Quant Section - GMAT; Manhattan Prep

If 𝘱 = 𝘹ᵃ⁺ᵇ／𝘹ᵇ, what is the value of positive integer 𝘱? (1) 𝘹 = 5 (2) 𝘢 = 0

0 𝘹ᵃ⁺ᵇ／𝘹ᵇ = 𝘹ᵃ⁺ᵇ⁻ᵇ = 𝘹ᵃ (2) SUFFICIENT: anything to the 0 power is 1.

Simplify: 0.00081／0.09

0.009

Simplify: (4 x 10⁻²) - (2.5 x 10⁻³)

0.0375

√(0.0081)

0.09

If 𝘺 > 0, what is the value of 𝘺? (1) 𝘺² ≤ 𝘺 (2) 𝘺 is an integer.

1 (1) AND (2) SUFFICIENT: The question indicates that 𝘺 cannot be negative. Statement (1) tells you that 𝘺 ≤ 1 Statement (2) tells you that 𝘺 could be 1, 2, 14, 192 or any integer.

If the length of an edge of a cube 𝘈 is one-third the length of an edge of cube 𝘉, what is the ratio of the volume of a cube 𝘈 to the volume of cube 𝘉?

1 to 27

Triangle 𝘈 has a base of 𝘹 and a height of 2𝘹. Triangle 𝘉 is similar to triangle 𝘈, and has a base of 2𝘹. What is the ratio of the area of triangle 𝘈 to triangle 𝘉?

1 to 4

200 is 16% of what number?

1,250

A florist has 2 azaleas, 3 buttercups, and 4 petunias. She puts two flowers together at random in a bouquet. However, the customer calls and says that she does not want two of the same flower. What is the probability that the florist does not have to change the bouquet?

13/18

Of 28 people in a park, 12 are children and the rest are adults. 8 people have to leave at 3pm; the rest do not. If, after 3pm, there are 6 children in the park, how many adults are still in the park?

14 adults

An empty bucket being filled with paint at a constant rate takes 6 minutes to be filled to 7／10 of its capacity. How much more time will it take to fill the bucket to full capacity?

2&4/7 minutes

Simplify: (12𝘢𝘣³ - 6𝘢²𝘣)／3𝘢𝘣 (given that 𝘢𝘣 ≠ 0)

2(2𝘣² - 𝘢) or 4𝘣² - 2𝘢

What is the maximum possible area of a parallelogram with one side of length 2 meters and a perimeter of 24 meters?

20 m²

The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, what is the difference between the number of hours worked by the person who worked for the longest time and the person who worked for the shortest time?

33 hours

If the sum of the last 3 integers in a set of 7 consecutive integers is 258, what is the sum of the first 4 integers?

330

What is 4,563,021／10⁵, rounded to the nearest whole number?

46

40% of all high school students hate roller coasters; the rest love them. 20% of those students who love roller coasters own chinchillas. What percentage of students love roller coasters but do not own a chinchilla?

48%

Solve for 𝘻: (4𝘻 - 7 )／(3 - 2𝘻) = -5

4／3

4𝘹 - 12 ≥ 𝘹 + 9, which of the following must be true? a. 𝘹 > 6 b. 𝘹 < 7 c. 𝘹 > 7 d. 𝘹 > 8 e. 𝘹 < 8

A

All of the following have the same set of unique prime factors EXCEPT: a. 420 b. 490 c. 560 d. 700 e. 980

A

If (B³)A < 0 and A > 0, which of the following must be negative? a. AB b. (B²)A c. B⁴ d. A/B² e. -B/A

A

If │𝘈│ > 19, which of the following could not be equal to 𝘈? a. 26 b. 22 c. 18 d. -20 e. -24

C

Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days? a. 2 b. 3 c. 4 d. 6 e. 9

C

For which of the following functions does f(𝘹) = f(2 - 𝘹)? a. f(𝘹) = 𝘹 + 2 b. f(𝘹) = 2𝘹 - 𝘹² c. f(𝘹) = 2 - 𝘹 d. f𝘹) = (2 - 𝘹)² e. f(𝘹) = 𝘹²

B

If f(𝘹) = (𝘹 + √(3))⁴, what is the range of the function f(𝘹)?

B

If 𝘹 is a positive integer, is 𝘹² + 6𝘹 + 10 odd? (1) 𝘹² + 4𝘹 + 5 is odd. (2) 𝘹² + 3𝘹 + 4 is even.

No (1) SUFFICIENT: If 𝘹 is even, you get even + even + odd = odd. If 𝘹 is odd, you get odd + even + odd = even. Therefore 𝘹 must be even. (2) INSUFFICIENT: Is even regardless of what is used for 𝘹.

If 𝘥 > 𝘢 and 𝘓 < 𝘢, which of the following cannot be true? a. 𝘥 + 𝘓 = 14 b. 𝘥 - 𝘓 = 7 c. 𝘥 - 𝘓 = 1 d. 𝘢 - 𝘥 = 9 e. 𝘢 + 𝘥 = 9

D

If 𝘮n = 3(𝘮 + 1) + 𝘯 and 𝘮 and 𝘯 are integers, 𝘮 could be any of the following values EXCEPT: a. 2 b. 3 c. 4 d. 5 e. 7

D

For each of these statements, indicate whether the statement is True of False: (a) If 𝘹² = 11, then 𝘹 = √(11). (b) If 𝘹³ = 11, then 𝘹 = ³√(11). (c) If 𝘹⁴ = 16, then 𝘹 = 2. (d) If 𝘹⁵ = 32, then 𝘹 = 2.

a. False b. True c. False d. True

Which of the following equations has no solution for 𝘢? a. 𝘢² - 6𝘢 + 7 = 0 b. 𝘢² + 6𝘢 - 7 = 0 c. 𝘢² + 4𝘢 + 3 = 0 d. 𝘢² - 4𝘢 + 3 = 0 e. 𝘢² - 4𝘢 + 5 = 0

E

Cost is expressed by the formula 𝘵𝘣⁴. If 𝘣 is doubled and 𝘵 remains the same, the new cost is how many times greater than the original cost? a. 1.2 b. 2 c. 6 d. 8 e. 16

E (smart numbers)

If 𝘹 = 𝘢 + 𝘣 and 𝘺 = 𝘢 + 2𝘣, then what is 𝘢 - 𝘣, in terms of 𝘹 and 𝘺? a. 2𝘺 - 3𝘹 b. 3𝘺 - 2𝘹 c. 2𝘹 - 3𝘺 d. 2𝘹 + 3𝘺 e. 3𝘹 - 2𝘺

E (smart numbers)

Tickets to a play cost $10 for children and $25 for adults. If 90 tickets were sold, were more adult tickets sold than children's tickets? (1) The average revenue per ticket was $18. (2) The revenue from ticket sales exceeded $1,750

D

Which of the following expressions has the largest value? a. (3⁴)¹³ b. [(3³⁰)¹²]¹ᐟ¹⁰ c. 30³ + 30³ + 30³ d. 4(3⁵¹) e. (3¹⁰⁰)¹ᐟ²

D

Which of the following is equal to (6 + √(5))／(2 - √(5))? a. 17 b. -17 c. 17 + 8√(5) d. -17 - 8√(5) e. 12 + 12√(5)

D

Simplify: (⅗ + ⅓)／(⅔ + ⅖)

⅞

Solve or simplify: √(150) - √(96)

√(6)

Solve for 𝘢, 𝘣, and 𝘤: 𝘢 + 𝘣 = 10, 𝘣 + c = 12, and 𝘢 + c = 16.

𝘢 = 7; 𝘣 = 3; 𝘤 = 9

Solve for 𝘺: 22 - │𝘺 + 14│ = 20

𝘺 = {-16, -12}

If f(𝘹) = 2𝘹² - 4 and g(𝘹) = 2𝘹, for what values of 𝘹 will f(𝘹) = g(𝘹)?

{-1, 2}

Which integer values of 𝘣 would give the number 2002／10⁻ᵇ a value between 1 and 100?

{-2, -3}

Which integer value 𝘫 would give the number -37,129 x 10ʲ a value between -100 and -1?

{-3, -4}

Given that (𝘱-3)² - 5 = 0, what is 𝘱?

{3 + √(5), 3 - √(5)}

Given that 𝘈~𝘉 = 𝘈² + 𝘉² + 2𝘈𝘉, what is 𝘈 + 𝘉, if 𝘈~𝘉 = 9?

{3, -3}

Solve for 𝘹: 𝘹(𝘹 - ((5𝘹 + 6 )／𝘹)) = 0

{6, -1}

A car loan is offered at 8% annual interest, compounded annually. After the first year, the interest due is $240. What is the principal on the loan?

$3,000

A stereo was marked down by 30% and sold for $84. What was the presale price of the stereo?

$120

A line has the equation 𝘹 = (𝘺/80) - 20. At which point does this line intersect the 𝘹-axis?

(-20, 0)

A line has the equation 𝘺 = 3𝘹 + 7. At which point does this line intersect the 𝘺-axis?

(0, 7)

Solve for 𝘹: 2(2 - 3𝘹) - (4 + 𝘹) = 7

-1

What is the probability that the sum of two number cubes will yield a 7, and then when both are thrown again, their sum will again yield a 7?

1/36

Solve or simplify: 10√(12)／2√(3)

10

48 : 2𝘹 is equivalent to 144 : 600. What is 𝘹?

100

Evaluate the following expression: (4／9)⁻³ᐟ²

27/8

If 𝘹² is divisible by 216, what is the smallest possible value for positive integer 𝘹?

36

Frank the Fencemaker needs to fence in a rectangular yard. He fences in three of the four sides of the yard. The unfenced side of the yard is 40 feet long. The yard has an area of 280 square feet. What is the length, in feet, of the fence that Frank installs?

54 feet

𝘹³ < 𝘹². Describe possible values of 𝘹.

Any non-zero number less than 1

When it is 2:01pm Sunday afternoon in Nullepart, it is Monday in Eimissaan. When it is 1:00pm Wednesday in Eimissaan, it is also Wednesday in Nullepart. When it is noon Friday in Nullepart, what is the possible range of times in Eimissaan?

Anywhere from 10pm Friday to 1am Saturday

Odd, even, or cannot be determined: If 𝘢 + 𝘣 is even, what is 𝘢𝘣?

Cannot Be Determined

Odd, even, or cannot be determined: If 𝘹/𝘺 yields an odd integer, what is 𝘹?

Cannot Be Determined

Odd, even, or cannot be determined: 𝘹𝘺 is even and 𝘻 is even, what is 𝘹 + 𝘻?

Cannot Be Determined

Is 𝘱 divisible by 168? (1) 𝘱 is divisible by 14. (2) 𝘱 is divisible by 12.

Cannot Be Determined (1) AND (2) INSUFFICIENT: You do not have enough information other than the prime factors of 14 and 12. Therefore 𝘱 could be any value beyond those prime factors

If 80 is a factor of 𝘳, is 15 a factor of 𝘳?

Cannot be determined

Simplify: 10𝘹／(5 + 𝘹)

Cannot simplify

Odd, even, or cannot be determined: If 𝘤, 𝘥, and 𝘦 are consecutive integers, what is 𝘤𝘥𝘦?

Even

True or False? ((𝘹 + 1)／𝘹)⁻² > (𝘹 + 1)／𝘹, if 𝘹 > 0.

False

Express as percent: a. 1,000／10 b. 25／8

a. 10,000% b. 312.5%

Express as fraction: a. 420% b. 8%

a. 21／5 or 4⅕ b. 2／25

Will it cause the original value to increase, decrease or stay the same? Add 1 to the numerator of a positive, proper fraction and subtract 1 from its denominator.

Increase

Will it cause the original value to increase, decrease or stay the same? Divide a positive, proper fraction by 3／13.

Increase

Mary and Nancy can each perform a certain task in 𝘮 and 𝘯 hours, respectively. Is 𝘮 < 𝘯? (1) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than 𝘮. (2)Twice the time it would take for both Mary and Nancy to perform the task together each working at their respective constant rates, is less than 𝘯.

No 𝘙 x 𝘛 = 𝘞 ((1／𝘮) + (1／𝘯)) x 𝘛 = (𝘮𝘯／(𝘮 + 𝘯)) = 1 (1) SUFFICIENT: 2(𝘮𝘯／(𝘮 + 𝘯)) > 𝘮 = 2𝘮𝘯 > 𝘮𝘯 + 𝘮² = 𝘮𝘯 > 𝘮² = 𝘯 > 𝘮

How many 𝘹-intercepts does 𝘧(𝘹) = 𝘹² + 3𝘹 + 3 have?

None

Order from least to greatest: a. 8／18 b. 0.8 c. 40%

c(40%) < a(8／18) < b(0.8)

Two sides of a triangle have lengths 4 and 10. If the third side has a length of integer 𝘹, how many possible values are there for 𝘹?

Seven

Express as fraction: a. 2.45 b. 0.008

a. 49／20 b. 1／125

Express as decimal: a. 1& 27／4 b. 12& 8／3

a. 7.75 b. 14.6.....

Express as percent: a. 80.4 b. 0.0007

a. 8,040% b. 0.07%

Did it take a certain ship less than 3 hours to travel 9 kilometers? (1 kilometer = 1,000 meters) (1) The ship's average speed over the 9 kilometers was greater than 55 meters per minute. (2) The ship's average speed over the 9 kilometers was less than 60 meters per minute.

Yes 𝘙 ≥ 50 m/min (1) SUFFICIENT: if the average speed was greater than 55 m/min it is greater than 50 m/min

Are √(3)／2 and (2√(3))／3 reciprocals?

Yes

Given that 6 is a divisor of 𝘳 and 𝘳 is a factor of 𝘴, is 6 a factor of 𝘴?

Yes

Which of the following fractions will terminate when expressed as a decimal? (choose all that apply.) a. 1／256 b. 27／100 c. 100／27 d. 231／660 e. 7／105

a, b, and d.

If 𝘳, 𝘴, and 𝘵 are consecutive positive multiples of 3, is 𝘳𝘴𝘵 divisible by 27, 54, or both?

both

`````````🏠 ``┎┬┬╔┑`` ``┣┼┼╫┫`` ``┣┼┼╫┫`` ``╔╪╪╝┫`` ``╟┼┼┼┫`` ``╙┴┴┴┛`` 🚌```````` Every morning, Grishma walks from her house to the bus stop; the placement of the house and bus stop are shown in the diagram above. She always travels exactly nine blocks from her house to the bus, but she varies the route she takes every day. (One sample route is shown.) How many days can Grishma walk from her house to the bus stop without repeating the same route?

126 days

A rectangular tank needs to be coated with insulation. The tank has dimensions of 4 feet, 5 feet, and 2.5 feet. Each square foot of insulation costs $20. How much will it cost to cover the surface of the tank with insulation?

$1,700

Rob spends ½ of his monthly paycheck, after taxes, on rent. He spends ⅓ on food and ⅛ on entertainment. If he donates the entire remainder, $500, to charity, what is Rob's monthly income, after taxes?

$12,000

A circus earned $150,000 in ticket revenue by selling 1,800 V.I.P. and Standard tickets. they sold 25% more Standard tickets than V.I.P. tickets. If the revenue from Standard tickets represents one-third of the total ticket revenue, what is the price of a V.I.P. ticket?

$125

The line represented by the equation 𝘺 = -2𝘹 + 6 is the perpendicular bisector of the line segment 𝘈𝘉. If 𝘈 has the coordinates (7, 2), what are the coordinates for 𝘉?

(-1, -2)

`╲``````````````∆`````````````` ```●𝘈 (-5,6)````````│`````````````` ``````╲`````````│``````````````` ⊲━━━━●━━━━│━━━━━━⊳``` ```(-2, 0) 𝘉``╲````│`````````````` ``````````````╲`│``````````````` `````````````````│╲ ```````````` `````````````````│```╲`````````` `````````````````∇``````╲``````` What are the coordinates for the point on line 𝘈𝘉 (see figure above) that is three times as far from 𝘈 as from 𝘉, what is in between points 𝘈 and 𝘉?

(-2.75, 1.5)

A line has a slope of ⅙ and intersects the 𝘹-axis at (-24, 0). At which point does this line intersect the 𝘺-axis?

(0, 4)

Is 𝘹 < 𝘺? (1) 1/𝘹 < 1/𝘺 (2) 𝘹/𝘺 < 0

(1) AND (2) SUFFICIENT: Statement (1) tells you (by rule of reciprocals) if 𝘹 and 𝘺 are both positive, both negative or 𝘹 is negative and 𝘺 is positive, 𝘹 > 𝘺. Statement (2) tells you that the quotient is negative (that they have different signs.

Order from least to greatest: a. ⅗／(8／10) b. 0.00751／0.01 c. (200／3) x 10⁻²

(200／3 x 10⁻²) < ⅗/(8／10) < (0.00751／0.01)

The line represented by the equation 𝘺 = 𝘹 is the perpendicular bisector of line segment 𝘈𝘉. If 𝘈 has the coordinates (-3, 3), what are the coordinates of 𝘉?

(3, -3)

A line has a slope ¾ and intersects the point (-12, -39). At which point does this line intersect the 𝘹-axis?

(40, 0)

Simplify: ((𝘮⁸)(𝘱⁷)(𝘳¹²))／((𝘮³)(𝘳⁹)(𝘱)) x (𝘱²)(𝘳³)(𝘮⁴)

(𝘮⁹)(𝘱⁸)(𝘳⁶)

If 4/𝘹 < -⅓, what is the possible range of values for 𝘹?

-12 < 𝘹 < 0

Given 𝘈𝘯 = 3 - 8𝘯: What is 𝘈₁₁ - 𝘈₉?

-16

If 𝘢𝘯 = (𝘢𝘯₋₁ x 𝘢𝘯₋₂)／2, 𝘢₅ = -6, and 𝘢₆ = -18, what is the value of 𝘢₃?

-2

Simplify (20 x (-7))／((-35) x (-2))

-2

Evaluate: ((4 + 8)／(2 - (-6))) - (4 + 8) / (2 - (-6)))

-25／2 or -12½

Evaluate: (4 + 12 ／ 3 - 18) - [-11 - (-4)]

-3

Simplify (-30／5) - ((18-9)／-3)

-3

Evaluate: -│-13 - (-17)│

-4

Given 𝘈𝘯 = 3 - 8𝘯: What is 𝘈₁?

-5

Simplify: (4 - 𝘺) - 2(2𝘺 - 3)

-5𝘺 + 10 (or 10 - 5𝘺)

On a particular exam, the boys in a history class averaged 86 points and the girls in the class averaged 80 points. If the overall class average was 82 points, what was the ratio of boys to girls in the class?

1 : 2

Two hoses are pouring water into an empty pool. Hose 1 alone would fill up the pool in 6 hours. Hose 2 alone would fill up the pool in 4 hours. How long would it take for both houses to fill up two-thirds of the pool?

1&3/5 hours

Amy and Adam are making boxes of truffles to give out as wedding favors. They have an unlimited supply of 5 different types of truffles. If each box holds 2 truffles of different types, how many different boxes can they make?

10 boxes

Randy can run π meters every 2 seconds. If the circular track has a radius of 75 meters, how many minutes does it take Randy to run twice around the track?

10 minutes

`````````𝘈``````` ````````/│\``````` ```````/`│`\`````` ``````/𝘹°│``\````` `````/```│```\```` ````/```╱╲```\``` ```/``╱`𝘋`╲20°\` ``/`╱````````╲`\` `/╱_________________60°╲\ 𝘉`````````````````𝘊 If triangle 𝘈𝘉𝘊, 𝘈𝘋 = 𝘋𝘉 = 𝘋𝘊 (see figure). If angle 𝘋𝘊𝘉 is 60° and angle 𝘈𝘊𝘋 is 20°, what is the value of 𝘹?

10°

Solve or simplify: √(20𝘢) x √(5𝘢)

10𝘢

What is the probability that the sum of two number cubes will yield a 10 or lower?

11/12

Solve for 𝘹: (3𝘹 - 6)／5 = 𝘹 - 6

12

What number is 62.5% of 192?

120

Simplify: (8(3)(𝘹)²(3))／6𝘹

12𝘹

If g(𝘹) = (𝘹³ - 𝘢𝘹)/4, and g(2) = ½, what is the value of g(4)?

13

What is the sum of 𝘹, 𝘺, and 𝘻? 𝘹 + 𝘺 = 8 𝘹 + 𝘻 = 11 y + 𝘻 = 7

13

Beginning in Town 𝘈, Biker Bob rides his bike 10 miles west, 3 miles north, 5 miles east, and then 9 miles north, to Town 𝘉. How far apart are Town 𝘈 and Town 𝘉? (Assume perfectly flat terrain.)

13 miles

A bowl is half full of water. Four cups of water are then added to the bowl, filling the bowl to 70% of its capacity. How many cups of water are now in the bowl?

14 cups of water

A rectangular solid has a square base, with each side of the base measuring 4 meters. If the volume of the solid is 112 cubic meters, what is the surface area of the solid?

144 m²

If 𝘢 and 𝘣 are integers and -4 ≤ 𝘢 ≤ 3 and -4 ≤ 𝘣 ≤ 5, what is the maximum possible value of 𝘢𝘣?

16

In a sequence of 8 consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?

16

In a diving competition, each diver has a 20% chance of a perfect dive. The first perfect dive of the competition, but no subsequent dives, will receive a perfect score. If Janet is the third diver to dive, what is her chance of receiving a perfect score? (Assume that each diver can perform only one dive per turn.)

16/125

What is the minimum possible perimeter of a quadrilateral with an area of 1,600 square feet?

160 ft

The average of 11 numbers is 10. When one number is eliminated, the average of the remaining is 9.3. What is the eliminated?

17

What is the longest diagonal of a rectangular box that is 120 inches long, 90 inches wide, and 80 inches tall?

170 inches

The △ is inside a ◯: ``𝘊_________15__________𝘉 ````╲``````````│ ``````╲````````│ `````````╲``````│8 ```````````╲```│ `````````````╲`│ ```````````````𝘈 Triangle 𝘈𝘉𝘊 is inscribed in a circle, such that 𝘈𝘊 is a diameter of the circle (see figure above). If 𝘈𝘉 has a length of 8 and 𝘉𝘊 has a length of 15, what is the circumference of the circle?

17π

Lisa spends ⅜ of her monthly paycheck on rent and 5／12 on food. Her roommate, Carrie, who earns twice as much as Lisa, spends ¼ of her monthly paycheck on rent and ½ on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate?

17／72 (smart numbers). Assign Lisa a monthly paycheck of $24 (since denominators are 8, 12, 2, and 4).

(3 - √(7))(3 + √(7)) =

2

Given the set of numbers {4, 5, 5, 6, 7, 8, 21}, how much higher is the mean than the median?

2

Hugo lies on top of a building, throwing pennies straight down to the street below. The formula for the height, "H", that the penny falls is H = Vt + 5t², where "V" is the original velocity of the penny (how fast Hugo throws it when it leaves his hand) and "t" is equal to the time it takes to hit the ground. The building is 60 meters high, and Hugo throws the penny down at an initial speed of 20 meters per second. How long does it take for the penny to hit the ground?

2

If 4ᵃ + 4ᵃ⁺¹ = 4ᵃ⁺² - 176, what is the value of 𝘢?

2

The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedy movies are there?

2

⁴√(64)／⁴√(4)

2

If 𝘹 and 𝘺 are integers, what is 𝘹 + 𝘺? (1) 3ᵡ = 81 (2) 5ᵡ = 25／5ᵞ

2 (2) SUFFICIENT: (5ᵡ)(5ᵞ) = 25 = 5ᵡ⁺ᵞ = 5² = 𝘹 + 𝘺 = 2

If 𝘹, 𝘺, and 𝘻 are prime numbers, and 𝘹 < 𝘺 < 𝘻, what is the value of 𝘹? (1) 𝘹𝘺 is even. (2) 𝘹𝘻 is even.

2 2 is the only even prime number (1) SUFFICIENT: if 𝘹𝘺 is even, then either 𝘹 or 𝘺 is even, and if 𝘹 < 𝘺, then x must be 2. (2) SUFFICIENT: Same process as statement (1)

What is the sum of all the possible three-digit numbers that can be constructed using the digits, 3, 4, and 5 if each digit can be used only once in each number?

2,664

The operation ==> is defined by 𝘹 ==> y = 𝘹 + (𝘹 + 1) + (𝘹 + 2) ... + 𝘺. For example, 3 ==> 7 = 3 + 4 + 5 + 6 + 7. What is the value of (100 ==> 150) - (125 ==> 150)?

2,800

In a bag of marbles, there are 3 red, 2 white, and 5 blue marbles. If Kia takes 2 marbles out of the bag, what is the probability that he will have 1 white and 1 blue marble? (Assume that Kia does not replace the marbles in the bag.)

2/9

Mario's Pizza has 2 choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers every pizza in extra cheese as well as regular. If Linda's volleyball team decides to order a pizza with 4 toppings, how many different choices do the teammates have at Mario's Pizza?

20 choices

A rectangular swimming pool has a length of 30 meters, a width of 10 meters, and an average depth of 2 meters . If a hose can fill the pool at a rate of 0.5 cubic meters per minute, how many hours will it take the hose to fill the pool?

20 hours

The lengths of the two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle?

200 square units

The velocity of a falling object in a vacuum is directly proportional to the amount of time the object has been falling. If after 5 seconds an object is falling at a speed of 90 miles per hours, how fast will it be falling after 12 seconds?

216 miles per hour

`````````𝘈____________________𝘉 ````````/|```````````/ ``````/`|``````````/ ````/___[__]______________/ ``𝘋 x 𝘌 ```3𝘹 ``𝘊 "𝘈𝘉𝘊𝘋" is a parallelogram (see the figure above). The ratio of "𝘋𝘌" to "𝘌𝘊" is 1 : 3. Height "𝘈𝘌" has a length of 3. If quadrilateral "𝘈𝘉𝘊𝘌" has an area of 21, what is the area of "𝘈𝘉𝘊𝘋"?

24

``````````````````````𝘋 ````````````````````╱│ 𝘉`````````````````╱``│ ```│╲```````````╱`````│16 ``│___`╲ ╱╲ ╱`````____│ ``│___│____ ╲╱____________│___│ 𝘈````````𝘊``````````𝘌 ⊲━━━━20━━━━⊳ What is the area of triangle 𝘈𝘉𝘊 (see figure above)? (1) 𝘋𝘊 = 20 (2) 𝘈𝘊 = 8

24 Any time two triangles each have a right angle and also share an additional right angle (90° at point 𝘊), they will be similar. Label angles 𝘋𝘊𝘌 as 𝘹, leaving 𝘊𝘋𝘌 as 90 - 𝘹, 𝘈𝘊𝘌 as 𝘹, leaving 𝘉𝘊𝘈 as 90 - 𝘹. (1) SUFFICIENT: Use Pythagorean theorem (or recognize that it is a 3-4-5 triangle) to solve for 𝘊𝘌. This gives you 𝘈𝘊 = 8 or (20 - 12). After matching the angles you find the smaller triangle to be one-half the size, making the dimensions of triangle 𝘈𝘉𝘊 6-8-10. (2) SUFFICIENT: gives you the same information as statement (1).

𝘈 is a set of even integers, while 𝘉 is a set of integers that are multiples of 3. There are 16 integers in set 𝘈, 22 integers in set 𝘉, and 7 integers in both sets. How many integers are in exactly one of the two sets?

24 integers

A cylinder has a surface area of 360π units square, and is 3 units tall. What is the diameter of the cylinder's circular base?

24 units

A professional gambler has won 40% of his 25 poker games for the week so far. If, all of a sudden his luck changes and he begins winning 80% of the time, how many more games must he play to end up winning 60% of all his games for the week?

25 more games

𝘉______________________________𝘈 `│╲``````````````│ `│```╲```````````│ `│``````╲16√(2)```│ `│````````╲``````│ `│```````````╲```│ `│________________________╲│ `𝘊````````````````𝘋 A square is bisected into two equal triangles (see figure). If the length of 𝘉𝘋 is 16√(2) inches, what is the area of the square?

256 in²

If f(𝘹) = 2𝘹⁴ - 𝘹², what is the value of f(2√(3))?

276

What is the maximum possible area of a triangle with a side of length 7 units and another side of length 8 units?

28 square units

A retailer sells only radios and clocks. If she currently has 44 items in inventory, how many of them are radios? (1) The retailer has more than 28 radios in inventory. (2) The retailer has less than twice as many radios as clocks in inventory.

29 𝘳(radios) + 𝘤(clocks) = 44 (1) AND (2) SUFFICIENT: (1) tells you that 𝘳 ≥ 29 and (2) tells you that 𝘳 < 2𝘤. If you isolate 𝘤 in the second equation 𝘤 = 44 - r, 𝘳 < 2, 𝘳 < 2(44 - 𝘳), 𝘳 < 88 - 2𝘳, 3𝘳 < 88, 𝘳 < 88/3 or 29&1/3. Statement (2) tells you that 𝘳 > 29.

If 𝘺 = 30𝘱, and 𝘱 is prime, what is the greatest common factor of 𝘺 and 14𝘱, in terms of 𝘱?

2𝘱

Simplify: 𝘹 - (3 - 𝘹)

2𝘹 - 3

Solve or simplify: √(𝘹²𝘺³ + 3𝘹²𝘺³)

2𝘹𝘺√(𝘺)

Solve or Simplify: √(18)/√(2)

3

Solve or simplify: (1／81)⁻¹ᐟ⁴

3

Alexandra needs to mix cleaning solution in the ratio of 1 part bleach for every 4 parts water. When mixing the solution, Alexandra makes a mistake and mixes in half as much bleach as she ought to have. The total solution consists of 27 mL. How much bleach did Alexandra put into the solution?

3 mL

Five A-list actresses are vying for the three leading roles in the new film "Catfight" in Denmark. The actresses are Julia Robards, Meryl Strep, Sally Fieldstone, Nicole Kadman, and Hallie Strawberry. Assuming that no actress has any advantage in getting any role, what is the probability that Julia and Hallie will star in the film together?

3/10

There is a rectangular parking lot with a length of 2𝘹 and a width of 𝘹. What is the ratio of the perimeter of the parking lot to the area of the parking lot, in terms of 𝘹?

3/𝘹 or 3 : 𝘹

If 𝘹 and 𝘺 are positive integers and x/y has a remainder of 5, what is the smallest possible value of x𝘺?

30

The class mean score on a test was 60, and the standard deviation was 15. If Elena's score was within 2 standard deviations of the mean, what is the lowest score she could have received?

30

A list kept at Town Hall contains the town's average daily temperature in Fahrenheit, rounded to the nearest integer. A particular completed month has either 30 or 31 days. How many days does the month have? (1) The median temperature is 73.5. (2) The sum of the average daily temperatures is divisible by 3.

30 (1) SUFFICIENT: Since every item in the list is an integer the only way for the temperature to be a non-integer is if there is an even number of days.

In how many different ways can the letters in the word "LEVEL" be arranged?

30 ways

````````````𝘈_____4______𝘉```` ``````````╱│ ```````\```` ```````╱```│5````````\``` `````╱`````│____````````\`` ``╱______________│____│______________\` 𝘋``````````𝘌`````````````𝘊 𝘈𝘉𝘊𝘋 is a quadrilateral, with 𝘈𝘉 parallel to 𝘊𝘋 (see figure). Point 𝘌 is between 𝘊 and 𝘋 such that 𝘈𝘌 represents the height of 𝘈𝘉𝘊𝘋, and 𝘌 is the midpoint of 𝘊𝘋. If 𝘈𝘉 is 4 inches long, 𝘈𝘌 is 5 inches long, and the area of triangle 𝘈𝘌𝘋 is 12.5 square inches, what is the area of 𝘈𝘉𝘊𝘋? (Note: figure not drawn to scale.)

35 in²

A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim?

36 arrangements

```\╲``````````````````` ````\``╲```````````````` `````\````╲````````````` `````7\``````/╲12```````` ```````\```3/ ╲/`╲`````` ````````\``/````````╲``` `````````\/_____________________╲ ``````````\```````````╱` ```````````\```````╱𝘹``` ````````````\╱\`╱`````` `````````````\╱````````` What is 𝘹 in the figure below?

36/7

Given that 𝘈 ♢ 𝘉 = 4𝘈 - 𝘉, what is the value of (3 ♢ 2) ♢ 3?2

37

``````𝘈_____________________________𝘉```` ``````│``𝘌__________________𝘍``│```` ``````│``│``````````│``│```` ``````│``│``````````│``│```` ``````│``│``````````│``│```` ``````│``│__________________│``│```` ``````│___𝘏___________________𝘎___│```` ``````𝘋`````````````````𝘊```` 𝘈𝘉𝘊𝘋 is a square picture frame (see figure). 𝘌𝘍𝘎𝘏 is a square inscribed within 𝘈𝘉𝘊𝘋 as a space for a picture. The area of 𝘌𝘍𝘎𝘏 (for the picture) is equal to the area of the picture frame (the area of 𝘈𝘉𝘊𝘋 minus the area of 𝘌𝘍𝘎𝘏). If 𝘈𝘉 = 6, what is the length of 𝘌𝘍?

3√(2)

If g(𝘹) = 3𝘹 + √(𝘹), what is the value of g(𝘥² + 6𝘥 + 9)?

3𝘥² + 19𝘥 + 30 OR 3𝘥² + 17𝘥 + 24

Given that (𝘹² + 6𝘹 +9)／(𝘹 + 3) = 7, what is 𝘹?

4

Solve or simplify: ³√(100 - 36)

4

What is the units digit of (2)⁵(3)³(4)²?

4

If 𝘹 and 𝘺 are nonnegative, what is the greatest common factor of 𝘹 and 𝘺? (1) 𝘹 and 𝘺 are both divisible by 4. (2) 𝘹 - 𝘺 = 4

4 (1) AND (2) SUFICIENT: Statement (1) tells you that 𝘹 and 𝘺 are both divisible by 4 and statement (2) 𝘹 - 𝘺 = 4. Therefore 𝘹 and 𝘺 are consecutive multiples of 4, making their GCF 4

If the perimeter of a rectangular flower bed is 30 feet, and its area is 44 square feet, what is the length of each of its shorter sides?

4 feet

2𝘹 : 𝘺 is equivalent to 4𝘹 : 8,500. What is 𝘺?

4,250

If 1,500 is increased by 20% and then reduced by 𝘺%, yielding 1,080, what is 𝘺?

40

United Telephone charges a base rate of $10.00 for service, plus an additional charge of $0.25 per minute. Atlantic Call charges a base rate of $12.00 for service, plus an additional charge of $0.20 per minute. For what number of minutes would the bills for each telephone company be the same?

40 minutes

inside of circle ◯: `````𝘉╭`````╮``````` ```╭`╲`````````╮``` ``╭````╲````````╮`` 𝘊╭_______40°╲𝘈___________╮𝘋 `╰`````````╲`````╯` ``╰```````````╲`╯`` ```╰````````````╯𝘌`` ``````╰``````╯`````` 𝘉𝘌 and 𝘊𝘋 are both diameters of circle with center 𝘈 (see figure). If the area of the circle is 180 units², what is the total area of the shaded regions?

40 units²

A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

40%

What is the maximum possible area of a quadrilateral with a perimeter of 80 centimeters?

400 cm²

Initially, the men and women in a room were in the ratio of 5 : 7. Six women leave the room. If there are 35 men in the room, how many women are left in the room?

43

Students are in clubs as follows: Science-20, Drama-30, and Band-12. No student is in all three clubs, but 8 are in both Science and Drama, 6 are in both Science and Band, and 4 are in Drama and Band. How many different students are in at least one of the three clubs?

44 different students

Caleb spends $72.50 on 50 hamburgers for the marching band. If single burgers cost $1.00 each and double burgers cost $1.50 each, how many double burgers did he buy?

45 double burgers

Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical Sun-and-Fun beach dolls, 1 Elegant Eddie dress-up doll, 1 G.I. Josie army doll, and 1 Tulip Troll doll. If the youngest niece does not want the G.I. Josie doll, in how many different ways can he receive gifts?

48

A bottle is 80% full. The liquid in the bottle consists of 60% guava juice and 40% pineapple juice. The remainder of the bottle is then filled with 200 mL of rum. How much guava juice is in the bottle?

480 mL

`````````4`╱╲`4``````````` _______________╱________╲__________________ ╲``````╱``4````╲``````╱` ``╲``╱```````````╲``╱``` ```╱╲`````````````╱╲```` ╱___________╲____________╱__________╲ `````````╲````╱`````````` ```````````╲╱```````````` The points of a six-pointed star consist of six identical equilateral triangles, with each side 4 cm (see figure). What is the area of the entire star, including the center?

48√(3) cm²

If 𝘹 and 𝘺 are integers, what is the value of 𝘹² + 2𝘹𝘺 + 𝘺²? (1) 𝘹 + 𝘺 = 7 (2) 2𝘹 = (28 - 47)／2

49 𝘹² + 2𝘹𝘺 + 𝘺² = (𝘹 + 𝘺)² (1) SUFFICIENT (2) SUFFICIENT

A cylindrical tank has a diameter of 14 meters and a height of 20 meters. A water truck can fill π cubic meters of the tank every minute. How long will it take the water truck to fill the water tank from empty to half full?

490 minutes, or 8 hours and 10 minutes

````𝘟`````𝘈``````````` `````╭●```\````╮````` ``𝘊╭```````\`````╮``` ``╭``╲`````\`````╮`` `╭`````╲````\`````╮` `╰```````╲40°\````╯` ``╰````````╲`\```╯`` ```╰`````````╲\`╯``` ``````╰```````╯𝘉```` Angle 𝘈𝘉𝘊 is 40° (see figure) and the area of the circle is 81π. If 𝘊𝘉 is a diameter of the circle, how long is arc 𝘈𝘟𝘊?

4π

If 𝘈 = (𝘹／3)／(2／𝘺), what is "A"? (1) 𝘹𝘺 = 8 (2) 𝘹／𝘺 = 2

4／3 𝘈 = 𝘹𝘺／6 (1) SUFFICIENT

Matt gets a $1,000 commission on a big sale. This commission alone raises his average commission by $150. If Matt's new average commission is $400, how many sales has Matt made?

5

A mixture of "lean" ground beef (10% fat) and "super-lean" ground beef (3% fat) has a total fat content of 8%. What is the ratio of "lean" ground beef to "super-lean" ground beef?

5 : 2

What is the ratio 𝘹 : 𝘺 : 𝘻? (1) 𝘹 + 𝘺 = 2𝘻 (2) 2𝘹 + 3𝘺 = 𝘻

5 : 3 : 1 (1) AND (2) SUFFICIENT: Use combined ratios and substitution method.

What is the sum of all the positive integers up to 100, inclusive?

5,050

𝘹 is 40% of 𝘺. 50% of 𝘺 is 40. 16 is what percent of 𝘹?

50%

_____________________________ \╲65°``````````/│ `\`╲``````````/`│ ``\``╲```````/``│ ```\```╲`𝘹°`/```│26 ``25\```╱╲/╲``│ `````\``╲╱╲/``│ ``````\╱7```10╲│ What is the value of 𝘹 in the figure above?

50°

How many terms are there in the set of consecutive integers from -18 to 33, inclusive?

52

Company 𝙕 only sells chairs and tables. What percent of its revenue in 2008 did Company 𝙕 derive from its sale of chairs? (1) In 2008, the price of tables sold by Company 𝙕 was 10% higher than the price of chairs sold by Company 𝙕. (2) In 2008, Company 𝙕 sold 20% fewer tables than chairs.

53% 𝘙𝘵(Revenue Tables) = 𝘗𝘵(Price Tables) x 𝘘𝘵(Quantity tables) (1) AND (2) SUFFICIENT: Replace 𝘗𝘵 with 1.1𝘗c and replace 𝘘𝘵 with 0.8𝘘𝘤. Rt = (0.88)𝘙c. 𝘙c／Revenue Total = 𝘙𝘤／𝘙𝘵 + 𝘙𝘤 = 𝘙𝘤／(0.88)𝘙𝘤 + 𝘙𝘤 = 1／1.88 ~ 53%

〇 inside of circle ◯: 𝘖````````````〳``````〳 `╲``````````〳``````〳 ```╲5``````〳``````〳` `````╲````〳╲4```〳`` ```````╲〳````╲〳```` A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide (see figure). What is the area of the walkway?

56π m²

A charitable association sold an average of 66 raffle tickets per member. Among the female members, the average was 70 raffle tickets. The male to female ratio of the association is 1:2. What was the average number of tickets sold by the male members of the association?

58 tickets

Solve or simplify: √(63) + √(28)

5√(7)

Carina has 100 ounces of coffee divided into 5- and 10-ounce packages. If she has 2 more 5-ounce packages than 10-ounce packages, how many 10-ounce packages does she have?

6

If -4 is a solution for 𝘹 in the equation 𝘹² + 𝘬𝘹 + 8 = 0, what is 𝘬?

6

If 𝘬 is an integer, and if 0.02468 x 10ᵏ is greater than 10,000, what is the least possible value of 𝘬?

6

Of 30 snakes at the reptile house, 10 have stripes, 21 are poisonous, and 5 have no stripes and are not poisonous. How many of the snakes have stripes AND are poisonous?

6

What is the length of the sequence of different digits in the decimal equivalent of 3／7?

6

What is the units digit of (6⁶／6⁵)⁶?

6

A pentagon has three sides with length 𝘹, and two sides with the length 3𝘹. If 𝘹 is ⅔ of an inch, what is the perimeter of the pentagon?

6 inches

Given that 𝘢𝘣 = 12 and (𝘤／𝘢) + 10 = 15, what is 𝘣𝘤?

60

If the average of 𝘹 and 𝘺 is 50, and the average of 𝘺 and 𝘻 is 80, what is the value of 𝘻 - 𝘹?

60

What is the greatest common factor of 420 and 660?

60

10% of all aliens are capable of intelligent thought and have more than 3 arms, and 75% of aliens with 3 arms or less are capable of intelligent thought. If 40% of all aliens are capable of intelligent thought, what percent of aliens have more than 3 arms?

60%

A feed store sells two varieties of birdseed: Brand 𝘈, which is 40% millet and 60% sunflower, and Brand 𝘉, which is 65% millet and 35% safflower. If a customer purchases a mix of the two types of birdseed that is 50% millet, what percent of the mix is Brand 𝘈?

60%

If the sum of the last 3 integers in a set of 6 consecutive integers is 624, what is the sum of the first 3 integers of the set?

615

The "luminous flux," or perceived brightness, of a light source is measured in lumens and is inversely proportional to the square of the distance from the light. If a light source produces 200 lumens at a distance of 3 meters, at what distance will the light source produce a luminous flux of 25 lumens?

6√(2) (or √(72) meters)

A line has the equation 𝘹 = -2𝘺 + 𝘻. If (3, 2) is a point on the line, what is 𝘻?

7

Velma has exactly one week to learn all 71 Japanese hiragana characters. If she can learn at most a dozen of them on any one day and will only have time to learn four of them on Friday, what is the least number of hiragana that Velma will have to learn of Saturday?

7

A skeet shooting competition awards prizes for each round as follows: the first-place winner receives 11 points, the second-place winners 7 points, the third-place finisher receives 5 points, and the fourth-place finisher receives 2 points. No other prizes are awarded . Johan competes in several rounds of the skeet shooting competition and receives points every time he competes. If the product of all of the points he receives equals 84,700, in how many rounds does he participate?

7 rounds

Estimate: √(60)

7.7

Three gnomes and three elves sit down in a row of six chairs. If no gnome will sit next to another gnome and no elf will sit next to another elf, in how many different ways can the elves and gnomes sit?

72

What is the least common multiple of 18 and 24?

72

The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square? (1) The height of the equilateral triangle is equal to 6√(3) (2) The area of the equilateral triangle is equal to 36√(3)

72 No calculation is needed to solve this problem. Both equilateral triangles and squares are regular figures--those that can change size, but never shape. Regular figures are those for which you only need one measurement to know every measurement. (1) SUFFICIENT: side equals 12 (in an equilateral the side and the height exist in a ratio of 𝘏 = ½𝘉√(3)), then using the 45-45-90 formula (half of a square is a right-triangle) each side equals 12／√(2). Then multiply: (12／√(2)) x (12／√(2)) = 144／2 = 72. (2) SUFFICIENT: Since 𝘏 = ½𝘉√(3), and the area = ½𝘉𝘏, you can substitute the former equation as the height. ½(𝘉 x ½𝘉√(3)) = 36√(3), (𝘉 x ½𝘉√(3)) = 72√(3), ½𝘉² = 72, 𝘉² = 144, 𝘉 = 12. Then use the 45-45-90 formula from statement (2) to solve for the area of the square.

Triangle 𝘈𝘉𝘊 is inscribed in a circle, such that 𝘈𝘊 is a diameter of the circle and angle 𝘉𝘈𝘊 is 45°. If the area of triangle 𝘈𝘉𝘊 is 72 square units, how much larger is the area of the circle than the area of triangle 𝘈𝘉𝘊?

72π - 72

What is the value of integer 𝘹? (1) The least common multiple of 𝘹 and 45 is 225. (2) The least common multiple of 𝘹 and 20 is 300.

75 (1) AND (2) SUFFICIENT: The factors of 45:(3² x 5¹) and 225:(3² x 5²). Therefore the 𝘹 must contain at least two 5's. (If 𝘹 contained fewer 5's the LCM would contain fewer 5's). In statement (2) by the same process 𝘹 must contain at least one 3. 3 x 5 x 5 = 75

If k(𝘹) = (4𝘹³)(𝘢), and k(3) = 27, what is k(2)?

8

If 𝘹 = (9𝘣 - 3𝘢𝘣)／((3／𝘢) - (𝘢／3)), what is 𝘹? (1) 9𝘢𝘣／(3 + 𝘢) = 18／5 (2) 𝘣 = 1

8 9𝘢𝘣／(3 + 8) (1) SUFFICIENT: gives you the exact value of the combo.

Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are 20 boys and 40 girls. If there are 6 happy bays and 4 sad girls, how many boys are neither happy nor sad?

8 boys

A hydrogenator water gun has a cylindrical water tank, which is 30 centimeters long. Using a hose, Jack fills his Hydrogenator with π cubic centimeters of his water tank every second. If it takes him 8 minutes to fill the tank with water, what is the diameter of the circular base of the gun's water tank?

8 cm

On the planet Flarp, 3 floops equal 5 fleeps, 4 fleeps equal 7 flaaps, and 2 flaaps equal 3 fliips. How many floops are equal to 35 fliips?

8 floops

800, increased by 50% and then decreased by 30%, yields what number?

840

There is a 30% chance of rain and a 70% chance of shine. If it rains, there is a 50% chance that Bob will cancel his picnic, but if the sun shining , he will definitely have his picnic. What is the chance that Bob will have his picnic?

85%

The size of a computer screen is measured by the length of its diagonal. How much bigger is the visible area of a square 24-inch screen than the area of a square 20-inch screen?

88 in²

If S𝘯 = (4ᵑ) + 3, what is the units digit of S₁₀₀?

9

If 𝘮 and 𝘯 are positive integers and (2¹⁸)(5ᵐ) = (20ᵑ), what is the value of 𝘮?

9

If 𝘹𝘺 ≠ 0 and √((𝘹𝘺)／3) = 𝘹, what is 𝘺? (1) 𝘹/𝘺 = ⅓ (2) 𝘹 = 3

9 𝘺 = 3𝘹 (2) SUFFICIENT

40 percent of Andrea's living room floor is covered by a carpet that is 4 feet by 9 feet. What is the area of her living room floor?

90 ft²

The first term in an arithmetic sequence is -5 and the second is -3. What is the 50th term? (Recall that an arithmetic sequence, the difference between successive terms is constant.)

93

``●◆ x ■◆ ﹉﹉﹉ ▲■◆ In the multiplication above, each symbol represents a different unknown digit, and ● x ■ x ◆ = 36. What is the three-digit integer ●■◆?

936

The number 𝘈 is a two-digit positive integer; the number 𝘉 is the two-digit positive integer formed by reversing the digits of 𝘈. If 𝘘 = 10𝘉 - 𝘈, what is the value of 𝘘? (1) The tens digit of 𝘈 is 7. (2) The tens digit of 𝘉 is 6.

99𝘺 (2) SUFFICIENT: 𝘈 = 10𝘹 + 𝘺 and 𝘉 = 10𝘺 + 𝘹. Therefore, 𝘘 = 2𝘣 - 𝘢 = 10(10𝘺 + 𝘹) - (10𝘹 + 𝘺) = 100y + 10𝘹 - 10𝘹 - 𝘺 = 99𝘺.

If 8 and -4 are the solutions for 𝘹, which of the following could be the equation? a. 𝘹² - 4𝘹 - 32 = 0 b. 𝘹² - 4𝘹 + 32 = 0 c. 𝘹² + 4𝘹 - 12 = 0 d. 𝘹² + 4𝘹 + 32 = 0 e. 𝘹² + 4𝘹 + 12 = 0

A

If │10𝘺 - 4│ > 7 and 𝘺 < 1, which of the following could be 𝘺? a. -0.8 b. -0.1 c. 0.1 d. 0 e. 1

A

If 𝘢 > 7, 𝘢 + 4 > 13, and 2𝘢 < 30, which of the following must be true? a. 9 < 𝘢 < 15 b. 11 < 𝘢 < 15 c. 15 < 𝘢 < 20 d. 13 < 𝘢 < 15

A

Which of the following is equivalent to -3𝘹 + 7 ≤ 2𝘹 + 32? a. 𝘹 ≥ -5 b. 𝘹 ≥ 5 c. 𝘹 ≤ 5 d. 𝘹 ≤ -5

A

If │𝘹 +1│ = │3𝘹 - 2│, what are the possible values for 𝘹? a. 1/4 and 3/4 b. 1/4 and 3/2 c. 2/3 and 3/2 d. 2/3 and 4/3 e. 3/4 and 4/3

B

If 𝘤 + 𝘥 = 11 and 𝘤 and 𝘥 are positive integers, which of the following is a possible value for 5𝘤 + 8𝘥? a. 55 b. 61 c. 69 d. 83 e. 88

B

If 𝘹𝘺 = 2, 𝘹𝘻 = 8, and 𝘺𝘻 = 5, then the value of 𝘹𝘺𝘻 is closest to: a. 5 b. 9 c. 15 d. 25 e. 75

B

Nicky and Cristina begin running a race at the same time, though Nicky begins the race 36 meters ahead of Cristina runs at a pace of 5 meters per second and Nicky runs at a pace of only 3 meters per second, how many seconds will Nicky have run before Cristina catches up to him? a. 15 seconds b. 18 seconds c. 25 seconds d. 30 seconds e. 45 seconds

B

S = {1, 2, 5, 7, 𝘹} If 𝘹 is a positive integer, is the mean of set 𝘚 greater than 4? (1) The median of set 𝘚 is greater than 2. (2) The median of set 𝘚 is equal to the mean of set (𝘚).

B

Simplify: (4ᵞ + 4ᵞ + 4ᵞ + 4ᵞ)(3ᵞ + 3ᵞ + 3ᵞ) a. 4⁴ᵞ x 3³ᵞ b. 12ᵞ⁺¹ c. 16ᵞ x 9ᵞ d. 12ᵞ e. 4ᵞ x 12ᵞ

B

Which of the following numbers is NOT prime: (Hint: avoid actually computing these numbers.) a. 6! - 1 b. 6! + 21 c. 6! + 41 d. 7! - 1 e. 7! + 11

B

Bradley owns 𝘣 video game cartridges. If Bradley's total is one-third the total owned by Andrew and four times the total owned by Charlie, how many video game cartridges do the three of them own altogether in terms of 𝘣? a. 16𝘣／3 b. 17𝘣／4 c. 13b／4 d. 19𝘣／12 e. 7𝘣／12

B (smart numbers). Use the 𝘣 = 4 and look for the one that yields 17

If 𝘈𝘉／7 > 1／14 and 𝘈 = 𝘉, which of the following must be greater than one? a. 𝘈 + 𝘉 b. 1 - 𝘈 c. 2𝘈² d. 𝘈² - ½ e. 𝘈

C

If 𝘎² < 𝘎, which of the following could be 𝘎? a. 1 b. 23/7 c. 7/23 d. -4 e. -2

C

If 𝘢 ≠ 2, which of the following is equal to (𝘣(𝘢² - 4))／(𝘢𝘣 - 2𝘣) a. ab b. a c. a + 2 d. a² e. 2b

C

If 𝘹² + 𝘬 = 𝘎 and 𝘹 is an integer, which of the following could be the value of 𝘎 - 𝘬? a. 7 b. 8 c. 9 d. 10 e. 11

C

Which of the following is equivalent to (⅓)⁻⁴(⅑)⁻³(1／27)⁻²? a. (⅓)⁻⁸ b. (⅓)⁻⁹ c. (⅓)⁻¹⁶ d. (⅓)⁻¹⁸ e. (⅓)⁻¹⁴⁴

C

Seamus has 3 times as many marbles as Ronit, and Taj has 7 times as many marbles as Ronit. If Seamus has 𝘴 marbles then, in terms of 𝘴, how many marbles do Seamus, Ronit, and Taj have together? a. 3𝘴/7 b. 7𝘴/3 c. 11𝘴/3 d. 7𝘴 e. 11𝘴

C (smart numbers)

Is the integer 𝘹 odd? (1) 2(𝘺 + 𝘹) is an odd integer. (2) 2𝘺 is an odd integer.

Cannot Be Determined. (1), (2), and (1) AND (2) INSUFFICIENT: Since 2 multiplied by an integer is even, 𝘺 + 𝘹 must not be an integer. Therefore 𝘺 must not be an integer. Statement (2) could be any odd integer divided by 2.

If 𝘫 is divisible by 12 and 10, is 𝘫 divisible by 24?

Cannot be determined

Every attendee at a monster truck rally paid the same admission fee. How many people attended the rally? (1) If the admission fee had been raised to $20 and twice as many people had attended, the total admission fees collected would have been three times greater. (2) If the admission fee had been raised to $30 and two-thirds as many people had attended, the total admission fees collected would have been 150% of the actual admission fees collected.

Cannot be determined. (1), (2) and (1) AND (2) INSUFFICIENT. 𝘛(total) = 𝘈(attendees) x 𝘗(price).. (1): 3𝘛 = 40𝘈. (2): 3𝘛 = 40𝘈.

If 𝘯 is a one-digit positive integer, what is 𝘯? (1) The units digit of 4ᵑ is 4. (2) The units digit of 𝘯⁴ is 𝘯.

Cannot be determined. (1), (2), & (1) AND (2) INEFFICIENT: Because both 1 and 5 work for both it cannot be determined.

Is 𝘮𝘯 > -12 (1) 𝘮 > -3 (2) 𝘯 > -4

Cannot be determined. (1), (2), and (1) AND (2) INSUFFICIENT

An eccentric casino owner decides that his casino should only use chips in $5 and $7 denominations. Which of the following amounts cannot be paid out using these chips? a. $31 b. $29 c. $26 d. $23 e. $21

D

For one roll of a certain number cube with six faces, numbered 1 through 6, the probability of rolling a two is ⅙. If this number cube is rolled 4 times, which of the following is the probability that the outcome will be a two at least 3 times? a. (⅙)⁴ b. 2(⅙)³ + (⅙)⁴ c. 3(⅙)³(⅚) + (⅙)⁴ d. 4(⅙)³(⅚) + (⅙)⁴ e. 6(⅙)³(⅚) + (⅙)⁴

D

{9, 12, 15, 18, 21} Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set? I. 14, 16 II. 9, 21 III. 15, 100 a. II only b. III only c. I and II d. II and III e. I, II, and III

D

A bookshelf holds both paperback and hardcover books. The ratio of paperback books to hardcover books is 22 to 3. How many paperback books are on the shelf? (1) The number of books on the shelf is between 202 and 247, inclusive. (2) If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1.

D 22 paperbacks, 3 hardcovers, 25 total (1) SUFFICIENT: there is only one multiple of 25 between 202 and 247, must be 225 (2) SUFFICIENT: (𝘱 - 18)／(𝘩 + 18) = 22/3

Will it cause the original value to increase, decrease or stay the same? Multiply a positive, proper fraction by ⅜.

Decrease

What is 𝘹? (1) 𝘹 = 4𝘺 - 4 (2) 𝘹𝘺 = 8

E

Odd, even, or cannot be determined: If 𝘯, 𝘱, 𝘲, and 𝘳 are consecutive integers, what is their sum?

Even

If 𝘱, 𝘲, and 𝘳 are integers 𝘱𝘲 + 𝘳 even? (1) 𝘱 + 𝘳 is even. (2) 𝘲 + 𝘳 is odd.

For 𝘱𝘲 + 𝘳 to be even, either both 𝘱𝘲 and 𝘳 need to be even or both 𝘱𝘲 and 𝘳 need to be odd (1), (2) and (1) AND (2) INSUFFICIENT: In statement (1) both could be odd or even. Statement two is the same. Neither give you enough information.

Which quadrants, if any, contain points on the line represented by 𝘹 + 18 = 2𝘺?

I, II, and III

Which quadrants, if any, do NOT contain any points on the line represented by 𝘹 - 𝘺 = 18?

II

Which quadrants, if any, do NOT contain any points on the line represented by 𝘹 = 10𝘺?

II and IV

Will it cause the original value to increase, decrease or stay the same? Multiply the numerator of a positive, proper fraction by 3／2.

Increase

Company 𝘟 has exactly two product lines and no other sources of revenue. If the consumer product line experiences a 𝘬% increase in revenue (where 𝘬 is a positive integer) in 2015 from 2014 levels and the machine parts line experiences a 𝘬% decrease in revenue in 2015 from 2014 levels, did Company 𝘟's overall revenue increase or decrease in 2015? (1) In 2014, the consumer products line generated more revenue than the machine parts line. (2) 𝘬 = 8

Increase Total Revenue = 𝘤(revenue) + 𝘮(revenue) (1) SUFFICIENT: This statement tells you that 𝘤(consumer products) is greater than 𝘮(machine parts). Thus, a 𝘬% increase in 𝘤 is greater than a 𝘬% increase in 𝘮, and the overall revenue went up.

Is the sum of the integers from 54 to 153, inclusive, divisible by 100?

No

Is the average of 𝘯 consecutive integers equal to 1? (1) 𝘯 is even. (2) If 𝘚 is the sum of the 𝘯 consecutive integers, then 0 < 𝘚 < 𝘯.

No (1) SUFFICIENT: The average of an even number of consecutive integers is never an integer. (2) SUFFICIENT: Average = Sum/number. 0 < 𝘚 < 𝘯 = 0／𝘯 < 𝘚／𝘯 < 𝘯／𝘯 = 0 < 𝘚／𝘯 < 1.

Odd, even, or cannot be determined: If 𝘩 is even, 𝘫 is odd, and 𝘬 is odd, what is 𝘬(𝘩 +𝘫)?

Odd

Life expectancy is defined by the formula 2𝘚𝘉／𝘎, where 𝘚 = shoe size, 𝘉 = average monthly electric bill in dollars and 𝘎 = GMAT score. If Melvin's GMAT score is twice his monthly electric bill, and his life expectancy is 50, what is his shoe size?

Size 50

Will it cause the original value to increase, decrease or stay the same? Multiply both the numerator and denominator of a positive, proper fraction by 3½

Stay the same

True or False? (-3／4)³ > -3／4

True

Given that 7 is a factor of 𝘯 and 7 is a factor of 𝘱, is 𝘯 + 𝘱 divisible by 7?

Yes

If 5𝘉 > 4𝘉 + 1, is 𝘉² > 1?

Yes

If 𝘢 is divided by 7 or by 18, an integer results, is 𝘢/42 an integer?

Yes

If 𝘴 is a multiple of 12 and 𝘵 is a multiple of 12, is 7𝘴 + 5𝘵 a multiple of 12?

Yes

If 𝘹, 𝘺, and 𝘻 are consecutive integers , is 𝘹 + 𝘺 + 𝘻 divisible by 3?

Yes

Is 𝘻 > 0? (1) (𝘻 + 1)(𝘻)(𝘻 - 1) < 0 (2) │𝘻│ < 1

Yes (1) AND (2) SUFFICIENT: If you test values other than -1, 0, 1, or 2 like -2 and ½ statement (1) works and ½ > 0.

⊲━|━━━|━━|━⊳ 𝘹`````𝘺```𝘻 On the number line above, is 𝘹𝘺 < 0? (1) Zero is to the left of 𝘺 on the number line above. (2) 𝘹𝘺 and 𝘺𝘻 have opposite signs.

Yes (1) AND (2) SUFFICIENT: Statement (1) tells you that zero is either to the left of 𝘹, making both 𝘹 and 𝘺 positive numbers or between 𝘹 and 𝘺, making 𝘹 a negative number and 𝘺 a positive number. Statement (2) tells you that zero must be between 𝘹 and 𝘺, making 𝘹𝘺 < 0.

Is 𝘱𝘲 divisible by 168? (1) 𝘱 is divisible by 14. (2) 𝘲 is divisible by 12.

Yes (1) AND (2) SUFFICIENT: by combining the primes from the two numbers, the question remains "does 𝘱𝘲 contain at least three 2's, one 3, and one 7". The answer a definitive yes

Eco Wildlife Preserve contains 5𝘹 zebras and 2𝘹 lions, where 𝘹 is a positive integer. If the lions succeed in killing "z" of the zebras, is the new ratio of zebras to lions less than 2 to 1? (1) 𝘻 > 𝘹 (2) 𝘻 = 4

Yes (1) SUFFICIENT: (5𝘹 - 𝘻)/2𝘹 < 2/1 and 𝘻 > 𝘹.

𝘢, 𝘣, and 𝘤 are integers in the set {𝘢, 𝘣, 𝘤, 51, 85, 72}. Is the median of the set greater than 70? (1) 𝘣 > 𝘤 > 69 (2) 𝘢 < 𝘤 < 71

Yes (1) SUFFICIENT: If the smallest possible value for 𝘉 is 71 than the median would at least be 70.5.

If 𝘹, 𝘺, and 𝘻 are integers, is 𝘹 even? (1) 10ᵡ = (4ᵞ)(5^𝘻) (2) 3ᵡ ⁺ ⁵ = 27ᵞ ⁺ ¹

Yes (1) SUFFICIENT: If you take the prime factors of statement (1); (5 x 2)ᵡ = (2 x 2)ᵞ(5)^z, (5)ᵡ(2)ᵡ = (2)²ᵞ(5)^z, this tells you that 𝘹 = 𝘻 and 𝘹 = 2𝘺. Since an even integer multiplied by an odd integer equals an even number, 𝘹 = 2𝘺 tells you that 𝘹 is even?

If 𝘤 and 𝘥 are integers 𝘤 - 3𝘥 even? (1) 𝘤 and 𝘥 are odd. (2) 𝘤 - 2𝘥 is odd.

Yes (1) SUFFICIENT: If 𝘤 and 𝘥 are odd, then 𝘤 - 3𝘥 equals O - (3 x O) = O - O = Even (2) INSUFFICIENT: 𝘤 must be odd because 2𝘥 is even, tells you nothing about 𝘥.

`````````∆````````````` `````````│````````````𝘓 `````````│`````````╱`` ⊲━━━━│━━━━●━⊳` `````````│````╱ `𝘚 (4,0) `````````│`╱`````````` `````````● 𝘙 (0, -5)`````` ``````╱`│````````````` `````````∇````````````` Line 𝘓 passes through 𝘙 (0, -5) and 𝘚 (4,0) (see figure above). Point 𝘗 with coordinates (𝘹, 𝘺) is a point on line 𝘓. Is 𝘹𝘺 > 0? (1) 𝘹 > 4 (2) 𝘺 > -5

Yes (1) SUFFICIENT: If 𝘹 > 4, then point 𝘗 is in Quadrant I, so 𝘹𝘺 > 0. (2) INSUFFICIENT: Point 𝘗 could be in either Quadrant I or Quadrant IV.

If 0 < 𝘢𝘣 < 𝘢𝘤, is 𝘢 negative? (1) 𝘤 < 0 (2) 𝘣 > c

Yes (1) SUFFICIENT: Statement (1) tells you that 𝘤 is negative. Therefore, 𝘢 is negative.

Express as decimal: a. 9／2 b. 3,000／10,000

a. 4.5 b. 0.3

_____𝘧°_╲𝘦°__________𝘣°│𝘢°______ ``````𝘩°╲𝘨°```𝘤°│𝘥°```` ``````````╲````│````` ````````````╲`𝘪°│````` ``````````````╲│𝘭°```` ```````````````𝘫°│╲``` ````````````````│𝘬°`╲ a. If 𝘤 + 𝘨 = 140, find 𝘬. b. If 𝘨 = 90, what is 𝘢 + 𝘬? c. If 𝘧 + 𝘬 = 150, find 𝘣.

a. 40° b. 90° c. 150°

````````````╱````````` 𝘈 ___________𝘹°╱_________________ 𝘉 𝘊 __________╱_____________________ 𝘋 ```𝘺°╱```````````````` ``╱``````````````````` a. If 𝘹 - 𝘺 = 10 what is 𝘹? b. If 𝘹 + (𝘹 + 𝘺) = 320, what is 𝘹?

a. 95° b. 140°

````````╲`````````````╱```````` ```````````╲```````╱``````````` _____________________𝘧°╲𝘢°╱𝘣°_____________________ ````````````𝘦°╱𝘥°╲𝘤°```````````` ```````````╱```````╲``````````` ````````╱`````````````╲```````` a. If 𝘢 is 95, what is 𝘣 + 𝘥 - 𝘦? b. If 𝘤 + 𝘧 = 70, and 𝘥 = 80, what is 𝘣?

a. 95° b. 65°

Given that (𝘥／4) + (8／𝘥) + 3 = 0, what is 𝘥?

{-8, -4}

In the decimal, 2.4𝘥7, 𝘥 represents a digit from 0 to 9. If the value of the decimal rounded to the nearest tenth is less than 2.5, what are the possible values of 𝘥?

{0, 1, 2, 3, 4}

Given that 𝘻² - 10𝘻 + 25 = 9, what is 𝘻?

{2, 8}

Al and Barb shared the driving on a certain trip. What fraction of the total distance did Al drive? (1) Al drove for ¾ as much time as Barb did. (2) Al's average driving speed for the entire trip was ⅘ of Barb's average driving speed for the trip.

⅜ 𝘙 x 𝘛 = 𝘞 (1) AND (2) SUFFICIENT: Statement (1) gives you Al's time in terms of Barb (¾). Statement (2) gives you Al's speed in terms of Barb (⅘). Al's Work = ⅗ of Barb's work. Totaling 8/5. (⅗)／(8/5) = ⅜.

The "competitive edge" of a baseball team is defined by the formula √(𝘞／𝘓), where 𝘞 represents the number of the team's wins and 𝘓 represents the number of the team's losses. This year, the GMAT All-stars had 3 times as many wins and one-half as many losses as they had last year. By what factor did their "competitive edge" increase?

√(6)

If 4/𝘹 < ⅓, what is the possible range of values for 𝘹?

𝘹 < 0 OR 𝘹 > 12

Solve for 𝘹 and 𝘺: 𝘺 = 2𝘹 + 9 and 7𝘹 + 3𝘺 = -51

𝘹 = -6; 𝘺 = -3

Solve for 𝘹: (𝘹 + 2)／(4 + 𝘹) = ⅖

𝘹 = ½

Inside of ◔: ````````╱... `````╱........... ``╱____________ 𝘖`````````` In the figure above, 𝘖 represents the center of a circular, 60-minute timer. If the minute-hand moves through the shaded region shown, does the shaded region represent more than 10 minutes on the timer? (1) The minute-hand has a length of 10. (2) The area of the sector is greater than 16π.

𝘹 > 10 minutes = 𝘹 > 1/6πr²(area of the circle) (1) AND (2) SUFFICIENT: