traps

1. rate problems: the problem deals with a lot, the choices deal with 10 bottles per min. Useless

2. % problems: Solving for PR - Qr = $54, instead of for PR/100 - Qr/100 = $54

3. overlapping sets:

3a at least 10 pc of people > 60 employed?
a choice splits this > 60 group into sub groups, and claims that 20 pc of subgrp A, 10 pc of subgrp B , 12 pc of subgrp C are employed.

3b the question is like "x pc of company A are masters; y pc of masters are idiots; z pc of these idiots got short nose". How many got short nose?
here a choice deals with: 12 pc of company employees play guitar and they are 100.

3c. Here, the stem split a group into multiple subgroups, and talks only about a select few; then asks the prob (all but the latter set).

4. whats the speed at the end of second hour? Choices deal with average speed of sorts. Useless

5. How many miles is it from George's house to the groceries store?

a. If George did not visit a gas station on his way to the groceries store, he would have driven 4 miles less.
b. The gas station is 8 miles from George's house

Both insuff.


6. How many children are there in Nancy's class?

1. Yesterday there were 14 kids in the class besides Nancy
2. Usually there are 2 kids who are sick and not present in the class

Insuff.

7. Train takes 2 hours from A and B. did it arrive on time y'day
1. y'day it started at 8 am and arrived at office 10:20 with 20 mins walking
2. Train took 2 hours (USELESS)


8. a. x + 2y = k, b. 2x+4y = 3k Insuff. twin trouble

9. if the membership of the drama club and music club are combined, what % of the combined membership will be male?

(1) of the 16 members of the drama club 15 are male
(2) of the 20 members of the music club, 10 are male

spy girl, useless

10. "On a loan, evil necromonger charges X% interest in the first year, and Y% interest in the second. If he loaned Rhyme 20,000$ in 2006, how much Rhyme pay by interest in 2008?"

A) X = 10
B) (X + Y + XY/100) = 100

Loan wolf, B suff.


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8. 12 midnight to 12 noon does not mention what days, and hence you cannot find out
the time period.

9. Clock angle: arccos(cos (5.5*x), where x = number of mins.
1:24 pm 60 mins + 24 = 84 mins. 5.5*x = 462 degress
462 - 360 = 102 degrees

10. If x = |largest divisor - smallest divisor|, whats the number? x + 1.

11. Whenever you hear difference between x and y, use |x - y|

12. Whenever you hear two variables in different eras, use variables like A_1980, B_1980, A_1981 and B_1982.

13. Standard deviation can't be negative, since it is a square root; second, if it is normally distributed, you can see the spread like (-sd, +sd), (-2sd, +2sd), (-3sd, +3sd), 34:14:2

14. Symmetric distribution ==> median = mean

15. Find the member in set A, such that sum of differences between that number and every number in that set is maximum. That is median

16. AM >= GM >= HM and GM^2 = AM*GM
a, b AM = (a+b)/2
GM = sqrt(ab)
HM = 2ab/(a+b)

There are all equal, when a = b

17. x + (1/x) >= 2 iff x is non-negative (?)
[sqrt(x) - sqrt(1/x)}^2 = x + (1/x) -2 >= 0 (proof)
x^2 + 1/x^2 >=2 iff x <> 0
x^3 + 1/x^3 >=2 iff x is non-negative

18. x^2 + y^2 +/- xy > 0 (?)

let a be x/y; in which case, we are left with a^2 + a + 1
a^2 + a +1 = (a+1/2)^2 + 3/4 >= 3/4
a^2 - a +1 = (a-1/2)^2 + 3/4 >= 3/4.

whats the minimum value of x^2 + y^2 ?
x^2 + y^2 >= 2*GM = 2xy, therefore x^2 + y^2 +/- xy > 0

19. x^2 + y^2 = xy
(x/y) + (y/x) = 1, which is impossible.
min value is 2*GM, which is 2.

20. a^2 + b^2 >= 2ab, therefore, a^2 + b^2 + c^2 >= ab +bc+ca

21. a/b + b/a > 2 when a <> b

22. a1 < a2 < a3 < ... < an
na1 < S(n) < na_n
a1 < Avg < an

23. Maximize smth, minimize everything else; and vice versa.
1 <= a < b < c < d < e < f
a + b + c + d + e + f = 75
f = 41, whats the range of e?
75 - (1+2+3+4 )- 41 = 24 (UPPER bound when pick least for a, b, c, d
lower bound, when spreading tightly around the average of a, b, c, d, e
a + b + c + d +e = 75 - 41 =34
avg = 34/5 = 6.8, take the [6.8+0.5] as the median, with least diff 1
a < b < c < d < e
5 6 7 8 9 mean 7
4 6 7 8 9 mean 6.8
LOWER BOUND = 9

24. if all memebers are >= avg, then every member is the same as avg.
if all members are <= avg, then every member is the same as avg.
these are extremums.
a < b < c < d
a + b + c + d = 4
4a <= 4 <= 4d
a <= 1 <= d

recall the problem
w > x > y > z > 0, is z < 4

(1) the sum of recipocals (note the pronununciatoin;) is 1
(2) 1/w > 1/4

Recast the problem 0 < (1/w) < (1/x) < (1/y) < (1/z)
Let s be the sum of the above.
Then 4/w < s < 4/z or 4/w < 1 < 4/z


25. #(members > avg): #(members < avg), assuming that there is no member with the avg. This can be done with allegation.

26. Inequalities. Is x^3 > x?
x^3-x = x(x^2-1) = (x+1)x(x-1)
When is this +ve? (-1,0) U (1,+inf) (note the alternation)

1. x^2 > 1 B =(-inf, -1) U (1, +inf)
The real test: is A superset of B? Definitely not. So, Insufficient.

The key is: find the domain of the functions, check whether it is superset or not.

27. looserish way of doing things. Is x/(x+2) > 1/x?

a looser picks up numbers. a smart one sets up differently.
let P = x/(x+2) - 1/x = (x^2-x-2)/(x(x+2)) = (x-2)(x+1)/(x(x+2))
or ask whether (x+2)(x+1)x(x-2) > 0

We can defnly answer when x < -2 or x > 2 (extreme sets)

28. Average again: 5 employees, rangin $5 to $20. Whats the range of average.
Min avg: all min but one max: 5, 5, 5, 5, 20, avg = 8 or 4:1
Max avg: all but one max: 20, 20, 20, 20, 5 avg = 17 or 1:4

There are 100 employees, ranging from $5 to $20. Range of average?
99:1 [99(5)+20]/100 = 5 + (15/100)
1: 99 [99(20)-5]/100 = 20 - (15/100)

iow, ( 5 + [range_wage/100], 20 - [range_wage/100] )

29. x is a multiple of y => x = ky
x is a factor of y => y = mx

30. traingles,
a, b, C(angle between a and b): unique triangle. law of cosines to get c.
a, b, c, unique trinagle: use cosines to find one angle.
a, b, A, two trinagles, thanks to Sin(A) = Sin(180-A)

31. averages.

quantity a; quantiy b
mixture's average price c.
c < (a+b)/2

quantity a < quantity b.

32. list of numbers.

lowest <= average <= highest

list of 7; is the lowest <= 5?
(a) sum of 7 < 27, average <= 4. Enough

33. overlapped sets: A and B. Rows A and not-A; colums B and not-B

34. a is 8 times as far from x as far from y.
|a-x| = 8|a-y|. Note which to multiply.

35. min and max problems. Use ceiling and floor.
Shaggy has to learn the same 71 hiragana characters, and also has one week to do so; unlike Velma, he can learn as many per day as he wants. However, Shaggy has decided to obey the advice of a study-skills professional, who has advised him that the number of characters he learns on any one day should be within 4 off the number he learns on any other day.

(A) What is the least number of hiragana that Shaggy could have to learn on Saturday?

(B) What is the greatest number of hiragana that Shaggy could have to learn on Saturday?

min, all others max.

s, s+4

s + 6(s+4) = 71
7s = 47
s = 6 5/7

Pick the ceiling of that integer, so that ceiling(s+4) gets maximized.

floor(6 5/7) = 7

7, 11,11,11,11,10,10,

2. max, all others min

s, s-4

s+6(s-4) = 71
7s = 95
s =13 4/7

pick the floor of s, since floor(s-4) gets minimized.

13, 10, 10,10,10,9,9

36. x% increase y % increase, total %
((x+y)+(xy/100))%
x% increase, y % decrease, total %
(1+x%)(1-y%)

37. a + b = 5000
if a > 3000, then b < 2000

38. a + b > k, whats the min value of a^2+b^2
a^2+b^2 >= 2ab
2ab's min value = 2(k/2)(k/2) = k^2/2
a^2 + b^2 >= k^2/2


29. ab = 5000, and a and b are +ve.
a > 500, b < 5000/500 = 10