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12 Easy Pieces (or not?) [#permalink]
21 Jan 2012, 05:10

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After posting some 700+ questions, I've decided to post the problems which are not that hard. Though each question below has a trap or trick so be careful when solving. I'll post OA's with detailed solutions after some discussion. Good luck.

1. There are 5 pairs of white, 3 pairs of black and 2 pairs of grey socks in a drawer. If four socks are picked at random what is the probability of getting two socks of the same color? A. 1/5 B. 2/5 C. 3/4 D. 4/5 E. 1

2. If x is an integer and 9<x^2<99, then what is the value of maximum possible value of x minus minimum possible value of x? A. 5 B. 6 C. 7 D. 18 E. 20

3. Fanny and Alexander are 360 miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and 65 mph respectively, how far apart will they be exactly 1.5 hours before they meet? A. 25 miles B. 65 miles C. 70 miles D. 90 miles E. 135 miles

4. If -3<x<5 and -7<y<9, which of the following represent the range of all possible values of y-x? A. -4<y-x<4 B. -2<y-x<4 C. -12<y-x<4 D. -12<y-x<12 E. 4<y-x<12

5. The angles in a triangle are x, 3x, and 5x degrees. If a, b and c are the lengths of the sides opposite to angles x, 3x, and 5x respectively, then which of the following must be true? I. c>a+b II. c^2>a^2+b^2 III. c/a/b=10/6/2

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

6. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same color and the first and the last marbles are of different colors. How many different arrangements are possible? A. 30 B. 60 C. 120 D. 240 E. 480

7. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the total ratio of negative numbers to non-negative numbers be 2 to 1? A. 11/14 B. 13/18 C. 4/7 D. 3/7 E. 3/14

8. There are 15 black chips and 5 white chips in a jar. What is the least number of chips we should pick to guarantee that we have 2 chips of the same color? A. 3 B. 5 C. 6 D. 16 E. 19

9. Julie is putting M marbles in a row in a repeating pattern: blue, white, red, green, black, yellow, pink. If the row begins with blue marble and ends with red marble, then which of the following could be the value of M? A. 22 B. 30 C. 38 D. 46 E. 54

11. The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p? A. 97 B. 151 C. 209 D. 211 E. 219

12. If {-\frac{1}{3}}\leq{x}\leq{-\frac{1}{5}} and {-\frac{1}{2}}\leq{y}\leq{-\frac{1}{4}}, what is the least value of x^2*y possible? A. -1/100 B. -1/50 C. -1/36 D. -1/18 E. -1/6

Notice that most of the problems have short, easy and elegant solutions, since you've noticed a trick/trap hidden in the questions.

1. There are 5 pairs of white, 3 pairs of black and 2 pairs of grey socks in a drawer. If four socks are picked at random what is the probability of getting two socks of the same color? A. 1/5 B. 2/5 C. 3/4 D. 4/5 E. 1

No formula is need to answer this one. The trick here is that we have only 3 different color socks but we pick 4 socks, which ensures that in ANY case we'll have at least one pair of the same color (if 3 socks we pick are of the different color, then the 4th sock must match with either of previously picked one). P=1.

2. If x is an integer and 9<x^2<99, then what is the value of maximum possible value of x minus minimum possible value of x? A. 5 B. 6 C. 7 D. 18 E. 20

Also tricky. Notice that x can take positive, as well as negative values to satisfy 9<x^2<99, hence x can be: -9, -8, -7, -6, -4, 4, 5, 6, 7, 8, or 9. We asked to find the value of x_{max}-x_{min}, ans since x_{max}=9 and x_{min}=-9 then x_{max}-x_{min}=9-(-9)=18.

6. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same color and the first and the last marbles are of different colours. How many different arrangements are possible? A. 30 B. 60 C. 120 D. 240 E. 480

Seems tough and complicated but if we read the stem carefully we find that the only way both conditions to be met for 5 red marbles, which are half of total marbles, they can be arranged only in two ways: R*R*R*R*R* or *R*R*R*R*R.

Here comes the next good news, in these cases BOTH conditions are met for all other marbles as well: no two adjacent marbles will be of the same color and the first and the last marbles will be of different colors.

Now, it's easy: 2 blue, 2 green and 1 yellow can be arranged in 5 empty slots in 5!/(2!*2!)=30 ways (permutation of 5 letters BBGGY out of which 2 B's and 2 G' are identical). Finally as there are two cases (R*R*R*R*R* and *R*R*R*R*R. ) then total # of arrangement is 30*2=60.

7. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the total ratio of negative numbers to non-negative numbers be 2 to 1? A. 11/14 B. 13/18 C. 4/7 D. 3/7 E. 3/14

If choose variable for set A there will be too many fractions to manipulate with, so pick some smart #: let set A contain 18 numbers.

"2/9 of the numbers in a data set A were observed" --> 4 observed and 18-4=14 numbers left to observe; "3/4 of those numbers were non-negative" --> 3 non-negative and 1 negative; Ratio of negative numbers to non-negative numbers to be 2 to 1 there should be total of 18*2/3=12 negative numbers, so in not yet observed part there should be 12-1=11 negative numbers. Thus 11/14 of the remaining numbers in set A must be negative.

4. If -3<x<5 and -7<y<9, which of the following represent the range of all possible values of y-x? A. -4<y-x<4 B. -2<y-x<4 C. -12<y-x<4 D. -12<y-x<12 E. 4<y-x<12

To get max value of y-x take max value of y and min value of x: 9-(-3)=12; To get min value of y-x take min value of y and max value of x: -7-(5)=-12;

Hence, the range of all possible values of y-x is -12<y-x<12.

5. The angles in a triangle are x, 3x, and 5x degrees. If a, b and c are the lengths of the sides opposite to angles x, 3x, and 5x respectively, then which of the following must be true? I. c>a+b II. c^2>a^2+b^2 III. c/a/b=10/6/2

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

According to the relationship of the sides of a triangle: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Thus I and III can never be true: one side (c) can not be larger than the sum of the other two sides (a and b). Note that III is basically the same as I: if c=10, a=6 and b=2 then c>a+b, which can never be true. Thus even not considering the angles, we can say that only answer choice B (II only) is left.

Answer: B.

Now, if interested why II is true: as the angles in a triangle are x, 3x, and 5x degrees then x+3x+5x=180 --> x=20, 3x=60, and 5x=100. Next, if angle opposite c were 90 degrees, then according to Pythagoras theorem c^2=a^+b^2, but since the angel opposite c is more than 90 degrees (100) then c is larger, hence c^2>a^+b^2. _________________

12. If {-\frac{1}{3}}\leq{x}\leq{-\frac{1}{5}} and {-\frac{1}{2}}\leq{y}\leq{-\frac{1}{4}}, what is the least value of x^2*y possible? A. -1/100 B. -1/50 C. -1/36 D. -1/18 E. -1/6

To get the least value of x^2*y, which obviously will be negative, try to maximize absolute value of x^2*y, as more is the absolute value of a negative number "more" negative it is (the smallest it is).

To maximize |x^2*y| pick largest absolute values possible for x and y: (-\frac{1}{3})^2*(-\frac{1}{2})=-\frac{1}{18}. Notice that: -1/18<-1/36<-1/50<-1/100, so -1/100 is the largest number and -1/18 is the smallest number (we cannot obtain -1/6 from x^2*y or else it would be the correct answer).

3. Fanny and Alexander are 360 miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and 65 mph respectively, how far apart will they be exactly 1.5 hours before they meet? A. 25 miles B. 65 miles C. 70 miles D. 90 miles E. 135 miles

Make it simple! The question is: how far apart will they be exactly 1.5 hours before they meet? As Fanny and Alexander's combined rate is 25+65 mph then 1.5 hours before they meet they'll be (25+65)*1.5=135 miles apart.

8. There are 15 black chips and 5 white chips in a jar. What is the least number of chips we should pick to guarantee that we have 2 chips of the same color? A. 3 B. 5 C. 6 D. 16 E. 19

Worst case scenario would be if the first two chips we pick will be of the different colors. But the next chip must match with either of two, so 3 is the answer.

10. If n is an integer and \frac{1}{10^{n+1}}<0.00737<\frac{1}{10^n}, then what is the value of n? A. 1 B. 2 C. 3 D. 4 E. 5

Also no need for algebraic manipulation. 1/10^(n+1) is 10 times less than 1/10^n, and both when expressed as decimals are of a type 0.001 (some number of zeros before 1) --> so the given expression to hold true we should have: 0.001<0.00737<0.01, which means that n=2 (1/10^n=0.01 --> n=2).

2 ) For x to be integer, maximum value of x^2 should be a perfect square i.e 81 and minimum value of x^2 16

so maximim value ox x^2 = 81 , hence x = 9 similarly minimum value of x^2 = 16 hence x = 4 maximim value ox x^2 - minimum value of x^2 = 9-4 =5

Answer A

3 ) Since Fanny and alexander are travelling towards each other they travel for the same amount of time say 't' and sum of the distance traveled by them is equal to 360

hence 65t+25t = 360 90t = 360 t = 4 hrs

hence when they meet each would have traveled for 4 hrs. 1.5 hrs before they meet, each would travel for 2.5 hrs.

distance traveled by Fanny in 2.5 hrs= 25* 5/2 = 125/2 miles distance traveled by Alex in 2.5 hrs= 65* 5/2 = 325/2 miles

therfore 1.5 hrs before they meet, distance traveled by them is 125/2+325/2 = 450/2 = 225 miles

Hence they will be 360-225 = 135 miles apart

Answer E

4) we can find the values of y-x at the boundaries of the given inequality

9. Julie is putting M marbles in a row in a repeating pattern: blue, white, red, green, black, yellow, pink. If the row begins with blue marble and ends with red marble, then which of the following could be the value of M? A. 22 B. 30 C. 38 D. 46 E. 54

There are total of 7 different color marbles in a pattern. Now, as the row begins with blue marble and ends with red marble (so ends with 3rd marble in a pattern) then M=7k+3. The only answer choice which is multiple of 7 plus 3 is 38=35+3.

6. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same color and the first and the last marbles are of different colors. How many different arrangements are possible? A. 30 B. 60 C. 120 D. 240 E. 480 .... as there are two cases (R*R*R*R*R* and *R*R*R*R*R. ) then total # of arrangement is 30*2=60.

Answer: B.

I am little confused ...after this bold part.. since we have 5 slots available to be filled by 5 marbles and we can pick any marbles given our setting of marbles (without disturbing any conditions in the stem ?). now # arrangements in either case is going to be 5! (=120) so total # arrangements = 2*120= 240...

what am I missing here

# of arrangements of 2 blue, 2 green and 1 yellow marbles (BBGGY) in 5 slots is 5!/(2!*2!*1!)=30 not 5!, since 2 B's and 2 G's are identical.

THEORY. Permutations of n things of which P_1 are alike of one kind, P_2 are alike of second kind, P_3 are alike of third kind ... P_r are alike of r_{th} kind such that: P_1+P_2+P_3+..+P_r=n is:

\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is \frac{6!}{2!2!}, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be \frac{9!}{4!3!2!}.

So, for our case # of permutations of 5 letters BBGGYout of which 2 B's and 2 G's are identical is \frac{5!}{2!*2!}.

5. The angles in a triangle are x, 3x, and 5x degrees. If a, b and c are the lengths of the sides opposite to angles x, 3x, and 5x respectively, then which of the following must be true? this means, that c >b>a

I. c>a+b nope. since in a triangle any side cant be more than the sum of the rest sides II. c^2>a^2+b^2 it is possible! for example- the sides are 4 2 5 . then 25>16+4 25>20 III. c/a/b=10/6/2 hmm bad idea. let c=10 a=6 b=2 then again c >a+b. and it is not right (as mentioned above)

the answer is B. II only

6. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same colour and the first and the last marbles are of different colours. How many different arrangements are possible? hm have doubts, but will choose E as an answer. if it is a right answer, I can explain my logic E. 480

7. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the ratio of negative numbers to non-negative numbers be 2 to 1? D. 3/7 is the answer

let assume that total=18, then observed=2/9*18=4 non-negative of observed =3/4*4=3

negative/3=2/1 negative =6

6/(18-4)=6/14=3/7

9. Julie is putting M marbles in a row in a repeating pattern: blue, white, red, green, black, yellow, pink. If the row begins with blue marble and ends with red marble, then which of the following could be the value of M?

it means that we have 5*7 (blue, white, red, green, black, yellow, pink)+3 ( blue, white, red)=38 the answer is C

10. If n is an integer and \frac{1}{10^{n+1}}<0.00737<\frac{1}{10^n}, then what is the value of n? B is the answer

0.001<0.00737<0.01

11. The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p? A. 97 B. 151 C. 209 D. 211 E. 219

D is the answer from answer choices we can assume, that the last digit of the sum of 3 numbers could be 7,1,9.

for the last digit being 7- 13 +67 +37=157 (not mentioned in the answ. choices.so eliminate it) for the last digit being 1 - 17+67+37=121 61+77+13=151 (not mentioned in the answ. choices.so eliminate it) 77+71+63=211(bingo!) for the last digit being 9 -couldnt find any number.

12. If {-\frac{1}{3}}\leq{x}\leq{-\frac{1}{5}} and {-\frac{1}{2}}\leq{y}\leq{-\frac{1}{4}}, what is the least value of x^2*y possible?

1/25<=x^2<=1/9 -1/2<=y<-1/4 to find min number we need to multiply min number of y and max number of x^2, i.e. (-1/2)*1/9=-1/18

D is the answer _________________

Happy are those who dream dreams and are ready to pay the price to make them come true

2. If x is an integer and 9<x^2<99, then what is the value of maximum possible value of x minus minimum possible value of x? A. 5 B. 6 C. 7 D. 18 E. 20

I didnt get why everyone solved for x^2 we are asked to find Xmax-Xmin, not x^2 my choice is D _________________

Happy are those who dream dreams and are ready to pay the price to make them come true

3. is already answered. no need to write down the same

4. If -3<x<5 and -7<y<9, which of the following represent the range of all possible values of y-x? (y-x )min=Y min -Xmax =-7-5=-12 (y-x )max=Y max -Xmin =9-(-3)=12

D. -12<y-x<12 is the answer _________________

Happy are those who dream dreams and are ready to pay the price to make them come true

11. The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p? A. 97 B. 151 C. 209 D. 211 E. 219

What is the largest possible sum of these three numbers that we can form? Maximize the first digit: 76+73+71=220=even, so not a prime. Let's try next largest sum, switch digits in 76 and we'll get: 67+73+71=211=prime.

7. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the total ratio of negative numbers to non-negative numbers be 2 to 1? A. 11/14 B. 13/18 C. 4/7 D. 3/7 E. 3/14

If choose variable for set A there will be too many fractions to manipulate with, so pick some smart #: let set A contain 18 numbers.

"2/9 of the numbers in a data set A were observed" --> 4 observed and 18-4=14 numbers left to observe; "3/4 of those numbers were non-negative" --> 3 non-negative and 1 negative; Ratio of negative numbers to non-negative numbers to be 2 to 1 there should be total of 18*2/3=12 negative numbers, so in not yet observed part there should be 12-1=11 negative numbers. Thus 11/14 of the remaining numbers in set A must be negative.

Answer: A.

Bunuel..how ans A?? i got D..

i took 90 instead of 18..

90*2/9=20...20 were observe , 20*3/4= 15 were no negative.. so from remaining 70...we should bring 30 negative so ratio would be 2 to 1..between negative and non-negative ..

70*3/7=30...so i got 3/7 ans..

where m wrong??

Thanks bunuel for posting all these question.. i gonna know that where i m in math now..

The ratio of negative numbers to non-negative numbers to be 2 to 1 there should be total of 90*2/3=60 negative numbers and 30 non-negative.

"2/9 of the numbers in a data set A were observed" --> 20 observed and 90-20=70 numbers left to observe; "3/4 of those numbers were non-negative" --> 15 non-negative and 5 negative; In not yet observed part there should be 60-5=55 negative numbers. Thus 55/70=11/14 of the remaining numbers in set A must be negative.

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