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Six math need to solve

by monirjewel » Sat Nov 13, 2010 11:18 pm
If x and y are integers, what is the value of 2x^6y-4 ?
(1) X^2y=16
(2) xy=4


The function f is defined for each positive three digit integer n by f(n)=2^x3^y5^z, where x, y and z are hundreds, tens, and units digits of n, respectively. If m and v are three digit positive integers such that f(m)=9 f(v),then m-v=
(A) 8
(B) 9
(C) 18
(D) 20
(E) 80

A rectangular floor that measures 8 meters by 10 meters is to be covered with carpet squares that each measure 2 meters by 2 meters. If carpet squares cost $12 apiece, what is the total cost for the number of carpet squares needed to cover the floor?
(A) $200
(B) $240
(C) $480
(D) $960
(E) $1920

Which of the following is closet to 10^180-10^30?
(A) 10^210
(B) 10^180
(C) 10^150
(D) 10^90
(E) 10^6

If x, y, and z are consecutive even positive integers, which of the following could be equal to x+y+z?
(A) 141
(B) 200
(C) 318
(D) 391
(E) 412

A sum of money was divided between John and Anne such that the ratio of John's share to Anne's share was 5 to 3. If John's share exceeded 5/9 of the sum of money by $50, what was Anne's share?
(A) $180
(B) $270
(C) $340
(D) $450
(E) $720
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by Rahul@gurome » Sat Nov 13, 2010 11:54 pm
Question Number 6:
A sum of money was divided between John and Anne such that the ratio of John's share to Anne's share was 5 to 3. If John's share exceeded 5/9 of the sum of money by $50, what was Anne's share?
(A) $180
(B) $270
(C) $340
(D) $450
(E) $720
Assume that the sum of money = $x
John's share = $(5x/8)
Anne's share = $(3x/8)

According to the question,
  • John's share = (5/9)*(Sum of the money) + $50
Therefore, (5x/8) = (5/9)*(x) + 50
=> x/8 = x/9 + 10
=> (x/8 - x/9) = 10
=> x/72 = 10
=> x = 720
=> (3x/8) = (3/8)*(720) = 270

Thus, Anne's share is $270.

The correct answer is B.
Last edited by Rahul@gurome on Sun Nov 14, 2010 12:01 am, edited 1 time in total.
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by Rahul@gurome » Sun Nov 14, 2010 12:00 am
Question Number 5:
If x, y, and z are consecutive even positive integers, which of the following could be equal to x+y+z?
(A) 141
(B) 200
(C) 318
(D) 391
(E) 412
Say, x = 2n, then, y = (2n + 2) and z = (2n + 4)

Therefore, (x + y + z) = 2n + (2n + 2) + (2n + 4) = (6n + 6) = 6(n + 1)

Thus (x + y + z) must be divisible by 6.
Among the options only 318 is divisible by 6.

The correct answer is C.
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by Rahul@gurome » Sun Nov 14, 2010 12:06 am
Question Number 4:
Which of the following is closet to 10^180-10^30?
(A) 10^210
(B) 10^180
(C) 10^150
(D) 10^90
(E) 10^6
(10^180 - 10^30) = (10^30)*(10^150 - 1) ≈ (10^30)*(10^150) = (10^180)

This is because, (10^150 - 1) ≈ (10^150)

The correct answer is B.
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by Rahul@gurome » Sun Nov 14, 2010 12:11 am
Question Number 3:
A rectangular floor that measures 8 meters by 10 meters is to be covered with carpet squares that each measure 2 meters by 2 meters. If carpet squares cost $12 apiece, what is the total cost for the number of carpet squares needed to cover the floor?
(A) $200
(B) $240
(C) $480
(D) $960
(E) $1920
Area of the rectangular floor = (8*10) sq. meters = 80 sq meters
Area of the square carpets = (2*2) sq. meters = 4 sq. meters

Number of carpets required to cover the floor = (80/4) = 20

Total cost of the carpets = ($12 per carpet)*(20 carpets) = $240

The correct answer is B.
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by Rahul@gurome » Sun Nov 14, 2010 12:21 am
Question Number 2:
The function f is defined for each positive three digit integer n by f(n)=2^x3^y5^z, where x, y and z are hundreds, tens, and units digits of n, respectively. If m and v are three digit positive integers such that f(m)=9 f(v),then m-v=
(A) 8
(B) 9
(C) 18
(D) 20
(E) 80
Say, m = (100A + 10B + C) => A is the hundredths, B is the tenths and C is the units digit of m.
and, v = (100X + 10Y + Z) => X is the hundredths, Y is the tenths and Z is the units digit of v.

Thus, f(m) = (2^A)*(3^B)*(5^C)
and, f(v) = (2^X)*(3^Y)*(5^Z)

Now, f(m) = f(v) => (2^A)*(3^B)*(5^C) = 9*(2^X)*(3^Y)*(5^Z) = (2^X)*(3^(Y + 2))*(5^Z)

Comparing the exponents of 2, 3 and 5,
  • A = X
    B = Y + 2
    C = Z
Now, (m - v) = (100A + 10B + C) - (100X + 10Y + Z) = 100*(A - X) + 10*(B - Y) + (C - Z) = 10*2 = 20

The correct answer is D.
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by ctur9967 » Mon Nov 15, 2010 1:31 am
Hi,

Thanks for the tips.

I do not, however, see the link between the hundreds digit, the tens digit and the unit digit and the ability to express them as part of the function.

Ie why does f(100A + 10B + C ) = 2^A*3^B^5^C

Cheers,

Chris

Rahul@gurome wrote:Question Number 2:
The function f is defined for each positive three digit integer n by f(n)=2^x3^y5^z, where x, y and z are hundreds, tens, and units digits of n, respectively. If m and v are three digit positive integers such that f(m)=9 f(v),then m-v=
(A) 8
(B) 9
(C) 18
(D) 20
(E) 80
Say, m = (100A + 10B + C) => A is the hundredths, B is the tenths and C is the units digit of m.
and, v = (100X + 10Y + Z) => X is the hundredths, Y is the tenths and Z is the units digit of v.

Thus, f(m) = (2^A)*(3^B)*(5^C)
and, f(v) = (2^X)*(3^Y)*(5^Z)

Now, f(m) = f(v) => (2^A)*(3^B)*(5^C) = 9*(2^X)*(3^Y)*(5^Z) = (2^X)*(3^(Y + 2))*(5^Z)

Comparing the exponents of 2, 3 and 5,
  • A = X
    B = Y + 2
    C = Z
Now, (m - v) = (100A + 10B + C) - (100X + 10Y + Z) = 100*(A - X) + 10*(B - Y) + (C - Z) = 10*2 = 20

The correct answer is D.
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by Rahul@gurome » Mon Nov 15, 2010 1:41 am
ctur9967 wrote:Hi,

Thanks for the tips.

I do not, however, see the link between the hundreds digit, the tens digit and the unit digit and the ability to express them as part of the function.

Ie why does f(100A + 10B + C ) = 2^A*3^B^5^C
Read the question properly. The function is defined in that way.

The function f is defined for each positive three digit integer n by f(n)=2^x3^y5^z, where x, y and z are hundreds, tens, and units digits of n, respectively
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by ctur9967 » Mon Nov 15, 2010 2:39 am
Thankyou.

I clearly glazed over the question.

keep up the great work

Rahul@gurome wrote:
ctur9967 wrote:Hi,

Thanks for the tips.

I do not, however, see the link between the hundreds digit, the tens digit and the unit digit and the ability to express them as part of the function.

Ie why does f(100A + 10B + C ) = 2^A*3^B^5^C
Read the question properly. The function is defined in that way.

The function f is defined for each positive three digit integer n by f(n)=2^x3^y5^z, where x, y and z are hundreds, tens, and units digits of n, respectively
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