Some of the people in Town X are left-handed, some are tall, some are both, and some are neither. InTown Y, three times as many people are left-handed as are left-handed in Town X, three times as many people are tall as are tall in Town X, three times as many people are both as are both in Town X, but no one is neither. If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in Town X who are neither left-handed nor tall? (A) 23 (B) 39 (C) 72 (D) 143 (E) 199
In the options, 143 is the only number divisible by 11.
A Triangular field X is bordered by three square fields A, B and C whose areas are 144, 81 and 225 respectively. The area of the triangular field X is:
Three types of pencils, J,K, and L, cost $0.05, $0.10, and $0.25 each, respectively. If a box of 32 of these pencils costs a total of $3.40 and if there are twice as many K pencils as L pencils in the box, how many J pencils are in the box? (A) 6 (B) 12 (C) 14 (D) 18 (E) 20
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Forty percent of the rats included in an experiment were male rats. If some of the rats died during the experiment and 30 percent of the rats that died were male rats, what was the ratio of the death rate among the male rats to the death rate among the female rats? (A) 9/14 (B) 3/4 (C) 9/11 (D) 6/7 (E) 7/8
How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number? a)1875 b)2000 c)2375 d)2500 e)3875
How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number? a)1875 b)2000 c)2375 d)2500 e)3875
three types of pencils, j,k, and l, cost $0.05, $0.10, and $0.25 each, respectively. If a box of 32 of these pencils costs a total of $3.40 and if there are twice as many k pencils as l pencils in the box, how many j pencils are in the box? (a) 6 (b) 12 (c) 14 (d) 18 (e) 20
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forty percent of the rats included in an experiment were male rats. If some of the rats died during the experiment and 30 percent of the rats that died were male rats, what was the ratio of the death rate among the male rats to the death rate among the female rats? (a) 9/14 (b) 3/4 (c) 9/11 (d) 6/7 (e) 7/8
How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number? a)1875 b)2000 c)2375 d)2500 e)3875
i dunno if this is the right approach ..i'm givin it a try
number -> EEOOO even - 4 choices (2,4,6,8,) odd - 5 choices (1,3,5,7,9)
Even 1) if 2 digits are diff -> 4*3 = 12 2) if both digits are same - 3 (as 4 cant be repeated) TOTAL POSS for Even : 12+3 = 15
Odd 1)all 3 digits are different - 5*4*3 = 60 2) all 3 dig are same - 5 3) 2 digits are same and 1 is different - 5*4*3 = 60 TOTAL POSS for Odd : 60 + 60 + 5 = 125
Diff types of numbers that can be formed : 125*15 = 1875
A Triangular field X is bordered by three square fields A, B and C whose areas are 144, 81 and 225 respectively. The area of the triangular field X is:
A) 150 B) 144 C) 80 D) 54 E) 36
sides of triangle - 12, 9, 15 (in the ratio 3:4:5) so area - 0.5*12*9 = 54
Some of the people in Town X are left-handed, some are tall, some are both, and some are neither. InTown Y, three times as many people are left-handed as are left-handed in Town X, three times as many people are tall as are tall in Town X, three times as many people are both as are both in Town X, but no one is neither. If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in Town X who are neither left-handed nor tall? (A) 23 (B) 39 (C) 72 (D) 143 (E) 199
Ones place can be filled by 1,3,5,7,9 = 5 llly, tens place can be filled by 1,3,5,7,9 = 5 and hundreds place can be filled by 1,3,5,7,9 = 5
Now two left most digits are left:-
Case I when 4 is NOT used in left most place, it can have only 2,6,8 = 3 the 2nd place from left can have = 2,4,6,8,0 So, Total ways = 3*5*5*5*5
Case II when only 4 is used in left most place, it is locked with value 4 = 1 the 2nd place from left can have = 2,6,8,0 So, Total ways = 1*4*5*5*5
Combining two results:- Total ways = 1875 + 500 => 2375 Answer (C)
How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number? a)1875 b)2000 c)2375 d)2500 e)3875
A Triangular field X is bordered by three square fields A, B and C whose areas are 144, 81 and 225 respectively. The area of the triangular field X is:
A) 150 B) 144 C) 80 D) 54 E) 36
OA: D
Three types of pencils, J,K, and L, cost $0.05, $0.10, and $0.25 each, respectively. If a box of 32 of these pencils costs a total of $3.40 and if there are twice as many K pencils as L pencils in the box, how many J pencils are in the box? (A) 6 (B) 12 (C) 14 (D) 18 (E) 20
OA: C ------------------
Forty percent of the rats included in an experiment were male rats. If some of the rats died during the experiment and 30 percent of the rats that died were male rats, what was the ratio of the death rate among the male rats to the death rate among the female rats? (A) 9/14 (B) 3/4 (C) 9/11 (D) 6/7 (E) 7/8
For any integer k > 1, the term length of an integer refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 2 2 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y A. 5 B. 6 C. 15 D. 16 E. 18
For any integer k > 1, the term length of an integer refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 2 2 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y A. 5 B. 6 C. 15 D. 16 E. 18
Nice one Nisha.
IMO: D
Here is the analysis:
x+3y First: to get maximum length, we have to chose the smallest prime factors. The greater the prime factor, the lesser will be the number of prime factors, and thereby the lesser is the length.
So lets chose 2 as the prime factor for consideration here. Anything more than 2, reduces the number of factors and thereby reduces the length.
So, what is the closest 2^p to 1000? 2^10 = 1024
So x+3y to maximize x, lets say x=2^9 = 512 512 + 3.y should be less than 1000. The maximum y for which this is possible is y=2^7 = 128 so, 2^9 + 3. 2^7 = 896 Any increase in the power of 2, will be greater than 1000.
Max. length and min. value of number. Means max. factors shd be 2
SO, by hit n trial:-
x = 512 => (2^9) y = 128 => (2^7) x+3y = 906 Sum of Lengths = 9 + 7 = 16 Answer (D)
I am not sure if I got the Q correctly..
For any integer k > 1, the term length of an integer refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 2 2 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y A. 5 B. 6 C. 15 D. 16 E. 18
For any integer k > 1, the term length of an integer refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 2 2 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y A. 5 B. 6 C. 15 D. 16 E. 18
I dont think x and y are necessarily distinct ... So we want to max length for the lowest no within constraint ... So, x and y have to be 2 ...cox 2^x will be the max length ...
So, to max, x shud be the highest power of 2 below 1000 ...i.e 2^9 =512 So, 3y So, sum of lengths is 9 + 7 = 16 ...Ans D
How many five-digit numbers are there, if the two leftmost digits are even, the other digits are odd and the digit 4 cannot appear more than once in the number? a)1875 b)2000 c)2375 d)2500 e)3875
Nice question!
Let the # be EEOOO.
For O digits, we have 5 options (1, 3, 5, 7, 9). The no. of possibilities for OOO are 5*5*5 = 125.
For the first E digit, we have 4 options (2, 4, 6, . But for the second E digit, we have 5 options (0, 2, 4, 6, 8 ... yes, '0' is even number). Hence, total possibilities for the two even digits, EE, are 4*5 = 20. But this includes one case where digit '4' is selected on both first and the second place (which is what we don't want). Hence, total possibilities = 20 - 1 = 19.
Total number of 5 digit numbers fulfilling the given constraints = 19*125 = 2375.
For O digits, we have 5 options (1, 3, 5, 7, 9). The no. of possibilities for OOO are 5*5*5 = 125.
For the first E digit, we have 4 options (2, 4, 6, . But for the second E digit, we have 5 options (0, 2, 4, 6, 8 ... yes, '0' is even number). Hence, total possibilities for the two even digits, EE, are 4*5 = 20. But this includes one case where digit '4' is selected on both first and the second place (which is what we don't want). Hence, total possibilities = 20 - 1 = 19.
Total number of 5 digit numbers fulfilling the given constraints = 19*125 = 2375.
so in this there is a possibility that a number is like -> 00357.. i mean, can the 1st 2 digits of the number be '0'
For O digits, we have 5 options (1, 3, 5, 7, 9). The no. of possibilities for OOO are 5*5*5 = 125.
For the first E digit, we have 4 options (2, 4, 6 and eight ). But for the second E digit, we have 5 options (0, 2, 4, 6, 8 ... yes, '0' is even number). Hence, total possibilities for the two even digits, EE, are 4*5 = 20. But this includes one case where digit '4' is selected on both first and the second place (which is what we don't want). Hence, total possibilities = 20 - 1 = 19.
Total number of 5 digit numbers fulfilling the given constraints = 19*125 = 2375.
so in this there is a possibility that a number is like -> 00357.. i mean, can the 1st 2 digits of the number be '0'
Hey Nisha,
Solution takes care not to include 0 for the leftmost digit ...
As in we just have included 4 possibilities for the leftmost digit (2,4,6, ..So there is no chance to start with a 0 .
For the next digit, we can include 0, so a total of 5 options ... and a repeat poss of 4 for first 2 digits is 1 way . Hence a total of 4*5 - 1 poss for first 2 digits...
so in this there is a possibility that a number is like -> 00357.. i mean, can the 1st 2 digits of the number be '0'
take it this way.. let the 5 digit number be ABCDE.. 2 Left most digits are EVEN, but since it is 5 digit number it cannot start with '0' and also we know that 4 cannot be repeated.. A - 2-4-6-8 B - 0-2-4-6-8
this makes 19 possible combinations.. (2x(0-2-4-6- 8 = 5) + (4x(0-2-6- 8 =4) + (6x(0-2-4-6- 8 =5) + (8x(0-2-4-6- 8 =5) = 5+4+5+5 = 19 ways
For the remaining 3 digits.. C - 1-3-5-7-9 D - 1-3-5-7-9 E - 1-3-5-7-9 = 5x5x5 = 125 ways
. But for the second E digit, we have 5 options (0, 2, 4, 6, 8 ... yes, '0' is even number). Hence, total possibilities for the two even digits, EE, are 4*5 = 20.