1. If 2 different representatives are to be selected at random from a group of 10 employees
and if p is the probability that both representatives selected will be women, is p > 1/2

(1) More than 1/2 of the 10 employees are women.
(2) The probability that both representatives selected will be men is less than 1/10

What is the probability of choosing 2 women out of 10 people \frac{w}{10}*\frac{w-1}{9} and this should be >1/2. So we have \frac{w}{10}*\frac{w-1}{9}>\frac{1}{2} --> w(w-1)>45 this is true only when w>7. (w # of women <=10)

So basically question asks is w>7?

(1) w>5 not sufficient.

(2) \frac{10-w}{10}*\frac{10-w-1}{9}<\frac{1}{10} --> (10-w)(9-w)<9 --> w>6, not sufficient

(1)+(2) w>5, w>6 not sufficient

Answer E.

You can use Combinations, to solve as well:

C^2_w # of selections of 2 women out of w employees;

C^2_{10} total # of selections of 2 representatives out of 10 employees.

Q is \frac{C^2_w}{C^2_{10}}>\frac{1}{2} --> \frac{\frac{w(w-1)}{2}}{45}>\frac{1}{2} --> --> w(w-1)>45 --> w>7?

(1) w>5, not sufficient.

(2) C^2_{(10-w)} # of selections of 2 men out of 10-w=m employees --> \frac{C^2_{(10-w)}}{C^2_{10}}<\frac{1}{10} --> \frac{\frac{(10-w)(10-w-1)}{2}}{45}<\frac{1}{10} --> (10-w)(9-w)<9 --> w>6, not sufficient

(1)+(2) w>5, w>6 not sufficient

Answer E.

3. What is the value of the integer k?

(1) k + 3 > 0 --> k>-3, not sufficient to determine single numerical value of k.

(2) k^4 ≤ 0, as number in even power is never negative k^4 can not be less than zero, thus k^4=0 --> k=0. sufficient.

Answer: B

4. If the units digit of the three-digit positive integer k is nonzero, what is the tens digit of k?
(1) The tens digit of k + 9 is 3.
(2) The tens digit of k + 4 is 2.

k=abc, c not zero, question b=?

(1) The tens digit of k + 9 is 3:
abc
+9
---
a3x

Tens digit of k+9, which is 3, gains 1 unit from c+9 (as c is not zero), hence b+1=3 --> b=2. Sufficient.

(2) The tens digit of k + 4 is 2
abc
+4
---
a2x

Tens digit of k + 4, which is 2, may or may not gain 1 unit from c+4, hence either b+1=2 or b+0=2. Two different values for b. Not sufficient.

Answer: A.

Check the statements for question 2.
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For your #2:

x & y positive. x^3 > y?

(1) x^(1/2) > y
i) take x = 1/4, and y = 1/3
so x^(1/2) = 1/2 > 1/3... but you can see that (1/4)^3 is NOT > 1/3...

ii) and x = 4 and y = 1
so x^(1/2) = 2 > 1... and 4^3 is obviously > 1.

we answered x^3 > y both NO and YES, so contradiction. Statement (1) not sufficient


(2) x > y
i) simply take fractions (x = 1/2 and y = 1/4... x^3 = 1/8) and you'll see that x^3 NOT > y

ii) take numbers > 1 and you'll get x^3 > y.

contradiction again...


so E
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doh, i forgot the (together): i.e., taking both statements together...

same story: (1) x>y and (2) x^(1/2) > y

take x = 1/4 and y = 1/5 satisfying both (1) and (2) above.
you can see that x^3 > y => Not true

take x = 4 and y = 1
implies x^3 > y => True

contradiction
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4.

1. Sufficient to know that there was a carry over to tens place. enough to answer the question.

2. Not sufficient . as we don't know whether there was a carry over from units place or not.

Answer A.

Clearly E,
after plugging in the numbers, there is an insufficiency in both the statements.

Eg: 1) Try x= 25, y= 4 (YES) & x= .04 & y =.3 (NO)
2) Try x= 25, y= 4 (YES) & x= .2, y= .1 (NO)
(Some of the values that i randomly tried)

When selecting two representatives only 3, mutually exclusive, cases are possible:
1. Both are men;
2. Both are women;
3. One is a man and another is a woman.

The sum of the probabilities of these cases must be 1. So, knowing that P(Both Men)<1/10 does not necessarily mean that P(Both Women)>9/10.
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You cross-multiply LHS to RHS and divide by 10. So you get left with the numerators.

w^2-19w+81<0 has two roots:
(19+-\sqrt{37})/2 w1=(almost)\frac{19+6}{2}=\frac{25}{2}=12.5
w2=(almost)\frac{19-6}{2}=\frac{13}{2}=6.5
w1=12.5 w2=6.5 and the solution is 6.5<w<12.5
Remember that w is the number of women, so because we have 10 employees, the solution is 6,5<w\leq{10}
But because we cannot select 6 women and 1/2, the solution of the equation is women>6.

Hope it makes sense now
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