GMAT math practice question]

In how many ways can three boys and four girls line up in a row, if girls take the first and last positions?

A. 28
B. 120
C. 720
D. 1,440
E. 5,040

=>

There are four ways (girls) to fill the first position and three ways (girls left over) to fill the last position. The remaining five positions can be filled by the five remaining children (three boys and two girls) in 5!=120 ways.

By the multiplication rule, there are 4 * 3 * 120 = 1,440 different lines possible.

Therefore, the answer is D.

Answer: D