GRE problems

If a positive integer m were increased by 20%, decreased by 25% and then increased by 60%, the resulting number would be what percent of m?

Choose m = 100 100 -> 120 120 - 0.25 * 120 = 90 90 x 90 x .6 = 144 144/100 x 100 = 144% BETTER APPROACH: Alternatively, this is the same as multiplying by 1.2, then multiplying by 0.75, then multiplying by 1.6 m x 1.2 x 0.75 x 1.6 = 1.44m -> 144%

If 4a + 2b < n and 4b + 2a > m, then b - a must be A. < (m - n)/2 B. ≤ (m - n)/2 C. > (m - n)/2 D. ≥ (m - n)/2 E. ≤ (m + n)/2

Convert one of the equation to have the same inequality direction -4a -2b > -n 4b +2a > m Add the like inequalities -2a +2b > m - n b - a > (m - n) / 2 C

A right circular cylinder has volume 24π A: The height of the cylinder B: The radius

D

The ratio of 16 to g is equal to the ratio of g to 49. Quantity A: g Quantity B: 28

D

The Sargon Corporation offers an optional stock-option buy-in program to its employees. Of the employees with salaries greater than or equal to $100,000, 85% choose to participate in this plan. Of the employees with salaries less than $100,000, 77% choose to participate in this plan. Which of the following could be the total number of employees? Indicate all possible values for the number of employees. 100 200 350 460 525 640 750 880

200, 460, 640, 880

The average (arithmetic mean) population in town X was recorded as 22,455 during the years 2000-2010, inclusive. However, an error was later uncovered: the figure for 2009 was erroneously recorded as 22,478, but should have been correctly recorded as 22,500. What was the average population in town X during the years 2000-2010, inclusive, once the error was corrected?

22457 USE THE SHORTCUT There were (22,500 - 22,478 = 22) people who weren't counted. "Spread this out" among the years to get the change in average. 22 / 11 years = 2 22,455 + 2 = 22,457

The number that is 50% greater than 60 is what percent less than the number that is 20% less than 150?

25% Simplify what you can first 1.5 x 60 = 90 .8 x 150 = 120 "90 is what percent less than 150" Percent less -> use the larger number as the original 120 - 90 / 120 = .25 = 25%

If a right triangle has area 28 and hypotenuse 12, what is its perimeter?

28 quadratic trick (see magoosh)

Simplify 2^k - 2^(k+1) + 2^(k-1)

2^k(1-2+1/2)

A: 30/31 of 31/32 B: 30/31

31/32 < 1 A is 30/31 multiplied by some number less than one B must be greater

4<x<y<8 The ratio of x to y is 4 to 7 Quantity A: y - x Quantity B: 3

A

a b^2 c^3 > 0 a^2 b^2 c^3 < 0 Quantity A: ab^2 Quantity B: ac

From the first equation, a c^3 > 0, which implies a and c have the same sign From the second equation, c must be negative A: must be negative, since a is neg and b^2 is pos B: must be pos, since both a and c are neg B

Greta's salary was x thousand dollars per year, then she received a y% raise. Annika's salary was y thousand dollars per year, then she received an x% raise. x and y are positive integers. A: The dollar amount of Greta's raise B: The dollar amount of Annika's raise

Greta's raise: x*(y/100) Annika's raise: y*(x/100) A = B C

At a widget factory, 60 workers produce 1,000 widgets per week using power from internal generators. If (f) cubic meters of fuel are required by (g) number of generators every day to power the factory, how long will (t) cubic meters of fuel last in days?

Ignore the first sentence. Each day requires f x g m^3 of fuel. (f cubic meters of fuel per generator) t m^3 fuel x (1 day / f x g m^3 fuel) t/fg

What fraction is equal to 7.58333333333...?

Separate into 7.58 + 0.00333333 (758/100 + 1)/300 758*3 + 1 / 300 227500 / 30000 don't need to simplify!

A: (sqrt(25) x sqrt(8)) / (sqrt(10) x sqrt(15)) B: (sqrt(51) x sqrt(23)) / (sqrt(46) x sqrt(34))

Because they're all under a square root, just ignore the square root! A: (25 x 8) / (15 x 10) = 200 / 150 B: (51 x 23) / (46 x 34) = 1173 / 1564 A > 1, B < 1 A

5/8, 1/3, 4/7, 3/10 Quantity A: The greatest of the four fractions given above Quantity B: The sum of 0.325 and the least of the four fractions given above

C

|3 + 3x| < -2x A: |x| B: 4

Split the inequality: 3 + 3x < -2x 5x < -3 -3 - 3x < -2x -x < 3 x < -3/5 *AND* x > -3 |x| < 4 A < B B

Quantity A: |x^3 + 2| Quantity B: |x|

TRY LOGIC, THEN ALGEBRA At first A seems greater, because x^3 should be greater than x in magnitude. However, consider the case of x = -1. Then the quantities are equal (-1 + 2 = 1, and 1). Thus the answer is D D: Unknown

4/5 of the women and 3/4 of the men in a group speak Spanish. If there are 40% as many men as women in the group, what fraction of the group speaks Spanish? (Fill in the fraction)

The simplest way to solve this problem is by assigning values. Say there are 40 men and 100 women. 4/5 women = 80 women 3/4 men = 30 men 110/140 Don't need to simplify!

While driving from city A to city B, a car got 22 miles per gallon and while returning on the same road, the car got 30 miles per gallon. A: The car's average gas mileage for the entire trip, in miles per gallon B: 26

Total Average = Total Miles / Total Gallons

The reciprocal of x's non-integer decimal part equals x + 1, and x > 0 Quantity A: x Quantity B: √2

Try setting the comparison (A and B) equal to each other and see if the equations given hold. C

x > y A: the average of x, x, x, y, and y B: the average of x, x, and y (A>B, B>A, equal, cannot know)

Use algebra!! A = (3x + 2y)/5 = (9x + 6y)/15 -> 9x + 6y -> y B = 2x + y/3 = (10x + 5y)/15 -> 10x + 5y -> x x > y B > A B

Working continuously 24 hours a day, a factory bottles Soda Q at a rate of 500 liters per second and Soda V at a rate of 300 liters per second. If twice as many bottles of Soda V as of Soda Q are filled at the factory each day, what is the ratio of the volume of a bottle of Soda Q to a bottle of Soda V? (A)3/10 (B)5/6 (C)6/5 (D)8/3 (E)10/3

Use smart number for the number of bottles filled per day. Q_b = 15 V_b = 30 Liters of one bottle Q 15 x Q = 500 x 60 x 60 x 24 Q = (500 x 60 x 60 x 24) / 15 30 x V = 300 x 60 x 60 x 24 V = (300 x 60 x 60 x 24) / 30 Q / V = ((500 x 60 x 60 x 24) / 15) / ((300 x 60 x 60 x 24) / 30) Q / V = (500 / 15) / (300 / 30) = (500 x 30) / (15 x 300) Q / V = 10 / 3 E

A gang of criminals hijacked a train heading due south. At exactly the same time, a police car located 50 miles north of the train started driving south toward the train on an adjacent roadway parallel to the train track. If the train traveled at a constant rate of 50 miles per hour, and the police car traveled at a constant rate of 80 miles per hour, how long after the hijacking did the police car catch up with the train? A. 1 hour B. 1 hour and 20 minutes C. 1 hour and 40 minutes D. 2 hours E. 2 hours and 20 minutes

Use the relative speed! The relative speed is 80 - 50 = 30 mph The distance the police car needs to travel is 50 miles (the difference between the two). Then just use the formula W = R x T 50 = 30 x T T = 5/3 hours = 1 and 2/3 hours = 1 hour 40 minutes C

Beverly just filled up her gas tank, which has enough gas to last her, at her usual driving rate, about 45 days. However, Beverly becomes extra busy and begins driving 66.6% more than she usually does. How many days does the tank of gas last Beverly at her new rate?

r1 = 1 tank / 45 days r2 = r1 x 1.666 (she's burning gas at a rate 1.66 x her old rate) r2 = (1 tank / 45 days) x 1.666 new rate = 0.03702222222 days = 1 / new_rate = 1 / 0.03702222222 = 27.01 27 days

If a set of data consists of only the first ten positive multiples of 5, what is the interquartile range of the set?

set : {5,10,15,20,25,30,35,40,45,50} IQ Range = Q3 - Q1 = 40 - 15 = 25

|x — 2| > 3 A: The minimum possible value of |x — 3.5| B: The minimum possible value of |x — 1.5|

x - 2 > 3 x > 5 -x + 2 > 3 -x > 1 x < -1 (Imagine on a number line) Can A be zero? |x — 3.5| = 0 when x = 3.5 (not in range) Closest to zero will be when x is close to 5 |x — 3.5| ~ 1.5 when x ~ 5 Can B be zero? |x — 1.5| = 0 when x = 1.5 (not in range) Closest to zero will be when x is close to -1 |x — 1.5| ~ 3 when x ~ -1.5 B > A B

x and y are integers such that |x|(y) + 12 < 0 and |y| <= 1 A: x B: -12

y = -1, 0, 1 |x|y < -9 y must be negative, so y = -1 |x| > 9 x= ...-12, -11, -10, 10, 11, 12... Some are greater, some are less D

1/2 < x < 2/3 , and y^2 < 100 What is the minimum and maximum of xy?

y^2 < 100 -10 < y < 10 1/2 < x < 2/3 Extrema: -10 x 1/2 , -10 x 2/3, 10 x 1/2, 10 x 2/3 -5, -20/3, 5, 20/3 Min: -20/3 (~7) Max: 20/3 (7)

Husain and Dino had an average (arithmetic mean) of $20 each. Dino then won a cash prize, which increased the average amount of money they had to $80. If no other changes occurred, how many dollars did Dino win?

$120 Avg = Sum / N 80 = (40 + p) / 2 p = 120

45% of 80 is x% more than 24. What is the value of x?

0.45x80 = 24 + 24x(x/100) Use calculator (((0.45x80)-24)/24)*100 = 50

Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?

1 hour = t1 + t2 1 = (d/6) + (d/14) 1 = (14d + 6d) / (6x14) 6x14 = 20d (calculator) d = 4.2

Find all common factors of 135 and 225

1. 135 = 5 x 3 x 3 x 3 225 = 5 x 5 x 3 x 3 2. Intersection: 5, 3, 3, 3. All common factors: 3, 5, (3x5 = 15), (3x3 = 9), (5x3x3=45)

Find the Least Common Multiple of 28 and 42

1. 28 = 2 x 2 x 7 42 = 2 x 3 x 7 2. Smallest Covering Set = 2 x 2 x 3 x 7 3. LCM = 2 x 2 x 3 x 7 = 84

Conference Ticket Discounts 5-29 days in advance --- 15% 30-59 days in advance --- 30% 60-89 days in advance --- 40% Helen paid $252 for a conference ticket. If she had purchased the ticket one day later, she would have paid $306. How many days in advance did she purchase the ticket?

1. Find the viable options. She could have purchase the ticket 5 days, 30 days, or 60 days in advance. p = full price 5 days: 0.85 x p = 252, so p should be 306 252/0.85 = 296 INCORRECT 30 days: 0.7 x p = 252, so 0.85 x p should be 306 252/0.7 x 0.85 = 306 CORRECT 60 days: .6 x p = 252, so .7 x p should be 306 252/.6 x .7 = 294 INCORRECT

If n is an integer, then the units digit of n 6 CANNOT be which of the following? 0, 2, 3, 5, 7, 8

2,3,7,8 Remember that when working with the units digit of a product we only care about the units digits of each piece we multiply. We can use that trick to save valuable seconds on this problem. Because we are looking for the units digit of n 6, we can save time by simply running through all the possible units digits raised to the 6 th power. Starting with 0, we can try a few powers: 0 squared has a units digit of 0, 0 cubed has a units digit of 0, and so on. Notice that a number ending in 0 will forever end in zero when raised to a power. We can continue with all of the possible integers 0 - 9: Notice that the pattern for all of the integers will repeat after the 4 th power, so we can treat the 6 th power as the same as the units digit on the 2 nd power to save time. Therefore, in the list given, 2, 3, 7 and 8 cannot be the units digit of any integer raised to the 6 th power. The correct answers are B, C, E, and F.

Dick takes twice as long as Jane to run any given distance.Starting at the same moment, Dick and Jane run towards each other from opposite ends of the schoolyard, a total distance of x, at their respective constant rates until they meet. In terms of x, how far does Jane run?

2/3 x

At a convention of monsters, 2/5 have no horns, 1/7 have one horn, 1/3 have two horns, and the remaining 26 have three or more horns. How many monsters are attending the convention? (A) 100 (B) 130 (C) 180 (D) 210 (E) 260

2/5 x + 1/7 x + 1/3 x + 26 = x While it works to convert the fractions to have a common denominator and then subtract, it's faster to just subtract estimates of the decimal from 1.0 26 = x - .4x - .14x - .33333x 26 = .12667 x x = 26/.12667 x = 205.25 The closest answer is 210 (D)

Over the past year, the number of men in Pleasantville increased by 20% while the number of women in Pleasantville decreased by 20%. If the number of men and women in Pleasantville are now equal to each other, what was the percentage change in the total population of Pleasantville over the past year? (specify increase or decrease)

4% decrease

Two cars started from the same point and traveled on a straight course in opposite directions for exactly 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed of each car for the 2-hour trip?

48 mph and 56 mph

Svetlana ran the first 5 kilometers of a 10-kilometer race at a constant rate of 12 kilometers per hour. If she completed the entire 10-kilometer race in 55 minutes, at what constant rate did she run the last 5 kilometers of the race, in kilometers per hour? (A) 15 (B) 12 (C) 11 (D) 10 (E) 8

55 = t1 + t2 t1 = 5/12 hours = 25 mins t2 = 55 - 25 = 30 mins = .5 hours 5 = r x .5 hours r = 10 D

Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A? 1 3 4 6 8

6 W_A must be multiple of 6, W_B must be multiple of 4 W_A + W_B = 30

An office supply store carries an inventory of 1,345 different products, all of which it categorizes as "business use," "personal use," or both. There are 740 products categorized as"business use" only and 520 products categorized as both "business use" and "personal use." How many are "personal use"?

605

What's the greatest prime factor of 2^99 - 2^96

7 2^99 - 2^96 = 2^96(2^3 - 1) = 2^96 x 7 -> 7

At a certain school, all 118 juniors have an average (arithmetic mean) final exam score of 88 and all 100 seniors have an average final exam score of 92. A: The average (arithmetic mean) final exam score for all of the juniors and seniors combined B: 90

90 is conveniently the midpoint between the two averages. Because there are more juniors than seniors, the overall average will be "weighted" further towards the juniors, or less than 90.

A and B are points on the circumference of the circle with center O (not shown). The length of chord AB is 15. Quantity A: Circumference of circle O Quantity B: 12π

A

A bag contains green, blue and yellow glass marbles. The ratio of green to blue glass marbles is 2 : 7. The ratio of green to yellow glass marbles is 3 : 5. Quantity A: Number of blue glass marbles Quantity B: Number of yellow glass marbles

A

The height of a rectangular 3D figure is increased by p percent, its depth is decreased by p percent and its width is unchanged. Quantity A: The volume of the new3D figure if p = 20 Quantity B: The volume of the new3D figure if p = 40

A

p + |k| > |p| + k Quantity A: p Quantity B: k

A

x=−10 x/y=5/7 Quantity A: x Quantity B: y

A

x, y, and z are the lengths of the sides of a triangle. Quantity A: x+ y+ z Quantity B: 2z

A Remember the shortest path is always going to be there and back -- adding a detour (an extra vertex) will only make it longer

Quantity A: (2x−11)(2x+11) / 4 Quantity B: (x−11)(x+11)

A Use matching operations https://greprepclub.com/forum/which-is-greater-2x-11-2x-13233.html

-1<a<0<|a|<b<1-1<a<0<|a|<b<1 Quantity A: ((a^2√b)/√a)^2 Quantity B: (ab^5)/(√b)^4

A https://greprepclub.com/forum/1-a-0-a-b-10349.html

3a/(a−2b)=2 Quantity A: (4a−2b)/(a−2b) Quantity B: a+4b

A https://greprepclub.com/forum/3a-a-2b-18239.html

x>y>w>0 Quantity A xy/ w Quantity B yw/x

A https://greprepclub.com/forum/x-y-w-9130.html

x<1/y, and x and y are positive Quantity A: (2+x−x^2)/x Quantity B: (2y^2+y−1)/y

A https://greprepclub.com/forum/x-1-y-and-x-and-y-are-positive-12968.html

A: (0.25)^-3 B: 1/(2^-6) (A>B, B>A, equal, cannot know)

A = (1/4)^-3 = 1/(1/4)^3 = 1/(1/64) = 64 B = 1/(2^-6) = 2^6 = 64 C: Equal

The radius of cylinder X is the 2/3 height. Quantity A: The surface area of the end caps of the cylinder Quantity B: The surface area of the curved surface of the cylinder

A = 2 π r^2 -> 1 B = 2 π r h = 2 π r (3/2) r -> 3/2 3/2 > 1 B > A B

If "x" is an integer, which of the following inequality(ies) have a finite range of values of "x" satisfying it(them)? A. x2 + 5x + 6 > 0 B. |x + 2| > 4 C. 9x - 7 < 3x + 14 D. x2 - 4x + 3 < 0 E. (B) and (D)

A. x2 + 5x + 6 > 0 (x+5)(x+1) > 0 (x+a)(x+b) > 0 is true when x is "outside" a and b (infinite) B. |x + 2| > 4 x+2 > 4 x > 2 -x -2 > 4 x < -6 (infinite) C. x < 6/21 (infinite) D. (x+a)(x+b) < 0 is true when x is "inside" a and b (finite) D

Health insurance Plan A requires the insured to pay the initial $1000, and the plan pays the remainder. Plan B requires the insured to pay the initial $300, but then pays 80% of the cost over $300. Which of the following is a cost level for which both insurance plans pay out the same amount?

A: Paid = Cost - 1000 B: Paid = 0.8(Cost - 300) Cost - 1000 = (4(Cost - 300))/5 5C - 5000 = 4C - 1200 C - 5000 = -1200 C = 3800

A four-person leadership committee is to be chosen from a student council that consists of seven juniors and five seniors. Q is the total number of different leadership committees that include three seniors and one junior. Quantity A: Q Quantity B: 75

B

A jar contains 4 marbles: 2 red and 2 white. 2 marbles are chosen at random. Quantity A: The probability that the marbles chosen are the same colour Quantity B: The probability that the marbles chosen are different colours.

B

|x+y| = 10 x ≥ 0 z < y-x Quantity A: z Quantity B: 10

B Positive case: x + y =10 y = 10 - x z < 10 - x -x = 10 - 2x x >= 0, so z < 10 Negative case: x + y = -10 z < -10 - x - x z < something less than -10 So z < 10 still

The height a ball would reach when thrown in the upward direction, is directly proportional to the square of its weight. An 'x' gram heavy ball when thrown upward, reaches to a height of 2 feet. Quantity A: the height at which x/2 gram ball would reach when thrown upwards Quantity B: 1

B https://greprepclub.com/forum/the-height-a-ball-would-reach-when-thrown-in-the-upward-14295.html

|x| y > x |y| Quantity A: (x+y)^2 Quantity B: (x−y)^2

B https://greprepclub.com/forum/x-y-x-y-16396.html

Alexa invited the same of boys and girls to her party. Everyone who was invited came on time, but 5 additional boys turned up making the new ratio of boys to girls 5:4. Quantity A: Number of people invited to the party Quantity B: 40

C

Rajesh traveled from home to school at 30 miles per hour. Then he returned home at 40 miles per hour, and finally he went back to school at 60 miles per hour, all along the same route. What was his average speed for the entire trip, in miles per hour? (A) 32 (B) 36 (C) 40 (D) 45 (E) 47

C

The ratio of x to y is 3 : 4, and the ratio of x + 7 to y + 7 is 4 : 5. Quantity A (x+14) / (y+14) Quantity B 5/6

C

When a cylindrical tank is filled with water at a rate of 22 cubic meters per hour, the level of water in the tank rises at a rate of 0.7 meters per hour. Which of the following best approximates the radius of the tank in meters? A. √10/2 B. √10 C. 4 D. 5 E. 10

Consider the state of the tank after one hour. There are 22 m^3 of water and the height of the water is 0.7 m. The radius of this cylinder of water will be the same as the radius of the tank V = π r^2 h = 22 22 / (π * 0.7) = r^2 10 = r^2 r = sqrt(10) B

A high school football team consists exclusively of sophomores, juniors, and seniors. If the ratio of sophomores to juniors on the team is 1:2, and the ratio of juniors to seniors on the team is 1:3, which of the following could be the total number of players on the team? 36 40 42 45

Construct the complete ratio: Sp : J = 1 : 2 J : Se = 1 : 3 Sp : J : Se = 1 : 2 : 6 Add these to get the number in the smallest unit 1 + 2 + 6 = 9 All possible total must be a multiple of 9 36, 45

At noon of a certain day, when 5 pens and 3 pencils were placed in a drawer, the ratio of the number of pens to the number of pencils in that drawer became 47 to 17. Quantity A: The ratio of the number of pens to the number of pencils in the drawer immediately before noon of that day Quantity B: 3/1

D

x is an integer. Quantity A: |23x| Quantity B: 2 √(x^2)

D https://greprepclub.com/forum/gre-math-challenge-41-if-x-is-an-integer-500.html

n is a positive integer Quantity A (1/3^n) Quantity B 3 ( 1/4^n)

D https://greprepclub.com/forum/n-is-a-positive-integer-15782.html

The solution set for the inequality |x+2| < |4x+1| (a) -3/5 < x < 1/3 (b) -1/3 < x < 3/5 (c) 0 < x < 1/3 (d) -1/3 < x < 1/3 (e) None of these

E https://gre-prep-blog.wizako.com/gre-quant-practice/algebra/inequalities-absolute-values/

What is the smallest integer that satisfies the inequality x−3 / (x^2−8x−20) > 0? A. -2 B. 10 C. 3 D. -1 E. 0

Factor the inequality: (x-3) / ((x-10) (x+2)) > 0 Identify the critical points. x = 3, 10, -2 On a number line: neg inf. ~~~~~ -2 ~~~~~ 3 ~~~~~~~ 10 ~~~~~ pos inf Test each of the regions x < -2 x = -10 (-10 - 3) / ((-10 - 10) (-10 + 2)) = (-13) / (-20 * -8) = -13 / pos = negative This region does not hold -2 < x < 3 x = 0 -3 / (-10)(2) = neg / neg = pos This region holds smallest int = -1 D

What is the average (arithmetic mean) of the integers from 1 to 30, inclusive?

If evenly spaced, mean = median Median = average of two extremes Median = (30+1)/2 = 31/2 = 15.5 15.5

Jennifer has $700 more than Brian has. If she were to give Brian 30% of her money, then Brian would have 4/5 of the amount of money that Jennifer would then have. How much money does Brian currently have (before exchanging money)?

J = 400 + B B + (J/5) = 2/3(J - (J/5)) B + 3J/15 = 10J/15 -2J/15 B + 3J/15 = 8J/15 B = 5J/15 = J/3 J = 3B 3B = 400 + B 2B = 400 B = 200

The number of fish caught by Sally each day this week is given in the set {8, 12, 10, 11, 9, 11, 10}. A: The average number of fish caught by Sally B: 10

Just count "overs" and "unders" {-2, +2, 0, +1, -1, +1, 0} The total is "over" (+1) so the average must be greater than 10.

Three printing presses, R, S, and T, working together at their respective constant rates, can do a certain printing job in 4 hours. S and T, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take R, working alone at its constant rate, to do the same job?

Let Total Work = W = 40 rate_RST = 40/4 = 10 rate_ST = 40/5 = 8 rate_R = rate_RST - rate_ST = 10 - 8 = 2 T = W / R = 40 / 2 = 20 hours

A and B together complete a work in 4 days, B and C together in 6 days, C and A together in 5 days. Working independently, who will finish the work in the least time and in how many days? A) A, 120/7 days B) A, 120/17 days C) B, 120/13 days D) B, 120/17 days E) C, 120/7 days

Let W = 6 x 4 x 5 = 120 a+b = 120/4 = 30 b+c = 120/6 = 20 c+a = 120/5 = 24 a+b + b+c + c+a = 2(a+b+c) = 74 a+b+c=37 a+b+c - a+b = c = 37 - 30 = 7 a+b+c - b+c = a = 37 - 20 = 17 a+b+c - c+a = b = 37 - 24 = 13 a is fastest! days = W / R = 120 / 17 A, 120/17

Ross, Harry, and Chris own a farm. Ross, working alone, can plow the farm in 10 hours. Harry and Chris, working independently, can plow the same farm in 6 hours and 5 hours respectively. Ross starts plowing the farm and works on his own for an hour. Harry then joins him, and they work together for 2 hours. Finally, Chris joins and decides to help his friends. Harry continues along with Chris in order to finish the rest of the job while Ross takes rest. What fraction of the farm did Harry plow?

Let total work = W = 60 R = 60/10 = 6 H = 60/6 = 10 C = 60/5 = 12 H_1 = 2 x 10= 20 H_2 = H_portion of rest Rest = 60 - 6 - 20 - 12 = 22 22 = (10+12) x T T = 1 H_2 = R x T = 10 x 1 = 10 H_1 + H_2 = 20 + 10 = 30 30/60 = 1/2

If -8 < x < 2 and -4 < y < 10 what represents the range of all possible values of xy?

Normally you can just multiply the bottoms and tops, but since there are negatives it's a little tricky. The min will be the product of the most negative and the most positive: (-8 x 10 = -80) The max will be the product of the maximum numbers of the same sign (-8 x -4 = 32) -80 < xy < 32

A: The units digit of 17 × 17^17 B: The units digit of 18 × 18^18 (A>B, B>A, equal, cannot know)

Only worry about the units digit! And find the units digit pattern! A: 17 x 17^17 = 17^18 -> 7^18 7^1 = 7 -> 7 7^2 = 49 -> 9 7^3 = 343 -> 3 7^4 = 2401 -> 1 7^5 = ...7 -> 7 ... 7^8 -> 1 7^12 -> 1 7^16 -> 1 7^17 -> 7 7^18 -> *9* B = 18 x 18^18 = 18^19 -> 8^19 8^1 = 8 -> 8 8^2 = 64 -> 4 8^3 = 512 -> 2 8^4 = 4096 -> 6 8^5 = ... 8 -> 8 7^8 -> 6 7^12 -> 6 7^16 -> 6 7^17 -> 8 7^18 -> 4 7^18 -> *2* A > B

n^5 - n^3 < 0 A: n B: n^2 (A>B, B>A, equal, cannot know)

Option one: n is negative Then n^2 > n, B is greater Option two: n is between 0 and one (fraction/decimal) then n > n^2, A is greater D: cannot know

Jim's dog, Tucker, eats every hour on the hour (when the time on the clock reads an exact hour) and eats exactly half of the number of dog food pellets in his bowl. If Jim fills Tucker's bowl at 7:30 a.m. with 80 dog food pellets, what percent of the original 80 pellets will there be in Tucker's bowl at 11:30 a.m. of the same day?

Remember he doesn't start eating until 8:00am!! ("...eats every hour on the hour (when the time on the clock reads an exact hour)") 8:30am = 40 pellets 9:30am = 20 pellets 10:30am = 10 pellets 11:30am = 5 pellets 5/80 = 6.25%

What values of 'x' will be the solution to the inequality 15x - 2/x > 1? A. x > 0.4 B. x < 1/3 C. −1/3 < x < 0.4, x > 15/2 D. −1/3 < x < 0, x > 2/5 E. x < −1/3 and x > 2/5

Rewrite (15x^2 - 2 - x)/x > 0 (15x^2 + 5x - 6x - 2)/x > 0 5x(3x+1) -2 (3x +1)/x > 0 (5x-2)(3x +1)/x > 0 Both top and bottom are positive or both top and bottom are negative Positive: (5x-2)(3x +1) > 0 AND x > 0 (outside) x < -1/3 U x > 2/5 AND x > 0 ---> x > 2/5 Negative: (inside) x > -1/3 U x < 2/5 AND x < 0 ----> -1/3 < x < 0 D

In a class with 20 students, a test was administered and was scored only in whole numbers from 0 to 10. At least 1 students got every possible score, and the average score was 7. Quantity A: the lowest score that could have been received by more than one student Quantity B: 4

SUM = AVG * NUM. TERMS SUM = 7 * 20 = 140 11 students got 0,1,2, . . . 10 Total for 11 students = 55 140 - 55 = 85 Remaining 9 students totaled 85 Maximize 8 of the students (score 10) to minimize the other student 85 - 80 = 5 Lowest possible score is 5 A > B A

Running on a 10-mile loop in the same direction, Sue ran at a constant rate of 8 miles per hour and Rob ran at a constant rate of 6 miles per hour. If they began running at the same point on the loop, how many hours later did Sue complete exactly 1 more lap than Rob? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

Use relative rate! The distance of a lap is 10-miles, so being ahead by 10 miles means traveling a relative distance of 10 miles. The relative speed is 2 miles per hour. 10 miles x (1 hour/ 2 miles) = 5 hours C

If a - b > a + b, where a and b are integers, which of the following must be true? Indicate all that apply. I a < 0 II b < 0 III ab < 0

Using algebra: a - b > a + b -a -a -b > b (b is negative) OR a - b > a + b +b +b a > a + 2b 0 > 2b 0 > b (b is negative) All we know is that b is negative, so II only is True

Bruce lays 200 bricks an hour and Wayne lays 300 bricks an hour, each working at a constant rate. Quantity A: The time needed for Bruce to lay 700 bricks Quantity B: The time needed for Bruce and Wayne, working together at their respective constant rates, to lay 1,700 bricks

W = R x T A: 700 = 200 x T T = 700/200 = 7/2 hours = 3.5 hours B: 1700 = 500 x T T = 1700/500 = 17/5 hours = 3.4 hours A

| x - 3 | < 5 A: The least possible value of x B: -2 (A>B, B>A, equal, cannot know)

| x - 3 | < 5 x - 3 < 5 x < 8 -(x - 3) < 5 -x + 3 < 5 -x < 2 x > -2 Image number line! Least possible value is > -2 A > B A