37.If Bob produces 36 or fewer items in a week, he is paid x dollars per item. If Bob produces more than 36 items in a week, he is paid x dollars per item for the first 36 items and 11/2 times that amount for each additional item. How many items did Bob produce last week? (1) Last week Bob was paid a total of $480 for the items that he produced that week. (2) This week Bob produced 2 items more than last week and was paid a total of $510 for the items he produced this week.
Solution:
Bob has a regular rate and an overtime rate. We know he gets paid $x per item for up to 36 items and 1.5($x) per item for additional items. We need to determine the exact number of items that Bob produced last week. We can't come up with a simple equation because of the split payment, so let's jump right in to the statements. (1) Since we don't know Bob's rate, there's no way to use this info to figure out how many items he made: insufficient. (2) Since we don't know Bob's rate, there's no way to use this info to figure out how many items he made: insufficient. Neither statement was good enough alone, so here we must combine. Now we know two things: -last week Bob made $480 -this week Bob made $510 and made 2 extra items Therefore, we know that Bob was paid $30 ($510 - $480) for those 2 extra items. However, do we know if Bob was at his overtime or regular rate? Can we figure out the total number of items? Now we have to use the most valuable tool for DS: logic and common sense.
Let's work through two scenarios. First, let's assume that Bob was working at his overtime rate for those two items. If that's the case, then his overtime rate is $15 and his regular rate is $10 (since OT = 1.5 regular). If that's the case, then could Bob have made $480 the previous week? Sure, he could have made 36 items at $10 (for $360) and made 8 items at $15 (for the remaining $120). Second, let's assume that Bob was working at his regular rate for the two items. If that's the case, could Bob have made $480 the previous week? Sure, he could have made 32 items at $15 (for $480). Since both scenarios are possible, we still cannot determine exactly how many items Bob made last week: together insufficient, choose (E).
38.How many positive integers less than 10,000 are there in which sum of digits equals 5? (a) 31 (b) 51 (c) 56 (d) 62 (e) 93
Solution:
Here, we have 4 digits (positive integer less than 10000 means that 9999 is the biggest and we can pretend that "5" is "0005") that must sum to 5. Since we have 4 digits, we'll have 3 partitions. We're summing to 5, so we have 5 "donuts". O O O O O Since we can use 0, we can have multiple partitions in the same spot. For example, we could have: |||OOOOO (which translates to 0005) we could have: ||O|OOOO (which translates to 0014, or 14). So, we view this as a permutation question: we have 8 total objects, 3 of which are identical to each other (the partitions) and 5 of which are identical to each other (the donuts). Using the permutation formula for which some objects are identical: Total permutations = n!/r!s! = 8!/3!5! = 8*7*6/3*2*1 = 8*7 = 56... Choose (C).
39.If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors?
a)Two b)Three c)Five d)Six e)Eight
Solution:
We know that 14n/60 is an integer; therefore, 14n is a multiple of 60. For 14n to be a multiple of 60, 14n must contain (at a minimum) the same primes as does 60. Let's start by breaking down 60: 60 = 2*30 = 2*2*15 = 2*2*3*5
The "14" part of 14n contains a "2". Therefore, n must be responsible for the remaining primes of 60: 2, 3 and 5. Therefore, the minimum possible value for n is 2*3*5 = 30. At this point there are two ways we can finish the question. First, we can use some logic. The answer choices are numbers and only one of them can be correct. Since we already know that n could have 3 different primes, 3 must be the correct answer to the question. Alternatively, we can use the info in the question stem. We know that 0<n<200. The next smallest prime is 7; since 30*7 = 210, we cannot add 7 to the prime factors of n without violating the rule. Therefore, n must have only 3 different prime factors. Note that "different" is a key word in the question; n could also be 60, 90, 120, 150 or 180 (i.e. any multiple of 30 that's less than 200); however, all of these numbers have the same 3 distinct primes, so all of them generate the same answer to the question.
40.A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted? A. 104 B. 213 C. 577 D. 705 E. 726
Solution:
You have to fill up 6 blocks, and each block could be white, black, or red, a total of 3 picks. So, 6 blocks could have a total of 3^6, or 729 different combinations. Since we only have 5 blocks of each color, you have to rule out the combinations of all 6 being white, black or red, so 729 - 3 = 726.
Solution:
Bob has a regular rate and an overtime rate. We know he gets paid $x per item for up to 36 items and 1.5($x) per item for additional items. We need to determine the exact number of items that Bob produced last week. We can't come up with a simple equation because of the split payment, so let's jump right in to the statements. (1) Since we don't know Bob's rate, there's no way to use this info to figure out how many items he made: insufficient. (2) Since we don't know Bob's rate, there's no way to use this info to figure out how many items he made: insufficient. Neither statement was good enough alone, so here we must combine. Now we know two things: -last week Bob made $480 -this week Bob made $510 and made 2 extra items Therefore, we know that Bob was paid $30 ($510 - $480) for those 2 extra items. However, do we know if Bob was at his overtime or regular rate? Can we figure out the total number of items? Now we have to use the most valuable tool for DS: logic and common sense.
Let's work through two scenarios. First, let's assume that Bob was working at his overtime rate for those two items. If that's the case, then his overtime rate is $15 and his regular rate is $10 (since OT = 1.5 regular). If that's the case, then could Bob have made $480 the previous week? Sure, he could have made 36 items at $10 (for $360) and made 8 items at $15 (for the remaining $120). Second, let's assume that Bob was working at his regular rate for the two items. If that's the case, could Bob have made $480 the previous week? Sure, he could have made 32 items at $15 (for $480). Since both scenarios are possible, we still cannot determine exactly how many items Bob made last week: together insufficient, choose (E).
38.How many positive integers less than 10,000 are there in which sum of digits equals 5? (a) 31 (b) 51 (c) 56 (d) 62 (e) 93
Solution:
Here, we have 4 digits (positive integer less than 10000 means that 9999 is the biggest and we can pretend that "5" is "0005") that must sum to 5. Since we have 4 digits, we'll have 3 partitions. We're summing to 5, so we have 5 "donuts". O O O O O Since we can use 0, we can have multiple partitions in the same spot. For example, we could have: |||OOOOO (which translates to 0005) we could have: ||O|OOOO (which translates to 0014, or 14). So, we view this as a permutation question: we have 8 total objects, 3 of which are identical to each other (the partitions) and 5 of which are identical to each other (the donuts). Using the permutation formula for which some objects are identical: Total permutations = n!/r!s! = 8!/3!5! = 8*7*6/3*2*1 = 8*7 = 56... Choose (C).
39.If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors?
a)Two b)Three c)Five d)Six e)Eight
Solution:
We know that 14n/60 is an integer; therefore, 14n is a multiple of 60. For 14n to be a multiple of 60, 14n must contain (at a minimum) the same primes as does 60. Let's start by breaking down 60: 60 = 2*30 = 2*2*15 = 2*2*3*5
The "14" part of 14n contains a "2". Therefore, n must be responsible for the remaining primes of 60: 2, 3 and 5. Therefore, the minimum possible value for n is 2*3*5 = 30. At this point there are two ways we can finish the question. First, we can use some logic. The answer choices are numbers and only one of them can be correct. Since we already know that n could have 3 different primes, 3 must be the correct answer to the question. Alternatively, we can use the info in the question stem. We know that 0<n<200. The next smallest prime is 7; since 30*7 = 210, we cannot add 7 to the prime factors of n without violating the rule. Therefore, n must have only 3 different prime factors. Note that "different" is a key word in the question; n could also be 60, 90, 120, 150 or 180 (i.e. any multiple of 30 that's less than 200); however, all of these numbers have the same 3 distinct primes, so all of them generate the same answer to the question.
40.A rectangular floor measures 2 by 3 meters. There are 5 white, 5 black, and 5 red parquet blocks available. If each block measures 1 by 1 meter, in how many different color patterns can the floor be parqueted? A. 104 B. 213 C. 577 D. 705 E. 726
Solution:
You have to fill up 6 blocks, and each block could be white, black, or red, a total of 3 picks. So, 6 blocks could have a total of 3^6, or 729 different combinations. Since we only have 5 blocks of each color, you have to rule out the combinations of all 6 being white, black or red, so 729 - 3 = 726.
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