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Question 1

Answer 1

Content/Rules/Strategies:

Exponents: (x^a)^b = x^ab

What did I need to “SEE” in order to____?

See that answer had to be less than 1 because the negative exponent flips the fraction.

What did I need to RECOGNIZE about that thing?

(2²)^-2 = 2^(2)(-2) = 2^-4 = 1/(2⁴) = 1/16 = .0625.

Question 2

Answer 2

Content/Rules/Strategies

Averages

What did I need to “SEE” to ____?

Average of a set of numbers is determined by dividing the sum of the numbers by the number of terms.

A = t/n

What did I need to “RECOGNIZE” about that thing?

Question 3

Answer 3

Content/Rules/Strategies

Prime #

What did I need to “SEE” to ____?

Actually solving for “what is x+y?” –> doesn’t matter if it is prime.

What did I need to “RECOGNIZE” about that thing?

Statements 1 and 2 tell you the same information, x and y are some #, and don’t help you answer what is x+y. Together, they tell you x+y is a # and that will be enough to tell you if it is prime or not (Always Yes or Always No).

Question 4

Answer 4

Content/Rules/Strategies

Fractions + Percents

What did I need to “SEE” to ____?

Actually asking “What is value of x?”

What did I need to “RECOGNIZE” about that thing?

Both statements tell you.

(1) SUFFICIENT:
3/8 of x is 24
(3/8)x = 24
x = (24)(8/3)
x = 64

(2) SUFFICIENT:
5% of x is 3.2
0.05x = 3.2
x = 3.2/0.05
x = 320/5
x = 64

Question 5

Answer 5

Content/Rules/Strategies

Geometry

(Triangle) A = hb/2

(Circle) C = 2πr

Strategy: Prove ABC is a right triangle, so that CB will be the base and AC is height.

What did I need to “SEE” to ____?

(1) You cannot assume 30/60/90 based on this information. Knowing one angle does not tell you enough. INSUFFICIENT
(2) C = 2πr = 18π –> diameter of the circle is 18 –> AB is the diameter. However, point C is still free to “slide” around the circumference of the circle giving different areas for the triangle, so this is still insufficient to solve for the area of the triangle. INSUFFICIENT

What did I need to “RECOGNIZE” about that thing?

(1) and (2) SUFFICIENT: Inscribed triangles with one side on the diameter of the circle must be right triangles.

Statement 2 tells us AB must be the diameter, so triangle ABC must be a right triangle. With Statement 1, we know this is a 30-60-90 triangle, side length ratios of 1:√3:2 (line opposite the angle).

Question 6

Answer 6

Content/Rules/Strategies

Exponents

a^b=a^c, can set exponents b=c to solve

What did I need to “SEE” to ____?

nⁿ is really saying (10^{<span>10</span>})^{<span>(10^10)</span>} because n = 10^{<span>10 </span>}and then you are raising it to itself.

Then you have to set that = 10^d.

–> (10^{<span>10</span>})^{<span>(10^10)</span>} = 10^d –> 10^{(10x10^10)} = 10^d

What did I need to “RECOGNIZE” about that thing?

You can pull down exponents with the same base and set them equal.

10x10¹⁰=d

10¹¹=d

Question 7

Answer 7

Content/Rules/Strategies

Slope-intercept equation: y = mx + b

What did I need to “SEE” to ____?

Given: ax + y + b = 0

Put in slope-intercept equation: y = –ax – b

What did I need to “RECOGNIZE” about that thing?

Slope intercept tells us that the y-intercept is –b, so we can rephrase the question as “What is b?”

(1) Sufficient, (2) Insufficient

Question 8

Answer 8

What did I need to “SEE” in order to____?

Recognize the star earned $8M more ($32M - $24M = $8M) when her film grossed $40M more ($100M - $60M = $40M).

She wants to earn $40M on her next film, or $8M more than she earned on the more lucrative of her other two films.
What did I need to recognize about that thing?

Thus, her next film would need to gross $40M more than $100M, or $140M.

(D)

Question 9

Answer 9

Content/Rules/Strategies:

Ratio

What did I need to “SEE” in order to____?

To determine this we do not have to know the actual amounts of the three liquids; we only need to know the relative amounts, or percentages

What did I need to recognize about that thing?

(1) INSUFFICIENT: This tells us that there was 1 liter of the original mixture but it tells us nothing about the amount or percentages of vinegar or water.
(2) SUFFICIENT: We can figure out the relative percentages before the water evaporated, which is enough to then figure out post-evaporation.

(B)

Question 10

Answer 10

Content/Rules/Strategies:

Consecutive integers

What did I need to “SEE” in order to____?

“a

What did I need to recognize about that thing?

Set up bcd=2abc –> d=2a –> a+3 = 2a –> 3=a

Plug back into a+1… 4*5 = 20

Question 11

Answer 11

Content/Rules/Strategies:

D = R*T

To determine the average speed for the trip from Townsend to Smallville and back again, we need to know the average speed in each direction.

What did I need to “SEE” in order to____?

Because distance in each direction is the same, if we have average speed in each direction we will be able to find the average speed of the entire trip by taking the total distance and dividing it by the total time (R = D/T)

What did I need to recognize about that thing?

(1) SUFFICIENT: If the return time was 2/3 the outgoing time, the return speed must have been 3/2 that of the outgoing. Whenever the distance is fixed, the ratio of the times will be the inverse of the ratio of the speeds.

Set a value for Distance (120).

Can then calculate the “going” time as 3 hours.

Since “going” trip took 50% longer than “return” trip, “returning time” is 2 hours. Thus, average rate for return trip is Distance/Time, or 120/2 = 60 miles per hour.

Can then use the table to calculate average speed for entire trip: total distance, 240, divide by total time, 5. Results in average speed of 240/5 = 48 miles per hour.

(2) INSUFFICIENT: If all we know is distance on the way there, we will have no way to find the values on the way back.

Question 12

Answer 12

Content/Rules/Strategies:

Absolute Value

What did I need to “SEE” in order to____?

We know a and b are equidistant from zero on the number line, but we don’t know anything about signs of a and b.

What did I need to recognize about that thing?

I-3I = I3I

I. NOT ALWAYS TRUE: a does not necessarily have to equal b. For example, if a = -3 and b = 3, then |-3| = |3| but -3 ≠ 3.
II. NOT ALWAYS TRUE: |a| does not necessarily have to equal -b. For example, if a = 3 and b = 3, then |3| = |3| but |3| ≠ -3.
III. NOT ALWAYS TRUE: -a does not necessarily have to equal -b. For example, if a = -3 and b = 3, then |-3| = |3| but -(-3) ≠ -3.

Question 13

Answer 13

Content/Rules/Strategies:

Smart Numbers

What did I need to “SEE” in order to____?

Only given fractions

What did I need to recognize about that thing?

Plugin number for Millicent total library and solve from there

Question 14

Answer 14

Content/Rules/Strategies:

Square roots

What did I need to “SEE” in order to____?

Even exponents “hide the sign” of the base, so there are two solutions to the equation: (x + 4) = 3 or (x + 4) = -3.

What did I need to recognize about that thing?

Watch out! Although -7 is an answer choice, it is not correct. The question does not ask for the value of x, but rather for the value of x – 4 = -7 – 4 = -11.
(A)

Question 15

Answer 15

Content/Rules/Strategies:

Exponents

What did I need to “SEE” in order to____?

(5² × 3⁴4⁴ × 7²) / (3² × 4³) –> 5² × 3² × 4 × 7²

What did I need to recognize about that thing?

The answer choice has to be a multiple of 25 and even.

Question 16

Answer 16

Content/Rules/Strategies:

Probability

What did I need to “SEE” in order to____?

To determine the probability that jelly donuts will be chosen on the first and second selections, we must find the probability of both events and multiply them together.

First pick: 4/12

Second pick: NOT 4/12, because one jelly donut has been chosen. If a jelly donut is selected on the first pick, the number of donuts in the box has decreased from 12 to 11, and the number of jelly donuts has decreased from 4 to 3. Thus, probability of picking a jelly donut on the second pick is 3/11.

What did I need to recognize about that thing?

Since overall probability is calculated by multiplying the probabilities of both events, the probability of picking two jelly donuts is 4/12 × 3/11 = 12/132 = 1/11.
(A)

Question 17

Answer 17

Content/Rules/Strategies:

Quadratics

What did I need to “SEE” in order to____?

(x-3)(x-2)

What did I need to recognize about that thing?

Results in two values of x, which is still NS even with x>2

Question 18

Answer 18

Content/Rules/Strategies:

Overlapping sets

What did I need to “SEE” in order to____?

“40% of the students in the orchestra” NOT “40% of the total students”

What did I need to recognize about that thing?

40% of the 50 students in orchestra, not the total 100 students

(C)

Question 19

Answer 19

Content/Rules/Strategies:

Geometry

What did I need to “SEE” in order to____?

A rhombus is a parallelogram with four sides of equal length: AB = BC = CD = DA.

Diagonals of a parallelogram bisect each other: AC and BD intersect at their midpoints, which we will call E.

What did I need to recognize about that thing?

AE = EC = 4 and BE = ED = 3

Since ABCD is a rhombus, diagonals AC and BD are also perpendicular to each other.

Labeling as above –> the rhombus is divided by the diagonals into four right triangles, each with side length of 3 and 4.

Common right triangle side ratio = 3:4:5 –> infer unlabeled hypotenuse has length 5.

Thus, the perimeter is 5*4 = 20.

(C)

Question 20

Answer 20

Content/Rules/Strategies:

Smart Numbers

What did I need to “SEE” in order to____?

Variables in statement and answer choices

What did I need to recognize about that thing?

We need to know about the relationship of x,y and z.

Testing numbers across 1, 2 and Together shows that these statements are insufficient.

(E)

Question 21

Answer 21

Content/Rules/Strategies:

Smart Numbers

What did I need to “SEE” in order to____?

Given the relationship between the manager’s salary and the other employees

Plug-in smart numbers for manager’s salary and solve for q

What did I need to recognize about that thing?

Total salary to divide amongst employees is total, minus the $100 that you set for the manager

(A)

Question 22

Answer 22

Content/Rules/Strategies:

Work backward

What did I need to “SEE” in order to____?

11 players scored max 7 points –> 11*7 = 77

77 + 23 = 90

Within 100 point limit

(E)

Question 23

If n is an integer, is (0.1)ⁿ greater than (10)ⁿ?

1. n > -10
2. n < 10

Answer 23

Content/Rules/Strategies:

Exponents

What did I need to “SEE” in order to____?

n > -10 includes values -9, -8… 0, 1, 2…

What did I need to recognize about that thing?

So the statement is insufficient

(E)

Question 24

Each person who attended a company meeting was either a stockholder in the company, an employee of the company or both. If 62 percent of these who attended the meeting were stockholders and 47 percent were employees. What percent were stockholders, who were not employees?

Answer 24

Content/Rules/Strategies:

Double matrix

What did I need to “SEE” in order to____?

“Each person who attended a company meeting was either a stockholder in the company, an employee of the company or both.”

What did I need to recognize about that thing?

Neither stockholder nor employee box = 0.

(53%)

Question 25

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails? (A) x/(x+y) (B) y/(x+y) (C) xy/(x+y) (D) xy/(x-y) (E) xy/(y-x)

Answer 25

Content/Rules/Strategies: Working Rates What did I need to "SEE" in order to\_\_\_\_? Simultaneous rates --\> will need to add rates What did I need to recognize about that thing? Set up a chart, use a variable for Machine B time, use that to create variable for rate, solve for that variable in summing up the rates. (E)

Question 26

On a certain nonstop trip, Marta averaged x miles per hour for 2 hours and y miles per hour for the remaining 3 hours. What was her average speed, in miles per hour, for the entire trip? (1) 2x + 3y = 280 (2) y = x + 10

Answer 26

Content/Rules/Strategies: Distance Average speed = total distance/total time What did I need to "SEE" in order to\_\_\_\_? total distance = 2x + 3y total time = 5 average speed = 2x + 3y / 5 What did I need to recognize about that thing? Solving for "What is 2x+3y?" (A)

Question 27

## Footnote In each game of a certain tournament, a contestant either loses 3 points or gains 2 points. If Pat had 100 points at the beginning of the tournament, how many games did Pat play in the tournament? (1) At the end of the tournament, Pat had 104 points (2) Pat played fewer than 10 games

Answer 27

Content/Rules/Strategies: Linear Equation 100 + 2W - 3L = points at the end of the game W + L = # games played What did I need to "SEE" in order to\_\_\_\_? (1) 100 + 2W - 3L = 104 --\> 2W - 3L = 4; does not tell us W+L. Can plug in numbers for W and L but multiple values possible. (2) Does not help you narrow down among the answer choices plugged in What did I need to recognize about that thing? E

Question 28

If n = 20! + 17, then n is divisible by which of the following? I. 15 II. 17 III. 19 (A) None (B) I only (C) II only (D) I and II (E) II and III

Answer 28

**Content/Rules/Strategies:** Divisibility & Primes **What did I need to "SEE" in order to\_\_\_\_?** A multiple of N plus a multiple of N is a multiple of N. A multiple of N plus a non-multiple of N is NOT a multiple of N. **What did I need to recognize about that thing?** 15, 17 and 19 are all factors of 20!, but once you add 17, only 17 remains a factor. (C)

Question 29

If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n? A. 8 B. 9 C. 10 D. 11 E. 12

Answer 29

**Content/Rules/Strategies:** **What did I need to "SEE" in order to\_\_\_\_?** If a number is divisible by 990, it must be divisible by all the factors of 990, so start there. **What did I need to recognize about that thing?** First step is to find the prime factors of 990: 2, 3, 3, 5, 11. Then, look to see which answer choices have these prime factors included, only 11! would include these prime factors.

Question 30

If x is a positive integer greater than 1, what is the value of x? (1) 2x is a common factor of 18 and 24 (2) x is a factor of 6

Answer 30

**Content/Rules/Strategies:** Divisibility & Primes **What did I need to "SEE" in order to\_\_\_\_?** Common factors of 18 and 24: 1,2,3,6 (find multiplication pairs, NOT prime factors) **What did I need to recognize about that thing?** (1) 2x = 1,2,3, or 6... know x has to be + and \> 1, so 2x = 6 (2) INSUFFICIENT: x = 1,2,3,6 (A)

Question 31

Kevin invested $8,000 for one year at a simple annual interest rate of 6 percent and invested $10,000 for one year at an annual interest rate of 8 percent compounded semiannually. What is the total amount of interest that Kevin earned on the two investments? A. $880 B. $1,088 C. $1,253 D. $1,280 E. $1,296

Answer 31

Content/Rules/Strategies: Percents What did I need to "SEE" in order to\_\_\_\_? 1. Calculate the two rates as if they are both compounded annually: $8,000\*6% = $480; $10,000\*8% = $800 2. 800+480 = $1280 BUT you know that the second investment was semi-annual, so had to give a little more than 800, so the answer must be E. What did I need to recognize about that thing? Look at answer choices and estimate

Question 32

## Footnote Jackie has two solutions that are 2 percent sulfuric acid and 12 percent sulfuric acid by volume, respectively. If these solutions are mixed in appropriate quantities to produce 60 liters of a solution that is 5 percent sulfuric acid, approximately how many liters of the 2 percent solution will be required? A) 18 B) 20 C) 24 D) 36 E) 42

Answer 32

**Content/Rules/Strategies:** Weighted Average **What did I need to "SEE" in order to\_\_\_\_?** 2% must be at least 50%, can remove A, B and C **What did I need to recognize about that thing?** D would mean 2% and 12% are still pretty evenly leveraged because 36% is just over half of the total solution (60 liters) but we know the final mixture is considerably closer to 2% (3 percentage points away) than 12% (7 percentage points away). Can also use teeter-totter (image).

Question 33

S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices? (1) The number of distinct points in S is 5. (2) No three of the points in S are collinear.

Answer 33

**Content/Rules/Strategies:** Geometry Vertices: the three points that make up the triangle **What did I need to "SEE" in order to\_\_\_\_?** If you draw three points in a straight line, you can't make a triangle (1) Some of the points made could be in a straight line (2) Collinear means in a straight line, but does not specify how many points are in set S (1) and (2): At most point 2 points can be on a line, which is always true. As a result, all points are possible candidates to create triangles and there must be one distinct number of triangles that can be created (don't need to calculate the number). (C) SUFFICIENT **What did I need to recognize about that thing?**

Question 35

In Country C, the unemployment rate among construction workers dropped from 16 percent on September 1, 1992, to 9 percent on September 1, 1996. If the number of construction workers was 20 percent greater on September 1, 1996, than on September 1, 1992, what was the approximate percent change in the number of unemployed construction workers over this period? (A) 50% decrease (B) 30% decrease (C) 15% decrease (D) 30% increase (E) 55% increase

Answer 35

**Content/Rules/Strategies:** Percents + approximation **What did I need to "SEE" in order to\_\_\_\_?** Has to be decrease because 16 to 9 unemployment 50% is homer answer (16 to 9) and total number of works simultaneously changed **What did I need to recognize about that thing?** Create a table of all inputs, use smart numbers and estimate percent change

Question 36

In the first week of the Year, Nancy saved $1. In each of the next 51 weeks, she saved $1 more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks? A. $1,326 B. $1,352 C. $1,378 D. $2,652 E. $2,756

Answer 36

**Content/Rules/Strategies:** Evenly spaced set **What did I need to "SEE" in order to\_\_\_\_?** sum = average \* # of terms average = (high + low) / 2 of terms = (high - low) / interval + 1 **What did I need to recognize about that thing?** (C)