OpenOffice-Quant Flashcards
A scientist is studying bacteria whose cell population doubles at constant intervals, at which times each cell in the population divides simultaneously. Four hours from now, immediately after the population doubles, the scientist will destroy the entire sample. How many cells will the population contain when the bacteria is destroyed?
(1) The population just divided and, since the population divided two hours ago, the population has quadrupled, increasing by 3,750 cells.
(2) The population will double to 40,000 cells with one hour remaining until the scientist destroys the sample.
We need two additional pieces of information to solve this problem, which can be rephrased as “How frequently does the population double, and what is the population size at any given time immediately after it has doubled?”
(1) SUFFICIENT: If the population quadrupled during the last two hours, it doubled twice during that interval, meaning that the population doubled at 60 minute intervals. Since it has increased by 3,750 bacteria, we have:
Population (Now) – Population (2 hours ago) = 3,750
Population (Now) = 4·Population (2 hours ago)
Substituting, we get 4·Population (2 hours ago) – Population (2 hours ago) = 3,750
Population (2 hours ago) = 1,250.
The population will double 6 times from that point to 4 hours from now
Population (4 hours from now) = (2 6)(1,250) = 80,000.
(2) INSUFFICIENT: This statement does not give any information about how frequently the population is doubling.
The correct answer is A.
A driver paid n dollars for auto insurance for the year 1997. This annual premium was raised by p percent for the year 1998; for each of the years 1999 and 2000, the premium was decreased by 1/6 from the previous year’s figure. If the driver’s insurance premium for the year 2000 was again n dollars, what is the value of p? 12 33.5 36 44 50
Note that you can pick a smart number for n, but not for p. The value of p must match one of the answers (since the answer choices represent p).
One approach, then, is to plug those answers into the problem. Let the 1997 premium be n = $100. Ordinarily, you’d start with answer B or D, but the numbers are pretty annoying. Make your life easy; start with 50%.
If p = 50, then the 1998 premium is 50% more than $100, or $150. One-sixth of $150 is $25, so the 1999 premium is $150 – $25 = $125. One-sixth of that amount is $20 plus some change, so the 2000 premium is $125 – about $20 = about $104.
This number doesn’t equal the starting point, $100, but it’s not off by very much, so try choice (D) next.
If p = 44, then the 1998 premium is 44% more than $100, or $144. One-sixth of $144 is $24, so the 1999 premium is $144 – $24 = $120. One-sixth of that amount is $20, so the 2000 premium is $120 – $20 = $100. Bingo!
The correct answer is D.
A researcher has determined that she requires a minimum of n responses to a survey for the results to be valid. If p% of the surveyed individuals fail to respond to the survey, how many individuals, in terms of n and p, must the researcher survey to produce twice the minimum required number of responses?
A. 200n/100-p B. 2n/100-p C. 200n/p D. 2n (100+p)/100 E. (2n+2np)/100
Algebraic Solution
To solve this problem with algebra, let x represent the desired number of people, i.e., the number of people the researcher must include in the survey to achieve the desired yield of 2n responses (= twice the minimum number). Since there is typically a non-response rate of p percent, it follows that, if x is reduced by p percent, the result must be 2n in order for the researcher to produce twice the minimum required number of responses.
x-xp/100 = 2n x(100-p)/100 = 2n x = 200n/ 100-p
The correct answer is A.
Six mobsters have arrived at the theater for the premiere of the film “Goodbuddies.” One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand, though not necessarily right behind him. How many ways can the six arrange themselves in line such that Frankie’s requirement is satisfied? 6 24 120 360 720
Ignoring Frankie’s requirement for a moment, observe that the six mobsters can be arranged 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways in the concession stand line. In each of those 720 arrangements, Frankie must be either ahead of or behind Joey. Logically, since the combinations favor neither Frankie nor Joey, each would be behind the other in precisely half of the arrangements. Therefore, in order to satisfy Frankie’s requirement, the six mobsters could be arranged in 720/2 = 360 different ways.
The correct answer is D.
Ben is driving on the highway at x miles per hour. (One mile equals 5,280 feet.) Ben's tires have a circumference of y feet. Which of the following expressions gives the number of revolutions each wheel turns in one hour? 5,280(x/y) 5,280(y/x) 5,280(xy) 5,280/(xy) (xy)/5,280
The number of wheel revolutions that occur when the car drives a particular distance is:
distance traveled per wheel revolution
The total distance traveled by the car in one hour is x miles, or 5280x feet. The distance traveled per wheel revolution is y feet. Therefore, the number of wheel revolutions that occur in one hour is 5280x / y.
Alternatively, because we have variables in the answer choices, we can choose numbers for x and y. Let’s set x equal to 2 miles per hour and y equal to 10 feet. In one hour, then, Ben travels 2 miles, or 10,560 feet. If every revolution takes him 10 feet, then the tires revolve 10,560/10 = 1,056 times. When we plug x = 2 and y = 10 into the answer choices, only A returns the correct answer of 1,056 revolutions.
If x and y are positive and x2y2 = 18 – 3xy, then x2 =
18 – 3y /y3
18/y2
18/y2 + 3y
9/y2
36/y2
The equation x2y2 = 18 – 3xy is really a quadratic, with the xy as the variable.
x2y2 + 3xy – 18 = 0
(xy + 6)(xy – 3) = 0
xy = 3 or -6
However, we are told that x and y are positive so xy must equal 3.
Therefore, x = 3/y and x2 = 9/y2.
What is the value of y?
(1) 3| x 2 – 4| = y – 2
(2) |3 – y| = 11
(1) INSUFFICIENT: Since this equation contains two variables, we cannot determine the value of y. We can, however, note that the absolute value expression | x 2 – 4| must be greater than or equal to 0. Therefore, 3| x 2 – 4| must be greater than or equal to 0, which in turn means that y – 2 must be greater than or equal to 0. If y – 2 > 0, then y > 2.
(2) INSUFFICIENT: To solve this equation for y, we must consider both the positive and negative values of the absolute value expression:
If 3 – y > 0, then 3 – y = 11
y = -8
If 3 – y
A student committee that must consist of 5 members is to be formed from a pool of 8 candidates. How many different committees are possible? 5 8 40 56 336
To find the total number of possible committees, we need to determine the number of different five-person groups that can be formed from a pool of 8 candidates. We will use the anagram method to solve this combinations question. First, let’s create an anagram grid and assign 8 letters in the first row, with each letter representing one of the candidates. In the second row, 5 of the candidates get assigned a Y to signify that they were chosen for a committee; the remaining 3 candidates get an N, to signify that they were not chosen:
A| B| C| D| E| F| G| H
Y| Y| Y| Y| Y| N| N| N
The total number of possible five-person committees that can be created from a group of 8 candidates will be equal to the number of possible anagrams that can be formed from the word YYYYYNNN = 8! / (5!3!) = 56. Therefore, there are a total of 56 possible committees.
If xy represents a positive two-digit number, where x and y are single digit integers, which of the following CANNOT be true? x + y = 9 (x)(y) = 9 x – y = 9 y – x = 9 x/y = 9
Try to find at least one pair of values for x and y that could work for each answer choice. This is possible for all answer choices except choice (D). (A) x + y = 9 x = 4, y = 5 (B) (x)(y) = 9 x = 1, y = 9 (C) x – y = 9 x = 9, y = 0 (D) y – x = 9 x = ..., y = ... (E) x/y = 9 x = 9, y = 1
There are no possible values for x and y that could satisfy the equation y – x = 9. The largest possible value of y is 9, but the smallest value for x is 1, not 0. If x were equal to 0, then xy would represent a single digit number, for example 08, which equals 8—but the problem indicates that xy is a two-digit number. The value of y – x, then, cannot be larger than 8.
The correct answer is (D).
At a certain company, some of the employees have advanced degrees and some have one year or more of work experience. If 20 percent of all employees have one year or more of experience and no advanced degree, what percentage of employees have advanced degrees and less than one year of experience?
(1) 35 percent of the employees do not have advanced degrees and 55 percent have less than one year of work experience.
(2) 15 percent of the employees lack an advanced degree and have less than one year of work experience; 45 percent have one year or more of experience.
Overlapping set problems involving two sets, as we have here (experience vs. degrees), can be solved most easily using a Double Set Matrix:
|ADV. DEG| NO DEG| TOTALS LESS THAN 1 YR. | ? | | 1 YR. OR MORE | | | TOTALS | | | (1) SUFFICIENT: (2) SUFFICIENT: The correct answer is D.
Triathlete Dan runs along a 2-mile stretch of river and then swims back along the same route. If Dan runs at a rate of 10 miles per hour and swims at a rate of 6 miles per hour, what is his average rate for the entire trip in miles per minute? 1/8 2/15 3/15 1/4 3/8
There is an important key to answering this question correctly: this is not a simple average problem but a weighted average problem. A weighted average is one in which the different parts to be averaged are not equally balanced. One is “worth more” than the other and skews the “simple” average in one direction. In addition, we must note a unit change in this problem: we are given rates in miles per hour but asked to solve for rates in miles per minute.
Average rate uses the same D = RT formula we use for rate problems but we have to figure out the different lengths of time it takes Dan to run and swim along the total 4-mile route. Then we have to take the 4 miles and divide by that total time. First, Dan runs 2 miles at the rate of 10 miles per hour. 10 miles per hour is equivalent to 1 mile every 6 minutes, so Dan takes 12 minutes to run the 2 miles. Next, Dan swims 2 miles at the rate of 6 miles per hour. 6 miles per hour is equivalent to 1 mile every 10 minutes, so Dan takes 20 minutes to swim the two miles.
Dan’s total time is 12 + 20 = 32 minutes. Dan’s total distance is 4 miles. Distance / time = 4 miles / 32 minutes = 1/8 miles per minute.
Note that if you do not weight the averages but merely take a simple average, you will get 2/15, which corresponds to incorrect answer choice B. 6 mph and 10 mph average to 8mph. (8mph)(1h/60min) = 8/60 miles/minute or 2/15 miles per minute.
The correct answer is A.
Two sides of a triangle have lengths x and y and meet at a right angle. If the perimeter of the triangle is 4x, what is the ratio of x to y ? 2 : 3 3 : 4 4 : 3 3 : 2 2 : 1
Perhaps the most straightforward way to solve this problem is to “plug” answer choices back into the problem and check the truth of the final condition (i.e., the perimeter of the triangle must work out to 4x). Given a ratio x : y, we can select two specific values x and y having that ratio, use the Pythagorean theorem to find the hypotenuse of the triangle, add up all three sides to find the perimeter, and, finally, check whether the perimeter is indeed equal to 4x as required.
(A) Let x = 2 and y = 3. Using the Pythagorean theorem, the hypotenuse of the triangle is . These values yield a perimeter of , which is not equal to 4x = 8.
(B) Let x = 3 and y = 4. Using the Pythagorean theorem, the hypotenuse of the triangle is . These values yield a perimeter of 12, which is equal to 4x.
(C) Let x = 4 and y = 3. Using the Pythagorean theorem, the hypotenuse of the triangle is . These values yield a perimeter of 12, which is not equal to 4x = 16.
(D) Let x = 3 and y = 3. Using the Pythagorean theorem, the hypotenuse of the triangle is . These values yield a perimeter of , which is not equal to 4x = 12.
(E) Let x = 2 and y = 1. Using the Pythagorean theorem, the hypotenuse of the triangle is . These values yield a perimeter of , which is not equal to 4x = 8.
Using the answer choices is the most efficient method on this problem, but there is an algebraic solution.
Set A contains three different positive odd integers and two different positive even integers; set B contains two different positive odd integers and three different positive even integers. If one integer from set A and one integer from set B are chosen at random, what is the probability that the product of the chosen integers is even? 6/25 2/5 1/2 3/5 19/25
We have several options for solving this problem. The most efficient way is via the 1 – x Strategy: that is, calculating the probability of the outcome that we do not want (odd) and subtracting from 1. This strategy is most efficient on this problem because there is only one way in which the product of 2 numbers can be odd: when the two starting numbers are also odd. If one or both of the starting numbers are even, then the product will also be even.
1 – x Strategy
There are 3 possibilities for choosing an odd number from from set A and 2 possibilities for choosing an odd number from set B. There are 3 × 2 = 6 possibilities, then, for obtaining an odd product. The total number of general possible outcomes is 5 × 5 = 25, so the probability of obtaining an odd result is 6/25. Don’t forget the last step! 1 - 6/25 = 25/25 - 6/25 = 19/25.
The manufacturer’s suggested retail price (MSRP) of a certain item is $60. Store A sells the item for 20 percent more than the MSRP. The regular price of the item at Store B is 30 percent more than the MSRP, but the item is currently on sale for 10 percent less than the regular price. If sales tax is 5 percent of the purchase price at both stores, how much more will someone pay in sales tax to purchase the item at Store A? $0 $0.09 $0.18 $0.30 $1.80
First, determine the price of the item at each store.
The MSRP is $60.
Store A adds 20%. Take 10%, $6, and multiply by 2 to get 20%, or $12. Store A’s price is $60 + $12 = $72.
Store B adds 30% and then discounts by 10%. First, find the original price: 10% multiplied by 3 is 6 × 3 = 18 (or 30%). The non-sale price is $60 + $18 = $78. Then subtract 10% of the new number: $78 - $7.80 = $70.20.
The sales price is 5% at both stores. The question asks how much more someone will pay in sales tax at Store A. Don’t calculate the full sales tax for the item at each store! The sales tax is identical at the two stores for the first $70.20 spent.
Instead, calculate only the sales tax on the difference between the two prices. The difference is $1.80. 10% of that number is $0.18. 5% is half of that figure, or $0.09.
The correct answer is (B).
If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}?
1,800
1,845
1,890
1,968
2,016For sequence S, any value Sn equals 6n. Therefore, the problem can be restated as determining the sum of all multiples of 6 between 78 (S13) and 168 (S28), inclusive. The direct but time-consuming approach would be to manually add the terms: 78 + 84 = 162; 162 + 90 = 252; and so forth.
The solution can be found more efficiently by identifying the median of the set and multiplying by the number of terms. Because this set includes an even number of terms, the median equals the average of the two ‘middle’ terms, S20 and S21, or (120 + 126)/2 = 123. Given that there are 16 terms in the set, the answer is 16(123) = 1,968.
The correct answer is D.
