**9**– Choices (d) negative : positive and (e) disordered : ordered are correct.

An imaginary number is a number such that when multiplied by itself it becomes a negative number. However, there is no way a real number multiplied by itself will give you a negative number. Thus, the term imaginary. For example, two times an imaginary number cannot be thought of as greater or less than three times the number. Therefore they are called “not ordered.” The real numbers are ordered. For example, three

times the number 25 is greater than two times the number 25.

**4**– (b) succeeding

PRE means before; CURS means to run. So, PRECURSORY means to run (or go) before. The opposite is running or going after, or “succeeding.”

**10**– 355 × 356 is greater than 354 × 357.

There are more than seven ways to answer this question depending on how your mind works. One strategy is to find a way you can divide to make the problem simpler. The simple way is to divide both quantities by 356 × 354. You then get to compare 357/356 with 355/354. 357/356 = 1 1/356 and 355/354 = 1 1/354. So, the original second quantity is greater than the original first quantity.

**12**– If the client died in his sleep, there would be no way of knowing what he was dreaming.

**13**– Look for something that does not link smoking to cancer directly, but indirectly. That is, something that causes one to smoke and the same thing that causes cancer. So research could find that there is a certain condition that causes one to smoke and the same condition causes one to have cancer. Thus, smoking does not cause cancer; it is the condition that causes it. Unfortunately, that is not the case.

**8**– Terry is 5 years old.

Translate words to math. “Alice was five years older than Terry is now” translates to a = 5 + T, where a is the age that Alice was. Now translate again. Terry is half as old as Alice was: T = (1/2)a. Substitute for a:T = (1/2)(5 + T). 2T = 5 + T, T = 5. So, Terry is 5 years old.

**7**– He could type “The file will be deleted.”

Think of a statement that will be in conflict with what is mentioned. If the user types “The file will be deleted” and the statement is true, the file’s name will change but not be deleted. Thus the statement “The file is deleted” cannot be true. So it is false. But if the statement “The file will be deleted” is false, according to what is mentioned in the question, the file will be deleted, making the user’s statement true, which would be contradictory. So if the user types “The file will be deleted,” the computer will not be able to perform a function with the file and probably shut down.

**17**– (c) 10 There are two alternating sequences: 0, 1, 2, 3, 4, 5 and 0, 2, 4, 6, 8.

**19**– (b) C minus B is always less than A. Subtract the same thing from both sides. Since the sum of two sides of a triangle must be greater than the third side, A + B > C, so subtracting B from both sides, A > C – B.

**23**– (a) 25 × 25 × 25 ×…(where there are fifty 25s). Try to find a connection between the numbers. You are comparing 50 × 50 × 50 × 50…(100 times) to 100 × 100 × 100 × 100… (50 times). Write 100 as 50 × 2, so now you are comparing 50× 50 × 50 × 50…(100 times) to 50 × 2 × 50 × 2 × 50 × 2 × 50 × 2…(50 times).You can now cancel 50 × 50 × 50 × 50…50 times from both sides of the comparison. You would then get 50 × 50 × 50 × 50…(50 times) compared with 2 × 2 × 2 × 2…(50 times). You can see the left side of the comparison is 25 × 25 × 25 × 25…(50 times) times the right side.

**24** – Say IN means NOT, then associate EXTRIC with another word, EXTRACT. So INEXTRICABLE means not extractable—incapable of being extracted—or INSEPARABLE. Association is a powerful strategy for getting the meaning of words.

**25**– The doctor is the son’s mother.

**26**– If many people paid attention to the ad, it would be very crowded in the early hours. Of course the reason why the ad was put in was because the store did not think that so many people would read the ad.

**29**– (e) thimble : finger. Create a sentence expressing a specific relationship between the capitalized words. HELMET is worn on the HEAD to protect the head, as THIMBLE is worn on the FINGER to protect the finger.

**31**– The knight could tear up the paper he picks and offer the other one to the king. He could then tell the king that since the untorn paper reads “Death,” the one torn must have read, “Marriage.”

**37**– (c) 40 percent. Translate words to math. Percent can be translated to 1/100, of to X, what to x, is to =. A = [250/100](B). x/100 (A) = B. Substitute: x/100[(250/100)]B = B. Cancel B: x/100[250/100] = 1. 250x/10000 = 1 so x = 10,000/250 = 40.

**39**– e, a, c, b, d

**43**– (b) 202/962 is greater. Add 208/962 to both quantities and compare.

**44**– (e) termite : house. Create a sentence expressing a specific relationship between the capitalized words. MOTH is a living thing that destroys CLOTHING, as termite is a living thing that destroys a house.

**45**– (a) No good person lives to an old age and (d) All bad people do not die young. “Only the good die young” means that those who are not good do not die young and no good person does not die young. Choice (c) Only bad people do not die young would have been correct if all people were either good or bad. Some may be neither.

**47**– (d) I and II only. If nobody loves nobody, then it is like saying nobody loves zero people. So they must love more than zero people. Then somebody loves somebody and everybody loves somebody.

**58** – AD × CD is greater than h × AB. To solve, label the sides of the parallelogram with letters like a and b, then cancel the like quantities. h × AB = h × a and AD × CD = b × a We are then comparing h × a with b × a. Since the a is the same in each expression, we are really comparing h with b, and h is less than b because b is the largest side (the hypotenuse) of a right triangle. Thus the original h × AB is less than AD × CD.

**63** – (c) 4,100. 1,000 + 40 + 1,000 + 30 + 1,000 + 20 + 1,000 + 10 = 4,100.

**65** – 666. The integers between 0 and 1000 that are exactly divisible by 3 are 3, 6, 9, 12, 15, 18… You can see that there are 999/3 = 333 integers between 0 and 1000 that are exactly divisible by 3. The integers between 0 and 1000 are 1, 2, 3… 999. This is a total of 999 – 1 + 1 = 999 integers. Thus 999 – 333 = 666.

**66** – 100. Don’t be careless. Watch what you are dividing. You’re not dividing by 2, you are dividing by 1/2. So you have to know your basic skills. Diving by 1/2 is like multiplying by two. So the answer is (40 × 2) + 20 = 80 + 20 = 100.

**68** – 1. Please, 2. Ground, 3. Around, 4. Knight, 5. Fright, 6. Danger. Here are some more:7. Trivet, 8. Strove, 9. Shovel, 10. Slight, 11. Mother, 12. Craven

**73** – 28. Lowest average score = [21 × 80 + (60 – 21) × 0]/60 = 28

**77** – Roots are a powerful tool in figuring out the meaning of words. The root MAN means “hand.” The root MIT means “to send.” So MANUMIT means to send by hand or set free.

**80** – (a) 11/20

**86** – (f) 36. The final result must be divisible by 9. It is very interesting that when you add the digits of a number and subtract that from the original number you get a number that is a perfect multiple of 9. Here’s the proof: Represent the two-digit number as 10t + u where t is the tens digit and u is the units digit. Adding the digits, we get t + u. Subtracting from the original number, we get 10t + u – t – u = 9t. Since t is an integer, the only choice that fits is (f) 36. What’s more interesting is that any number that is a multiple of 9 has digits that add up to a multiple of 9. Here’s a great parlor trick: Have someone choose a number. Then have that person add the digits. Then have the person subtract that result from the original number to get a final result. Have the person now cross out one of the digits in the final answer. You will be able to tell the person what digit he crossed out if he tells you the remaining digit. For example, he starts with 23. 2 + 3 = 5. 23 – 5 = 18. He crosses off the 8. He is left with 1. You will be able to tell the digit crossed out. Why? Any number that is a multiple of 9 has digits that add up to a multiple of 9. So whatever number you crossed out, the crossed-out number must be 9 minus the number left!

**88** – (e) a country. MEANYRG spells GERMANY.

**89** – (d) Spaceship. Look for similarities in the messages: Avion occurs in (1) and (2) and so does Serious. So Avion must mean Serious. Balcon appears in (1) and (3) and so does Fumes. So Balcon must mean Fumes. So looking at (1), Sondor must mean Spaceship.

**91** – (c) cannot be determined as true or false Translate to something you know. For example, let Blips be Students and Plips be Teachers. And let Jips be Actors. So “Some students are actors” is neither false nor true.

**99** – 9. You may have thought of subtracting 9 from 17, giving you 8. But the question says “all but 9 die.” That means you have 9 left!

**121** – It confuses people with playing styles.

**126** – He or she should open the remaining door. The probability would be 2/3, not 1/2! Many people have thought that there is a 1/2 chance of having the car behind the door the contestant chose since there are two doors. However, because you have chosen a door first, out of three, there is a 1/3 probability there is a car behind it, and there would be a 2/3 probability any other door has the car behind it. So if you chose two of the other doors, you would have a 2/3 probability of the car. And if one door was eliminated by the host, since that door had the goat, if you switched doors, then it would be in effect as if you were choosing two doors, which would give you the 2/3 probability that the car was behind the remaining door. Another explanation: Interestingly enough, the answer is that the contestant should switch doors, because there is actually a 2/3 chance of winning the car by switching, while there is only a 1/3 chance of winning the car if the contestant opens his or her original door! This is how most people’s minds approach the problem: The host eliminates one door with the goat, so there is a goat or a car behind the contestant’s door. So the probability is 1/2. And it doesn’t matter whether the contestant opens his or her original door or changes doors. But let’s see what is really happening. The probability of having a car behind the contestant’s door when he or she originally chooses the door is 1/3, since there are three doors and only one door that has a car behind it. And there is a 2/3 probability that the car is behind one of the other two doors (since there are two ways a car can be behind one of the remaining doors [car-goat; goat-car]out of a total of three possibilities: car-goat; goat-car; goat-goat). Now, if the host eliminates a door, there would still be the 1/3 probability that the car was behind the contestant’s original door—so once a door is eliminated, there is a 2/3 probability the car was behind the remaining door. I have pondered the Let’s Make a Deal problem for some time, and I came to the conclusion that it is 2/3 when I took a more dramatic variation of the problem. Suppose I had 100 doors and 1 of them had a car behind it. The rest had goats. If I chose one door, there would no doubt be a 1/100 chance that there was a car behind it. So the remaining doors combined would have a 99/100 chance of the car. Now no matter what was done to the other doors, there would still be a 1/100 chance of the car’s being behind the door that was chosen. So if the host knew that 98 of the doors did not have the car, and then eliminated those doors, the remaining door then must have had a 99/100 chance that there was a car behind it. Suppose you had 100 doors. The chance of the car’s being behind the contestant’s chosen door is 1/100, obviously a very slim chance that won’t change even when the host eliminates 98 doors. When the doors are eliminated, and all but two are left, the contestant still has chosen a door with a very, very slim chance the car is behind it, namely 1/100. Because the host knows where the car is (and where the goats are), and because 98 of the goats were eliminated, the remaining door has a (98 + 1)/100 chance there is a car behind it. Note that initial reasoning may indicate that when there is only one door left, since 98 doors have been eliminated, the total number of possibilities for what’s behind the contestant’s door is two, a car or a goat. The favorable number of possibilities is one, the car. So by strict definition of probability, the probability is 1/2 no matter what the host knows or did. But according to the previous paragraph, that is not the case. I think probability has a somewhat ambiguous definition, especially if there is “conditional” probability, but it seems that if you tried the 100 doors out you would in fact find you’d get a 1/100 chance of getting the car.

**131** – (b) m. Write the sequence as follows: a l b (c, d) e m f (g, h) i n j (k,l). So the next letter is m.

**140** – (b) The equivalent discount of each of the three items is between 70 percent and 80 percent. Suppose we start with a price of $100 for each item. After the first discount, the first item will be $40. After the second discount, the item will be $24. This would represent an equivalent discount of 76 percent. Use the same process for the second and third item. For the second item, we would find that the price after the second discount is $25, which would represent an equivalent discount of 75 percent. For the third item, we would find the equivalent discount to be 79 percent. Thus, (b) is correct.

HOPE YOU HAD FUN WITH THEM!

REFERENCE:

Gruber, GR. (2010). *The World’s 200 Hardest Brain Teasers: Mind –boggling puzzles, problems, and curious questions to sharper your brain*. Sourcebooks