Problem Solving
The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4pi/3 what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
D. If we know the interior angle, then we know what the length of RU is. Make sure that RU is a direct line segment. It is 'not' a line segment. (4pi/3) / (8pi) = 1/6 So, the internal angle is 60 degrees. Knowing this, and that an arc segment is an isoceles triangle, we know that all angles are 60 degrees, an equilateral triangle. So, the length equals the radius.
Is 107 prime?
Divide your number by the primes lower than the square root. 7^2 is the lowest square root. Try dividing by 2, 3, 5, 7. You'll find that it is a prime.
How many hours would it take Pump A and Pump B working together, each at its own constant rate, to empty a tank that was initially full? (1) Working alone at its constant rate, Pump A would empty the full tank in 4 hours 20 minutes. (2) Working alone, Pump B would empty the full tank at its constant rate of 72 liters per minute.
E Obviously 1 and 2 alone cannot do this problem. Unfortunately, no matter how hard we want to equate the fact that B in hours per job, we cannot because we don't know the total liters a job finishes.
During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? A) [180-x]/2 B) [x+60]/4 C) [300-x]/5 D) 600/[115-x] E) 12,000/[x+200]
E Total Distance divided by total time. Total Distance = 100 Total Time is d = r*t summed up.
divisible by 8?
If the last 3 digits themselves are divisible by 8, then divisible by 8
divisible by 4?
If the last two digits themselves are divisible by 4, then it is divisible by 4
terminating decimals
If the prime number of the denominator has anything other than 2 or 5, it will not terminate.
divisible by 9?
If the sum of the digits is divisible by 9, then it is divisible by 9
Factor this expression: [ (X^4 + 1)(x^2 + 1)(x + 1) ] / 3y
Realize that this is a cascading difference of squares. if we multiply by 1 = (x - 1) / (x - 1) then we know that we can simplify this expression to... (x^8 - 1) / [(3y)(x-1)
Sum of equally spaced numbers.
Sum of numbers = average of numbers * number of numbers. avg = ( biggest + smallest ) / 2 # of # = 1 + (( biggest - smallest ) / spacing)
Divisible by 3?
Take all the digits and add them up. Are they divisible by 3?
Two numbers differ by 2 and sum to S. Which of the following is the greater of the numbers in terms of S ? A (S / 2) - 1 B (S / 2) C (S / 2) + (1 / 2) D (S / 2) + 1 E (S / 2) + 2
The key word here is greater of the numbers. This means that if we choose two numbers. 3 and 5. We are looking for the answer to be 5. D = 5 when S = 8. D
Ada and Paul received their scores on three tests. On the first test, Ada's score was 10 points higher than Paul's score. On the second test, Ada's score was 4 points higher than Paul's score. If Paul 's average (arithmetic mean) score on the three tests was 3 points higher than Ada's average score on the three tests, then Paul's score on the third test was how many points higher than Ada's score? (A) 9 (B) 14 (C) 17 (D) 23 (E) 25
This one can be solved by using your own numbers. P = [10, 10, x] A = [20, 14, y] (20 + x) / 3 = [(34 + y) / 3] + 3 We are looking for y - x = 23. D
Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? A. 100p / (125-p) B. 150p / (250-p) C. 300p / (375-p) D. 400p / (500-p) E. 500p / (625-p)
This one can get a little tricky. The best thing we can do here is first identify the objective. We are searching for what r is. We can then go on to plug numbers in. A = 20; Pa = 1 B = 20; Pb = 1.25 r = 100 * 20 / (20 (1) + 20 (1.25) = 4/9 p = 100 * 20 / (40) = 50 Now, we just need to see what matches. Also, realize that there is only 1 denominator (D) that divides by 9. D
3r <= 4s + 5 abs(s) <=5 Given the inequalities above, which of the following CANNOT be the value of r? A -20 B -5 C 0 D 5 E 20
This one really doesn't make sense. s can only be from -5 to 5 inclusive. How the hell is it A?
In a recent election, Ms. Robbins received 8,000 votes cast by independent voters, that is, voters not registered with a specific political party. She also received 10 percent of the votes cast by those voters registered with a political party. If N is the total number of votes cast in the election and 40 percent of the votes cast were cast by independent voters, which of the following represents the number of votes that Ms. Robbins received? (A) 0.06N + 3,200 (B) 0.1N + 7,200 (C) 0.4N + 7,200 (D) 0.1N + 8,000 (E) 0.06N + 8,000
This question requires us to understand the constituent parts of the voting system described and is looking for the total votes that were received by Ms. Robbins. N total votes, split by independent and partied. This can be described in a punnett square, a tree diagram, or whatever algebra. N = 60 % N + 40 % N Ms. Robbins Indep = 8000 Ms. Robbins Party = 60% N * 10 % E: 6% N + 8000
How many factors? 54?
Use prime factorization with powers and multiply all the pow + 1. 54 = 2 * 3^3 --> (1+1) * (3 + 1) = 8
A total of s oranges are to be packaged in boxes that will hold r oranges each, with no oranges left over. When n of these boxes have been completely filled, what is the number of boxes that remain to be filled? (A) s - nr (B) s - (n/r) (C) rs - n (D) (s/n) - r (E) (s/r) - n
Use your own numbers with variables in the answer choices! if there are 100 oranges (s) and each box holds 20 (r). Dividing those gives us the total boxes (t). We are looking for the answer choice that gives us t - n. That is E.
How many solutions in a quadratic?
if sqrt(B^2 - 4AC) = 0 then 1 solution.
1/8 in decimals
.125
1/6 in decimals
0.166666666...
3/8 in decimals
0.375
5/8 in decimals
0.625
5/6 in decimals
0.833333333...
7/8
0.875
sqrt(2)
1.4
sqrt(3)
1.7
In 2010, company J's revenues were 100k. 2011 revenues were 200% of 2010 revenues. What were 2011 revenues?
100k * 2 = 200k
In 2010, company K's revenues were 100k. 2011 revenues were 200% greater than 2010 revenues. What were 2011 revenues?
100k + 200k = 300k
Prime numbers under 40
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
sqrt(5)
2.2
23.07 * 5.348 has how many decimal spots?
5. Add the decimal spots of the two. 123.37836
M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true? (A) 1/3 < M < 1/2 (B) 1/5 < M < 1/3 (C) 1/7 < M < 1/5 (D) 1/9 < M < 1/7 (E) 1/12 < M < 1/9
A A way to approximate the maximum is by taking the highest reciprocal and multiply it by the number of iterations. A way to approximate the minimum is by taking the lowest reciprocal and multiply it by the number of iterations. 100*(1/201) > M > 100*(1/300)
If money is invested at r percent interest, compounded annually, the amount of the investment will double in approximately 70/r years. If Pat's parents invested $5,000 in a long-term bond that pays 8 percent interest, compounded annually, what will be the approximate total amount of the investment 18 years later, when Pat is ready for college? (A) $20000 (B) $15000 (C) $12000 (D) $10000 (E) $9000
A If an investment will double in 70/r years, we know that Pat's parent's investment of 8% will double in 8.7 years. Over the 18 years, we will see doubling around twice. $5k(2)^2 = $20k
What is the thousands digit in the decimal equivalent of 53 / 5000?
A Just make the denominator equal 1 by starting with a thousand. 53/5 = 10.6 10.6/1000 = .0106
A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys? A) 3/8 B) 1/4 C) 3/16 D) 1/8 E) 1/16
A Think of probability questions in terms of acceptable / total. There are 6 total ways 2 boys and 2 girls could have been born into a 4 children family. There are 2*2*2*2 possibilities for the total. 6/12 = 3/8
If n is positive, is sqrt(n) > 100? 1) sqrt(n - 1) > 99 2) sqrt(n + 1) < 101
B Just square 99 and 101 and see how that works on the number line..
If a and b are positive integers, is cubert(ab) an integer? 1) sqrt(a) is an integer 2) b = sqrt(a)
B Statement 1 alone does not tell us anything. Statement 2 works though: substitute a as b^2 to get cubert(b^3) to get b. Just realize, when one way doesn't work, try the other way.
For each student in a certain class, a teacher adjusted the student's test score using the formula y = 0.8x + 20, where x is the student's original test score and y is the student's adjusted test score. If the standard deviation of the original test scores of the students in the class was 20, what was the standard deviation of the adjusted test scores of the students in the class? A. 12 B. 16 C. 28 D. 36 E. 40
B The standard deviation is the same as taking the standardized value minus the expected value. So, we can use algebra to get to our answer. [0.8*(mu + 20) + 20] - [0.8*mu + 20] 0.8mu + 16 - 0.8mu = 16
A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary costs rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993 ? (A) 16.5% (B) 20% (C) 35% (D) 55% (E) 65%
B Weighted average problem. Draw a number line and attribute each value. Because there are 5 total parts and avg is closes to sal, we say that salary has 4/5 of the weight while fb has 1/5 of the weight which is the same as the beginning costs. sal avg fb 3 3.5 4 4.5 5 5.5 |------|------|------|------|------|
A salesperson who had been driving at a speed of 100 kilometers per hour slowed down to a speed of 47 kilometers per hour. Approximately how many miles per hour was the speed reduced? (1 kilometer ~ 0.625 mile) A) 29 B) 33 C) 53 D) 63 E) 75
B When you see the word 'approximately' take that as seriously as you can. 100kph ~ 62.6 mph. This means that 47kph is about half of 62.6. B is the closest answer.
The outer dimensions of a closed rectangular cardboard box are 8 centimeters by 10 centimeters by 12 centimeters, and the six sides of the box are uniformly 1/2 centimeter thick. A closed canister in the shape of a right circular cylinder is to be placed inside the box so that it stands upright when the box rests on one of its sides. Of all such canisters that would fit, what is the outer radius, in centimeters, of the canister that occupies the maximum volume? A. 3.5 B. 4 C. 4.5 D. 5 E. 5.5
C Only 3 cylinders could fit within the dimensions 7x9x11. (4.5)(pi)(7) (3.5)(pi)(11) (3.5)(pi)(9) We only need to compare the first two.
On May first, in order to reduce the number of overdue books, a children's library instituted a policy of forgiving fines and giving bookmarks to children returning all of their overdue books. On July first there were twice as many overdue books as there had been on May first, although a record number of books had been returned during the interim. Which of the following, if true, most helps to explain the apparent inconsistency in the results of the library's policy? (A) The librarians did not keep accurate records of how many children took advantage of the grace period, and some of the children returning overdue books did not return all of their overdue books. (B) Although the grace period enticed some children to return all of their overdue books, it did not convince all of the children with overdue books to return all of their books. (C) The bookmarks became popular among the children, so in order to collect the bookmarks, many children borrowed many more books than they usually did and kept them past their due date. (D) The children were allowed to borrow a maximum of five books for a two-week period, and hence each child could keep a maximum of fifteen books beyond their due date within a two-month period. (E) Although the library forgave overdue fines during the grace period, the amount previously charged the children was minimal; hence, the forgiveness of the fines did not provide enough incentive for them to return their overdue books.
C Read carefully. borrowing more books lead them past their due date.
The sides of a square region, measured to the nearest centimeter, are 6 centimeters long. The least possible value of the actual area of the square region is (A) 36.00 (B) 35.00 (C) 33.75 (D) 30.25 (E) 25.00
D 'nearest centimeter' means the value could be from 5.5 centimeters to under 6.5 centimeters. 5.5 squared is 30.25
There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of 5 books from the shelf contain at least 1 paperback and at least 1 hardback? A) 75 B) 120 C) 210 D) 246 E) 252
D Queue the words 'at least'. This likely points to us using the compliment to our benefit. To calculate no paperbacks or no hardbacks, we see that we can only create a scenario where there are no paperbacks. 6 choose 5 hardbacks = 6. Out of the 10 choose 5 = 252 possibilities, we know that 6 would not work so the answer is 252 - 6 = 246
If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers? 1) The range of the n integers is 14 2) The greatest of the n integers is 17
D There are 2 ways to solve this problem. 10n = x + ... [x + 2(n-1)] where n is some odd number. OR. Just put the numbers on a number line and see how they fall in. 1) If the range of the n integers is 14, then the equation above is solvable for n as 8, then sum being 80, the rest is easy. 2) Using the same equation, or even without it, we can reverse it to know the rest of the numbers in the chain backwards
The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter? A) 12 B) 18 C) 24 D) 36 E) 48
D There are two ways to solve this. Total - exceptions, or take it in a scenario approach. scenarios: 1 letter in between: 3 x 3 x 2 = 18 2 letters in between: 2 x 3 x 2 = 12 3 letters in between: 3 x 2 = 6 sum = 36 Total - exceptions: Total ways to express the letters but I not counting twice is 5x4x3x2!/2! = 5x4x3 = 60. Total times I can be put together is 4 times the orderings of D, G, and T which is 6. 60 - (4x6) = 36
If x is to be chosen at random from the set {1, 2, 3, 4} and y is to be chosen at random from the set {5, 6, 7}, what is the probability that xy will be even? A) 1/6 B) 1/3 C) 1/2 D) 2/3 E) 5/6
D Think of probability questions in terms of acceptable / total. There are obviously 12 different possibilities. To get the number of acceptable, just manually calculate out the number of evens. Be smart on these problems. Don't try to do it for the foundational knowledge.
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities? (A) 10 (B) 45 (C) 50 (D) 55 (E) 65
D Think of the problem as 3 overlapping sets. We are searching for exactly 2 of the amenities, so we want to subtract out the innermost overlap + the outer rings + the outside of the rings. First, convert the numbers into the target unit (not percent). [Aircn] A = 90 [Porch] B = 75 [swims] C = 45 [nones] D = 5 [althre] E = 5 [overlaps] abs((150 - 5) - (90 + 75 + 45 )) = 65 # the middle was counted twice, so subtract 2*5 65 - 10 = 55
Of the 300 subjects who participated in an experiment using virtual-reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects? A. 105 B. 125 C. 130 D. 180 E. 195
D Think of the problem as a 3 sets. We are searching for all that experienced only 1. So, we need to subtract out the innermost overlap + the second overlaps (35%). First, convert the numbers into the target unit (not percent). [sweaty] A = 120 [vomits] B = 90 [dizzies] C = 225 [2 cond] D = 105 A + B + C = 435. [Overlap] 435 - 300 = 135 [Inner] (135 - 105) / 2 = 15 # innermost was counted 1 in 135, so divided by 2 more. 300 - 105 - 15 = 180
Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope? (A) 82 (B) 118 (C) 120 (D) 134 (E) 152
D To get the maximum of possible length of the last piece, we need the minimum possible values from the entire set. {a, b, c, 84, e, f, 4a+14} To do this and keep the median, we can make sure that a, b, and c are the same values. Also, we can equate e and f to be the same as the median, yielding: {a, a, a, 84, 84, 84, 4a+14} Now we can solve for a and plug it in for the longest piece. [7a + 14 + 3(84)]/7 = 68 a = 30; 4a + 14 = 134
A photography dealer ordered 60 Model X cameras to be sold for $250 each, which represents a 20 percent markup over the dealer's initial cost for each camera. Of the cameras ordered, 6 were never sold and were returned to the manufacturer for a refund of 50 percent of the dealer's initial cost. What was the dealer's approximate profit or loss as a percent of the dealer's initial cost for the 60 cameras? A. 7% loss B. 13% loss C. 7% profit D. 13% profit E. 15% profit
D Total Profit / Total Initial Cost. Initial Cost = 250(5/6)(60) = 250(5)(10) Revenue = 250(54) Refund = 250(1/2)(5/6)(6) = 250(5/2) Do NOT double count the refund for the cost. Total Profit = Revenue + Refund - Initial Cost 250(6.5) = 250[54 + 5/2 - 50] Answer = 250(6.5) / 250(5)(10) 6.5/50 --> 13/100
A windows is in the shape of a regular hexagon with each side of length 80 cm. If a diagonal through the center of the hexagon is w centimeters long, then w =? A. 80 B. 120 C. 150 D. 160 E. 240
D We know that the total angle measurement = 180(6-2)/6 = 120. If we were to create a sector, we know that each of the angles, cut in half is 60 degrees, so an equilateral triangle. So, the diameter = 2(80)=160
On a scale drawing of a triangular piece of land, the sides of the triangle have lengths 5, 12, and 13 centimeters. If 1 centimeter on the drawing represents 3 meters, what is the area, in square meters, of the piece of land? A. 90 B. 180 C. 240 D. 270 E. 540
D We should immediately realize that this is a 5, 12, 13 right triangle. However, we cannot simply apply 3 times the area because we are adding. We must multiply each length by 3. (5)(3)(12)(3)/2 = 270
