Quantitative Reasoning - DD

Equation for a circle using the Pythageoren Theorem

- (h,k) is the center of the circle; r=radius - (x - h)^2 + (y - k)^2 = r^2 - Horizontal radius = a - Vertical radius = b

Q19. According to a survey, the cost of a tooth extraction is normally distributed with a mean of $220 and a S.D of $30. If a dentist is chosen at random, use the Emπrical Rule to determine the probability that he or she charges b/w $250 and $280 for an extraction?

Ans. Empirical Rule: 68% - 95% - 99.7% 95% lies b/w 2 S.D and 68% lies b/w 1 S.D. To calculate b/w 250 (1) and 280 (2) and the data lied b/w two S.D above the mean so 95% - 68% = 27% above and below the mean. Since we are only looking at one side above the mean, we would divide 27% by 2, so the 13.5% is the Ans.

14. While touring through Europe, Alfred finds that $1 is equal to 7 drachmas, and that 5 drachmas is equal to 3 quilos. What is the ratio of dollars to quilos? .

Ans. Let $ = dollars, D = drachmas, Q = quilos. We then have $1 = 7D and 5D = 3Q. multiplying both sides of the last equation by (7/5) we have 7D = 21/5 Q = 1$. The ratio of dollars to quilos is then 21/5.

Q32. Speedboat travels 9,000 feet across a lake. It travels part of the trip at a speed of 100 feet per second, and the remaining part at 200 feet per second. If the entire trip takes 70 seconds, what fraction of the total distance did the boat travel at a speed of 100 feet per second?

Ans. Let x represents the time spent travelling at 100 feet per second and 70 - x the time spent at 200 fps. Solve for x by adding the two equations: Total Distance = V1t1 + V2t2 9,000 = 100x + 200(70 -x) X = 50 seconds. The distance traveled by the boat in 50 seconds at 100 fps is 50*100 = 5000 feet, which represents 5/9 of the total distance of 9,000 feet.

Q9. What is the probability of rolling two sixes in a row in two rolls of a fair die?

Ans. P of rolling 1 six is 1/6 and P of rolling two sixes consecutively is (1/6) (1/6) = 1/36.

Q2. Zev invests $36,000 in two stocks, A and B. Stock A earns an annual dividend of 11% while Stock B sustains a loss of 4%. At the end of the year, Zev realizes a net profit of $1.560 from the two stocks. How much did he invest in Stock A?

Ans. This is a standard mixture problem. Let x be the amount invested in A, and then 36,000-x is the amount invested in B: 0.11x-0.04(36,000-x) = 1560 11x - 144, 000 + 4x = 156, 000 15x = 300,000 X=20,000

Q7. A and B are two spherical balloons. Balloon A holds 8 times as much air as balloon B. By how many times does the surface area of balloon A exceed that of balloon B?

Ans. Volume of sphere: 4/3πR^3, a sphere with 8 times the volume of another sphere has a radius. Surface Area of a sphere is 4πR^2. So, volume is 8 times the volume of sphere B. Surface Area is 4 times the SA of sphere B.

Q35. How many chords on a circle can be drawn from 6 distinct points lying in the circle?

Ans. a given line can start at any one of the six points, and terminate at any one of the five remaining points. This would give 6*5 = 30 lines if order mattered. However, it does not matter whether we begin drawing a line at point A and terminate it at point B or vice versa. To compensate for the fact that we have counted both possibilities for each line, we divide the 30 by 2 to get the answer: 15.

10. A square of side 20 is inscribed in a circle. What is the area of the circle?

Ans. a square of 20 has a diagonal of length 20/-2 hence the radius is 10/-2 thus the area would be πr^2 = π(10/-2)^2 = 200π Ans.

Q39. How many different rectangular prisms having a volume of 30ft^2 are there with sides that are each an integer number of feet?

Ans. all we need to do is list the five possible ways to express 30 as the product of three distinct integers: 2*3*5, 1*2*15, 1*3*10, 1*5*6, 1*1*30. Answer: 5

Q12. If a=log (3) and b = log (7), which expression represents log (63)?

Ans. because of the given information, we seek to express 63 in terms of powers of 3 and 7. This we do handily: 63 = 7*9 or 3^2 * 7. We now take the log of 63 in this form, making the required substitution: log (63) = log (3^2 * 7) = log (3^2) + log (7) = 2log(3) + log(7) = 2a + b.

Q16. A 120-gallon vat has a drain that pumps out liquid at a rate of 8 gallons per minute. If the vat is empty and the drain is open, at what rate, in gallons per hour, must water be pumped in so that the vat fills is exactly one hour?

Ans. converting 8 gallons per minute to gallons per hour, we get 480 gal/hour. Let x be the rate at which water must be pumped in: We then have x-480=120 gal/hour, leading us to x = 600 gal/hour.

Q25. Which pair of angles are solutions to the equation 2cos(0) = 1?

Ans. cos(0) = ½. For which the first quadrant solution 0 = 60. We know that the cosine is also positive in the fourth quadrant. It gives us the reference angle of 60 and in the 4th Q it gives us 300.

Q21. A piece of paper has 10 distinct dots drawn on it. if a pair of dots determines a line, how many different ways are there to draw a line on this paper?

Ans. here we seek the no. of ways that a pair of distinct points can be matched from the 10, where order does not count (since two points form a line without any preference to order). The answer is thus C (10, 2) = 10!/(8!*2!) = 45

5. Uncle Isador sells a bag of 300 jellybeans at a price of x dollars. At the same rate, How many jellybeans can his nephew Linus purchase for 50 cents?

Ans. if 300 jellybeans cost x dollars, then 300/x jellybeans cost 1 dollar. Taking half of this amount, 150/x jellybeans would cost half a dollar, or 50 cents

Q8. James can paint a room in 4 hours, while Mike can paint the same room in 5 hours. Working together, how many hours will it take them to paint the room if James only works for one hour?

Ans. if James works for 1 hour, then he does ¼ of the job. That leaves ¾ of the job left to Mike, who can do the whole job in 5 hours. So, Mike would do it in: ¾ x 5 = 15/4 = 3 ¾ hours.

9. The mean and standard deviation of the test scores in Ms. Levy's chemistry class is m and s respectively, where no two students received the same score. If Ms. Levy adds one point to every score in the class, how does this affect the mean and the standard deviation?

Ans. if she adds 1 point to each of the test scores, the mean of the class score has to increase by exactly by 1 point. Since the relative distance b/w each of the scores and the mean has not changed, the standard deviation remains exactly the same.

Q6. Sam travels the 40 miles from Albany to Brickville at 60 mph. At what speed must he travel back from Brickville to Albany so that the average velocity for the round trip is 75 mph?

Ans. if the average velocity over the 80-mile round trip is 75 mph then the total time it takes is t=D/v: 80/75 = 16/15 hours. We easily calculate the time taken for the first 40 miles: 40/60 = 10/15 hours. Thus, the time taken for the return trip must be equal to the remaining time: 16/15 - 10/15 = 6/15 = 2/5 hours. But this is also equal to t=D/v = 40/v. hence we set up the equation as: 40/v = 2/5 V = 100 mph.

Q20. A theater has 5 doors. In how many ways can an actor enter one door and leave in a different door?

Ans. in this case, we are simply looking at the no. of ways that 2 different doors can be paired, but where order matters! There are 5 choices for the first door, the entering door, and then only 4 choices for the second door, the exiting door. This gives us a total no. of 5*4 = 20 ways.

7. Mr. Kay's class scored an average of 75 on the most recent math test. If Mr. Kay gives three students an extra five points on each on their tests, the class average will be brought up to 76. How many students took the test?

Ans. let n be the number of students in Mr. Kay's class. An average of 75 means that the sum of the scores of the entire class must be 75n. after adding 5 points to each of the 3 students, the sum of marks increases to 75n+15. The new class average being 76, gives us the new equation: [75n + 15]/n = 76 N = 15 Ans.

13. A store carries bags of oats in 7 pounds and 4-pound bags. If the total amount of oats in stock is 80 pounds, what is the maximum number of 7-pound bags?

Ans. our first impulse might be to divide 80/7 and guess that we can have as many as eleven 7-pound bags. However, 7*11 = 77 leaves 3 pounds left over, not enough to accommodate a 4-pound bag. We then examine how many pounds of oats are left over when we fill 10, 9, 8 etc seven-pound bags until we reach a left-over amount that is divisible by 4. The largest number of 7-pound bags for which this occurs is 8, since 7*8 = 56 we have 24 left that is divisible by 4.

4. Oswaldina invests $4,000 at 6% simple interest for 5 years. How much interest will Oswaldina's investment earn in this time?

Ans. simple interest means that each year the interest accrued is based on the principal only, not on the amount accumulated from the previous years' interest. Each year, the amount of interest earned is a uniform 6%*4000 = $240. After 5 years this adds up to 5*240 = $1200.

Q11. If 2sin(x) - 1 = 0, what is the value of x if 0<x<π/2?

Ans. sin(x) = ½, which is in the first Quadrant and gives a value of x=30 or Π/6 radians.

Q37. Which value is equal to cos(-120) ?

Ans. since -120 places us in the 3rd Quadrant , the value of cos(-120) is negative. Since the reference angle for -120 is 60, the value we seek is cos(60).

Q24. Two work crews, crew A and crew B, start digging a 300-foot tunnel through a mountain, but working toward each other from opposite sides. If crew A digs at a rate of 12 feet a day and crew B digs at a rate of 18 feet a day, how many feet will crew A have dug when the two crews meet?

Ans. since both crews are digging the same tunnel, but from opposite ends, the rate at which the tunnel is being completed is the sum of the digging rates of each crew: 12 feet/day + 18 feet/day = 30 feet/day. Since the tunnel is being dug at the rate of 30 feet/day, it will then take 300/30 = 10 days for the two crews to meet, thus completing the 300-foot tunnel.

Q15. A pleasure boat travels 240 miles downstream from City A to City B in 12 hours. If the speed of the stream is 4 mph, how long will it take the same boat to make the return trip upstream from City B back to City A?

Ans. since it takes the boat 12 hours to travel the 240 miles from A to B, its average downstream speed is v = D/t = 240mi/12h = 20 mph. going back from B to A, the boat's speed will be 8 mph slower, or 12 mph (why? Because the speed of the stream increases the boat's speed by 4 mph on the way down, and decreases it by the same amount on the way up, a difference of 8 mph). Therefore, it will take the boat t = D/V = 240mi/12mph = 20 hours to make the return trip.

Q23. Ted can mow a lawn in 5 hours while Eilot can mow the same lawn in 8 hours. How long will it take them to mow the lawn if they work together, but if Ted only mows for 2 hours?

Ans. ted mows for 2 hours mean that he has moved 2/5 of the lawn (since he can mow the entire lawn in 5 hours), leaving 3/5 of the lawn to be moved by Eilot. Since Eliot can mow the lawn in 8 hours, he finishes the 3/5 of the remaining portion in 3/5*8 = 24/5 = 4 4/5hours, or 4 hours 48 minutes.

Q10. What is the probability of rolling a sum of six and then a sum of six again in two consecutive rolls of a pair?

Ans. the P of rolling a sum of six in a pair of dice is 5/36. Thus, the P of doing this twice in a row is (5/36) (5/36) = 5^2/36^2 = 5^2/6^4.

Q31. The even numbers on one otherwise white regular die are painted blue, while the perfect squares on a second regular die are painted blue. What is the probability that both dice will turn up blue when rolled together?

Ans. the P that one of the even numbers 2, 4 or 6 will turn up in the roll of the first die is 3/6 = ½. The P that a perfect square, i.e. 1 or 4 will turn up on the second die is 2/6 = 1/3. The P of both of these events occurring, and producing a blue face in both cases, is thus (1/2)*(1/3) = 1/6.

21. If two sides of a rhombus meet at an angle of 60 and one of the sides is of length 20, what is the area of the rhombus?

Ans. the area of a rhombus is b*h. the base is of length 20 (hypotenuse). And its height is 10/-3 (30-60-90 triangle). The area of the rhombus is then = 20 * 10/-3 = 200/-3.

Q14. An order comes in to a printing company to make standard labels for 100 soup cans. If each soup can is 3 inches in diameter and 4 inches in height, what is the smallest amount of paper, measured in square inches, that is needed to make the labels?

Ans. the area of a soup can label is represented by the area of a hollow cylinder (top and bottom excluded), which is the circumference of the cylinder times its height: 2πrh. In this case, one label has an area of 3π*4 = 12π so 100 cans would be 1200π.

Q1. A glass holds 6 ounces of 60-proof rum (i.e 30% alcohol). How much fruit juice must be added to the rum so that the mixture is diluted to 40-proof (20% alcohol)?

Ans. the fraction composed of the following: [Amount of pure alcohol]/ [total amount of liquid] Must be equal to 0.20 Let x be the amount of fruit juice to be added: We have: [6*(0.3)]/(6+x) = 0.20 Solving for x, x=3

Q34. A 40-liter tank is filled with 5 liters of acid and the rest water, while a 60-liter tank is filled with 15 liters of acid and the rest water. A mixture of two solutions is used to fill a third 20-liter tank. What quantity of solution from the 40-liter must be used of the mixture must be a 20% acid solution?

Ans. the numbers lead us to the fact that the 40-liter tank is 1/8 acid; and the 60-liter tank is the ¼ acid. We seek a 20-liter solution that is 20% or 1/5 acid. If x represents the amount of solution taken from the 40-liter tank, then 20-x represents the amount taken from the 60-liter tank. We use an expression that sets the sum of pure acid from each part equal to the amount of pure acid in the final mixture: (1/8)x + (1/4)(20-x) = (1/5)(20) Multiplying through by the LCD = 40 5X + 10(20 - X) = 8*20 X = 8 liters

Q29. What is the probability of obtaining four consecutive heads or tails in four tosses of a fair coin?

Ans. the probability of obtaining four consecutives heads in four tosses of a fair coin is (1/2)^4 = 1/16. The probability of obtaining four tails is the same. 1/16. We thus add the two possibilities to find the likelihood of either event: 1/16 + 1/16 = 1/8.

Q27. Jim reaches into a basket containing the letters of the word CANTEEN and randomly pulls out two letters. What is the probability that one letter is a consonant and the other is a vowel in any order?

Ans. the probability that Jim pulls out a vowel first and a consonant second is (3/7)(4/6), while the probability of a consonant first and a vowel second is (4/7)(3/6). We are interested in either of these outcomes (since order does not matter), thus we take the sum of these probabilities: (3/7)(4/6) + (4/7)(3/6) = 4/7.

18. A quadrilateral is inscribed inside an ellipse such that each of their four vertices coincides, as shown. If the equation of the ellipse is x^2/25 + y^2/16 = 1, find the area of the quadrilateral.

Ans. the quadrilateral is composed of equally sized triangle, one in each quadrant. The equation of the ellipse indicates a semi major axis of 5 and semi minor axis of 4. Which also describes the lengths of the legs of each of the 4 right triangles. The area of each of the triangle is thus (1/2)(4)(5) = 10 and thus the area of entire Q is 40.

Q36. If 4n+1, 5n and 7n-6 are three consecutive terms in an arithmetic sequence, what is the value of n?

Ans. the successive terms of an arithmetic sequences are found by adding a fixed amount to the previous term. If we take the fixed amount as k, the relationship b/w the three given terms are expressed as 4n+1+k = 5n, and 5n+k = 7n - 6. This gives us two equations and two unknowns. From the first equation k = n - 1, and from the second equation k = 2n - 6. Setting the two expressions for k, we get n-1 = 2n - 6 and n=5.

Q4. Five chairs are lined up in a row. How many ways three boys and two girls can sit in their chairs if the girls always sit together?

Ans. there are four cases to consider if the girls are to always sit together. GGbbb, bGGbb, bbGGb, bbbGG. In each case, there are 3! = 6 ways the boys can sit and 2! =2 ways the girls can sit. All together we have 4x6x2 = 48 ways.

22. Mike can paint a fence is 4 4/9 hours. However, working together with Sam, he can finish it in one hour and 40 minutes. How long would it take Sam to paint the same fence if he were working alone?

Ans. we follow the standard form of the solution. 1/t1 + 1/t2 = 1/T. so T1 = 4 4/9 = 40/9. T2 = 1 2/3 = 5/3. Hence, we plug in the values in the equation: we get = 9/40 + 1/t2 = 3/5. T2 = 8/3 = 2 2/3 = 2 hours 40 minutes.

If h is an even number and k is an odd number, which if the following is always odd?

Ans. we know that h is even and k is odd. Running through the choices, we see that h/k will not always be an integer, and even if it is, its not always odd. The expressions h+k+1, hk, and h^k are always even. But an odd number raised to an even power, k^h, will always be odd.

Q26. A quadrilateral is inscribed within an ellipse such that the vertices of each object coincide. If the ellipse's major axis of length 20 and its minor axis is of length 15, what is the area of the inscribed quadrilateral?

Ans. we look at the inscribed quadrilateral is being composed of 4 right triangles. Each right triangle has base = 10 (which is half major axis), and height = 15/2 (half the minor axis). The area of each triangle is this (1/2)bh = (1/2)(10)(15/2) = 75/2. Four of these triangles comprise the total area of the quadrilateral, A = 4(75/2) = 150.

11. A lotto winning of $54, 000 was originally to be divided among 9 ticket holders. If 3 more ticket holders step forward with winning numbers, how much less will each of the 9 original winners receive if the winnings are redistributed?

Ans. we subtract the revised apportionment from the original apportionment as follows: 54000/9 - 54000/12 = 6000 - 4500 = $1,500 Ans.

15. If a = log7b7, then what is the value of 7^a?

Ans. when 7 is raised to the power lob7b^7, the result is b^7.

If x > 10 and must be an integer number, which of the following expressions is the largest?

The easiest way to solve this equation is by plugging in a test value into all the answer choices (since the expressions are relatively simple). We will use an "easy" test value of 20.

Number of Solutions to linear equations (KA)

Determine the no. of solutions for an equation? Usually 3 cases 1. When x=5 or any number (1 solution) 2. When 3=5 (no solution) 3. When 13=13 (infinity solution)

Q5. You have 50 ounces of a 25% saline solution. How many ounces of a 10% saline solution must you add to make a new solution that is 15% saline?

Equation: 12.5 +0.1x = 0.15(50 +x) Solve for x: x= 100 Ans

Fahrenheit to Celsius

F= 9/5 C+32 C= 5/9 (F-32)

SPECIAL TRIANGLES

Remember: SOH CAH TOA 1. 30 - 60 - 90 1 - \3 - 2 2. 45 - 45 - 90 1 - 1 - \2

5 people are applying for the positions of president and treasurer. How many different ways can the two positions be filled if each position can only be filled by one person?

This problem can be solved using permutations and combinations. In order to determine which method to use, you must determine if order is important. Let's say that you and a friend were both running for President and Treasurer. In the end, you are President and your friend is Treasurer. Would the outcome be different if the opposite event happened? Of course it would be because you would no longer be President. Therefore, order does matter and a permutation must be used. The sample size (n) is 5, while the number of choices (r) is 2.

1 kg to pounds?

1 kg = 2.2 pounds 1 pound = 0.45 kg

miles to yards

1 mile = 1760 yard

yards to inches

1 yard = 36 inches

Radian and degree conversion

2πRadian = 360 degrees Π radian = 180 degrees 1 radian = 180/π Π/180 radians = 1 degree 1. 30 degrees is π/6 radians 2. Π/3 radians is 60 degrees

Volume of a Sphere

4/3πR^3

8. If the volume of a cube is 1, what is its surface area?

Ans. if the volume of a cube is 1, then each of its side is 1. Each of its six faces must have an area of 1. The total surface area is then 6.

12. The Barnett family buys enough propane to heat their house for 60 days during a normal winter. During to a particularly harsh winter this year, their propane use increased by 25%. For how many days will the same amount of propane last for the Barnett's?

Ans. let a = the amount of daily propane used during a normal winter. Then 1.25a is the daily amount used during the harsh winter. The total amount of propane is thus 60a, which 60a/a = 60 days during a normal winter and 60a/1.25a during the high-usage winter. Simplifying, we have 60a/1.25a = 60/1.25 = 48 days.

Q13. What is the largest possible value of x in the expression |16 - 2x| < 4?

Ans. recall the general pattern: |a-b| = |b-a| Or in this case |16 - 2x| = |2x - 16| Since 2x - 16 < 4, 2x < 20 or x < 10, the maximum possible value for x is 10.

20. What is the smallest positive number that is divisible by both 14 and 49?

Ans. since the prime factorization of 14 = 2*7, and that of 49 = 7*7, the least common multiple is then 2*7*7 = 98.

Q28. Find the value of x that satisfies the equation cox(x-30) = 0?

Ans. since we know that cos(90) = 0, then x-30 = 90, and x = 120.

Ellipse

Ans. the general form of ellipse with a horizontal major axis is: x^2/a^2 + y^2/b^2 = 1, where a is the semi major axis and b is the semi minor axis. The two foci of an ellipse, located along the major axis, are a distance of 2c apart, where c^2 = a^2 - b^2. Area of an ellipse: π(a)(b)

16. What is the largest distance b/w any two points inside a rectangular box whose dimensions are 3, 4, and 5?

Ans. the largest distance b/w two points in a rectangular box is the span from one corner to the diagonally opposite far corner, in this case from (0, 0, 0) to (3, 4, 5). We use the Pythagorean theorem in 3-D to find the distance: /-3^2 + 4^2 + 5^2 = /-50 = 5/-2.

Q40. If logb30 - logb2x = lobb5, the value of x is?

Ans. using the law of logs, logb30 - logb2x = logb(30/2x) = logb(15/x) = logb5. Taking the antilog, we see that 15/x = 5, or x=3.

Q17. Joan is 11 years younger than Sam, who is twice as old as Birdie. If Birdie will be twice as old as Joan in 4 years, how old is Sam now?

Ans. we can make 3 equations: J = S - 11 S = 2B B + 4 = 2(J+4) B = 2J + 4 = ½ S Since J = S-11 and S = 2B We get: 2(S-11) + 4 = ½ S 3S = 36 S = 12, hence Sam is 12 years old.

2. An unscrupulous hotel manager decides to tag on a 0.3% TV tax to his customer's bills in order to help pay for a new digital screen for his daughter. How much is the TV tax on a hotel bill of $240?

Ans. we know that 1% of $240 is $2.40. multiplying by 0.3 we see that it is $0.72.

Q18. For which value of k would the following system of linear equations yield no solution?

2y + kx = 6 Y + 5x = 4 Ans. a system of linear equations yields no solution when the lines are parallel. This occurs when the coefficients of x and y in one equation are identical to or multiple of the coefficients of x and y in the second equation, but where the constant term is different. The two equations of course must be expressed in the same form. 2y + kx = 6 2y + 10x = 8 Clearly, when k=10, the coefficients of x and y match up exactly, yielding no solution.

Q30. What is the length of one side of an isosceles right triangle whose area is 49 square cm?

Ans. (1/2)s^2 = 49, thus s = 7/-2(under root)

6. Car A sets out on a journey at a steady speed of 45 mph. Two hours later Car B sets out from the same starting point along the same route, travelling at a steady rate of 80 mph. Approximately how long will it take Car B to catch up with Car A?

Ans. Car B distance: 80t How long Car A has been on the road: t+2 (started 2 hours earlier) Distance Car A has been on the road: 45(t+2) Hence, 80t = 45(t+2) Solving for t we then get 80t = 45t + 90 We get t=2.5 hours.

Q22. How many solutions to the equation 3*tan(x) = 10 are there in the interval 0<x<2π?

Ans. No calculations are required, only the knowledge that tan(x) = a positive number has two solutions in the interval from 0 - 2π radians.

17. If the probability that it rains is 30%, the probability that Martha's flight will be delayed in 25%, and the probability that it simultaneously rains while Martha's flight is delayed is 6%, what is the probability that either it rains or Martha's flight will be delayed?

Ans. the probability that it either rains or Martha's fight will be delayed is the sum of the probabilities of each event minus their overlap. In this case it is 30% + 25% - 6% = 49%.