Rules and Foundations

Question 1

Given u < t, b > r, f < t, and r > t, is b >u?

Answer 1

Yes

Trick: line up the variables in inequality questions:
u <  t
           r < b
 f < t    
      t < r
 Then rewrite u<b>u.

Question 2

Is a + 2b < c + 2d?

(1) a < c
(2) d > b

Answer 2

C - add inequalities, but never multiply or divide.

a     < c \+    2b <    2d = a+2b<c+2d

Question 3

1. | x-2| < 5, what’s x?

2. x^2(1/2) < 4 ^(1/2), what’s x?

Answer 3

1. • (x-2) -3 and x < 7
2. |x| < 2
   -x < 2
   x> -2 and x < 2

Question 4

if a = 3bc, what’s the value of c?

(1) a = 10-b
(2) 3a = 4b

Answer 4

B. Combos and Equations - an equation with multiple variables in the question stem, it is probably a Combo in disguise. (usually don’t need both statement to be sufficient)

from question c = a/3b
from statement (2):  3a/4b = 1, a/b = 3/4, sufficient

Question 5

For all evenly spaced sets, what is arithmetic mean (average) and median?

Answer 5

They are the same. To quickly find average/median is using (first term + last term)/2.

To find sum of evenly spaced set - average x # of terms.

Question 6

To count integers between a number set - ex: how many integers between 6 and 10, all inclusive?

Answer 6

5 - always add one. To find multiple of 4 or even numbers => (last - first) / increment +1.

Question 7

How many multiples of 7 are there between 100 and 150?

Answer 7

Here is easier to list out the terms:

105, 112, 119, 126, 133, 140, 147. Total 7 terms.

Question 8

Is K^2 odd?

(1) K-1 is divisible by 2
(2) the sum of k consecutive integers is divisible by k.

Answer 8

D.

(1) k-1 is even, then k must be odd.
(2) (sum of k integers)/k = average of k integers = integer the only way to be true where average is an integer is to have an odd consecutive integer set. Therefore, k must be odd.

Question 9

How many books, each with a volume of 100 in^3, can be packed into a crate with a volume of 5,000 in^3?

Answer 9

Not sufficient to answer. When you are fitting 3 dimensional objects into other 3 dimensional objects, knowing the respective volume is not enough. You must know specific dimensions (length, width, and height) of each subject.

Question 10

What’s the length of the third side of a triangle that two sides are 3 and 4?

Answer 10

4-3 < c < 4+3
1 < c < 7

The length of 3rd side must like between the difference and the sum of the two sides given

Question 11

What is Pythagorean theory?

Answer 11

a^2 + b^2 = c^2
common on GMAT
3-4-5, 6-8-10, 9-12-15, 12-16-20
5-12-13
8-15-17

Question 12

What’s the length ratio for 45-45-90 triangle?

Answer 12

x : x : 2^1/2 x

Also called isosceles

Question 13

What’s the length ratio for the 30-60-90 triangle?

Answer 13

x : 3^1/3 : 2x

Question 14

If 2 triangles have 2 pairs of equal angles, you know the triangles are similar. Their corresponding sides are in proportion.

What’s the ratio for area?

Answer 14

If similar triangles have corresponding lengths in ratio a : b, then their area will be in ratio a^2 : b^2.

And their volumes will be in ratio a^3 : b^3

Question 15

Concrete value vs. relative values

A company sells two kinds of pie: apple pie and cherry pie. What fraction of the total pies sold last month were apple pies?

(1) company sold 460 pies.
(2) the company sold 30% more cherry pie than apple pies.

Answer 15

B.

If change question to “how many apple pies did the company sell?” answer is C.

Rule:
- if a DS question asks for relative value of two pieces of a ratio, ANY statement that gives the relative value of ANY two pieces of the ratio will be sufficient.

• if a DS question asks for a concrete value of one element of a ratio, you will need BOTH the concrete value and the relative value of two elements of the ratio.

Question 16

800, increase by 50% and decrease by 30%, yield what number?

Answer 16

800 x 150% x 70% = 840.

Question 17

Evenly spaced sets

What’s mean, median, average and sum?

Answer 17

mean = median
average = (first + last) / # of items
sum = average x # of items.

Question 18

Count integers

Answer 18

last - first +1

Question 19

Any set of consecutive integers with odd numbers, sum of all integers is always a multiple of the number of items.

sum of all integers is never a multiple of even number of items.

Answer 19

true

Question 20

If m=−2,

-m^−m=-(-2)^-(-2) is ?

Answer 20

First deal with the exponent: -(-2) comes out as positive 2.

Now deal with the resulting power:

-m−m=-(-2)^-(-2)=-(-2)^2

The minus inside the parentheses is eliminated by the even power. Thus, you are left with just the (-) outside the parentheses, which is imposed after the parentheses and even power are dealt with.

-(-2)^2 = -(4) = -4