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Brain Teasers

This 10-question challenge, developed by Gary R. Gruber, the author of more than 30 test-prep books (drgarygruber.com), is designed to test your problemsolving skills. There is no time limit for questions 1 through 9. Just remember: Creative people look outside the realm of ordinary thinking to solve problems.

  1. You have three playing cards lying face up, side by side. A five is just to the right of a two, a five is just to the left of a two, a spade is just to the left of a club, and a spade is just to the right of a spade. What are two possibilites for the three cards?
  2. Fill in the blanks to complete the sentence.
    In spite of the ____ of the speaker’s presentation, many people were ____ with her concepts and ideas.
    (A) interest / enthralled
    (B) power / taken
    (C) intensity / shocked
    (D) greatness / gratified
    (E) strength / bored
  3. Find a four-digit number where the first digit is one-third of the second, the third is the sum of the first and second, and the last is three times the second.
  4. Bill bought four times as many apples as Harry and three times as many apples as Martin. If Bill, Harry, and Martin purchased less than a total of 190 apples, what’s the greatest number of apples that Bill could have purchased?
    (A) 168
    (B) 120
    (C) 119
    (D) 108
    (E) 90
  5. A bus can hold x people. It is half full, and y people now get off. How many people can get on the bus?
  6. Suppose you have 50 U.S. coins, totaling exactly $1.00. If you drop a coin at random, what is the probability that it is a penny?
  7. If the following four statements are presented, which one of them is true?
    (A) The number of false statements here is one.
    (B) The number of false statements here is two.
    (C) The number of false statements here is three.
    (D) The number of false statements here is four.
  8. Fill in the blank to complete the sentence.
    Richard Wagner was frequently intolerant; moreover, his strange behavior caused most of his aquaintances to ____ the composer whenever possible.
    (A) contradict
    (B) interrogate
    (C) shun
    (D) revere
    (E) tolerate
  9. A four-sided figure ABCD contains interior right angle C. AB equals 12, BC equals 3, CD equals 4, and AD equals 13. What is the area of the figure ABCD?
    (A) 36
    (B) 108
    (C) 156
    (D) 1,872
    (E) It cannot be determined.
  10. In five seconds or less, determine which is greater, 1 or 77/99 divided by 99/77.
Answers: Click here to reveal extended quiz answers and explanations.
  1. One possibility is 5 of spades, 2 of spades, 5 of clubs; the other is 2 of spades, 5 of spades, 2 of clubs.
  2. (E) strength / bored
  3. 1, 349 Represent digits as a, b, c, d. So (1) a = 1/3 b, (2) c = a + b, and (3) d = 3b. From (1) we get (4) 3a = b. From (3) and (4) we get (5) d = 9a. The only way (5) can be true is if a = 1, since d is a single digit and not equal to 0, making d = 9. Thus, from (1), b = 3, and from (2), c = 4. So the number abcd is 1,349.
  4. (D) 108 Represent B = the number of apples that Bill bought, H = the number of apples that Harry bought, and M = the number of apples that Martin bought. “Bill bought 4 times as many apples as Harry” translates to B = 4H. Similarly, B = 3M. “Bill, Harry, and Martin purchased less than a total of 190 apples” translates to B + H + M < 190. You will find that manipulating these equations yields B<120. However, because H and M are integers, B = 108 and not 119!
  5. x/2 + y You may have gotten the answer x/2 – y, but that’s how many people remain on the bus! x – (x/2 – y), which is x/2 + y, is the answer.
  6. 80% if I have no quarters, 90% if I have one quarter. Represent n = nickels, p = pennies, d = dimes, and q = quarters.
    We have 5n + p + 10d + 25q = 100 (since there are 100 cents total) and n + p + d + q = 50 (since there are 50 coins total).
    Subtract the two equations:
    We get 4n + 9d + 24q = 50.
    Suppose q = 0. Then 4n + 9d = 50. The only way this can be possible is if d = 2 and n = 8.
    Suppose q = 1. Then 4n + 9d + 24 = 50, and so 4n + 9d = 26. The only way this is possible is if d = 2 and n = 2.
    So we either have
    (1) 0 quarters, 2 dimes, and 8 nickels, which leaves 40 pennies (since the total number of coins is 50).
    or (2) 1 quarter, 2 dimes, and 2 nickels, which leaves 45 pennies (since the total number of coins is 50).
    If I drop 1 penny we have the probability in (1) as 40/50 = 80%.
    If I drop 1 penny we have the probability in (2) as 45/50 = 90%.
    So if I have no quarters in my pile of coins the probability is 80%, if I have one quarter in my pile, the probability is 90%.
  7. (C) The number of false statements here is three. If there are three false statements, then (C) is true and (A), (B), and (D) are false.
  8. (C) shun
  9. (A) 36 Draw the figure with angle C as a right angle. The key strategy is to draw the extra line BD. Triangle BCD is then a right triangle and is a 3-4-5 right triangle. You also have triangle BAD, a 5-12-13 triangle that is also a right triangle. You can use the Pythagorean Theorem to prove they are right triangles, but it also works to remember the ones like 3-4-5 and 5-12-13. So if triangle BAD is a right triangle, (where angle ABD is a right angle) you can show the area is (5 x 12)/ 2 = 30. The area of triangle BCD is (3 x 4)/2 = 6, so the total area of the figure is 36.
  10. 1 is greater. Multiply 1 by 99/77 and compare that with 77/99.

 

 

 





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