GMAT Quant

( √2 - √3 )² = A. 5 - 2√6 B. 5 - √6 C. 1 - 2√6 D. 1 - √2 E. 1

( √2 - √3 )( √2 - √3 )= 2-√2√3 -√2√3+3 5 - 2(√2√3), 5-2(√6) A

Six years ago Anita was P times as old as Ben was. If Anita is now 17 years old, how old is Ben now in terms of P ? A. 11/P + 6 B. P/11 +6 C. 17 - P/6 D. 17/P E. 11.5P

(A - 6) = P(B - 6) But A is 17 and therefore 11 = P(B - 6) 11/P = B-6 (11/P) + 6 = B A When solving initially, I set up (P*B)-6 instead of P*(B-6)

For how many integer values of n will the value of the expression 4n + 7 be an integer greater than 1 and less than 200? A. 48 B. 49 C. 50 D. 51 E. 52

1 < 4n + 7 < 200 n can be 0, or -1 n cannot be -2 or any other negative integer or the expression 4n + 7 will be less than 1 The largest value for n will be an integer < (200 - 7)/4 193/4 = 48.25, hence 48 The number of integers between -1 and 48 inclusive is 50 C

Given that: 4m + n = 20 and |n| ≤ 20 How many ordered pairs (m,n) exist in which m and n both are integers? A. 5 B. 6 C. 10 D. 11 E. 41

1) Possible integer values of n are anything from -20 to 20, including 0 2) Possible integer values of m 4m+n=20, 4m=20-n, m=(20-n)/4, m=5-(n/4) This means that n/4 must be an integer, so n must be 0 or a multiple of 4 3) The number of shared values is 11 including 0 D

5A BC --- D43 In the above correctly worked addition sum, A,B,C and D represent different digits, and all the digits in the sum are different. What is the sum of A,B,C and D?

1) the sum of two 2-digit numbers cannot be more that 198 (99 + 99), so D must be 1 2) Use trial and error to satisfy 5A + BC = 143 A + C must give 3 in the units place, but neither can be 1 since all the digits have to be different. Therefore A + C = 13. With one to carry over into the tens column, 1 + 5 + B = 14, and B = 8.A + C + B + D = 13 + 8 + 1 = 22 B

(6^5 - 6^4)/5=? A. 1/5 B. 6/5 C. 6³ D. 64 / 5 E. 6^4

6^5 can be rewritten as 6^4x 6 Factor (6^4 x 6) - 6^4 = 6^4(6 - 1) = 6^4 x 5 Now, dividing by 5 will give us 6^4 E Remember to look for opportunities to factor

A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube? A. 8p B. pq C. pq + 27 D. -p E. (p - q)^6

A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5 Consider the answers in turn A) 8 is cube of 2 and p is cube, so the answer is a cube B) p and q are both cubes C) pq and 27 are both cubes, but their sum will not necessarily be a cube D) -p is a cube E) since the parentheses is raised to the power of 6, the answer will always be a cube

2^30 + 2^30 + 2^30 + 2^30

All four terms are identical therefore we have 4 (2^30).But 4 = 2^2, and so we can write 2^2 * 2^30 Which is equivalent to 2^32

There are 120 saplings to be planted in an orchard. Ben and Sue working together without a break can complete the job in six hours. How long would it take Ben working alone to complete the job? I) Sue plants 3 saplings in the time it takes Ben to plant 2. II) Sue working alone would take 10 hours to do the job. Data Sufficiency

B+S=6 I) Since Sue works faster, out of the 120 saplings she would plant 72 while Ben planted 48. Ben plants 48 trees in 6 hours or 48/6= 8 saplings/hour We can then find that 120 saplings/ 8 per hour = 120/8=15 hours SUFF II) Sue alone takes 10 hours, meaning that 120trees/10hours=12 trees/hour 12trees/h * 6 hours = 72 trees, meaning Ben planted 120-72=48 trees in 6 hours 48trees/6h= 8saplings/hour, meaning we can calculate the number of hours SUFF D

A box contains 20 balls: 12 red, 8 blue. If two balls are drawn at random without replacement, what is the probability that one will be red and the other will be blue? A. 1/96 B. 6/25 C. 24/95 D. 48/95 E. 1

Combination Approach: ((# ways of drawing red)*(# ways of drawing blue))/(# ways to draw two balls of any color) (12*8)/((20*19)/(1*2) (12*8)/(10*19) 96/190 48/95 Permutation Approach: Prob red first = P(R1) Prob red second = P(R2) Prob blue first = P(R1) Prob blue second = P(B2) [P(R1)*P(B2)] + [P(R2)*P(B1) [12/20 * 8/19] + [8/20 * 12/19] Factor because these amount to the same value 2(12/20 + 8/19) 2*(96/360) 96/180 48/95 D

A wheel has a diameter of x inches and a second wheel has a diameter of y inches. The first wheel covers a distance of d feet in 100 revolutions. How many revolutions does the second wheel make in covering d feet? A. 100xy B. 100y - x C. 100x - y D. 100y / x E. 100x / y

Distance of first wheel is d=100*circumference, d=100*xπ Circumference of second wheel = yπ Distance=revolutions*circumference Revolutions(wheel 2)= Distance/circumfrence(wheel 2) Since distance=100xπ R=100(xπ/yπ). 100y/y E

If x² - y² = 55, and x - y = 11, then y =? A. 8 B. 5 C. 3 D. -8 E. -3

Distribute x² - y² as (x+y)(x-y) Since x-y is 11, the first equation can be written (x+y)*11=55, or x+y=5 To solve for x we can add x-y=11 and x+y=5 to get 2x+16, or x=8 Plug in this value of x: 8-y=11, -y=3, y=-3 (Check: 8+y=5, y=-3) E

A 3 by 4 rectangle is inscribed in circle. What is the circumference of the circle? A. 2.5π B. 3π C. 5π D. 4π E. 10π

Draw the diagram. The diagonal of the rectangle is the diameter of the circle. The diagonal is the hypotenuse of a 3,4,5 triangle, and is therefore, 5.Circumference = π.diameter = 5π

Parallelogram with two adjacent sides 1:2. If area is 36√2 square units and the angle between the two sides is 45 degrees, what is the area, in square units, of the rectangle whose smaller side = the smaller side of the parallelogram and the larger side = the larger side of the parallelogram? A. 36 B. 36√2 C. 72 D. 96 E. 144

Find the area of the parallelogram in terms of x 1) Draw a triangle on either side and a rectangle in the middle Since two angles of each triangle are 45 degrees, the third is 180-45-45=90. 2) Use Pythagorean's Theorem to get height AP. AD is a short side of parallelogram with length x (hypotenuse of triangle), and AP and DP are the two shorter lengths of the triangle. AD^2 = AP^2 + DP^2, AD = √(AP^2 + DP^2) Factor because AP and DP are the same length AD = √(2 * AP^2), AD = √(2) * √(AP^2), AD = √(2) * AP, AP = AD/√(2) Since AD = x, replace it with this value AP = x/√(2) b=2x, h=x/√(2) 3) Area of the parallelogram = b*h = 36√(2). Solve for x. 2x * x/√(2) = 36√(2), 36√(2) = (2x^2)/√(2), 36 = (2x^2)/2, 2x^2 = 72, x^2 = 36, x = 6 4) Area of the rectangle = small side * large side, or b*h b=x, h=2x, A=x*2x, A=6*12, A=72 C

One number, n, is selected at random from a set of 10 integers. What is the probability that ½ n + 13 = 0 ? I) The largest integer in the set is 13. II) The arithmetic mean of the set is zero.

For the equation to equal 0, n has to equal -26. I) This tells us the largest integer, but not the smallest, so we do not know if -26 is in the set. INSUF II) We know that some numbers are positive and some negative, but we do not know the actual numbers. INSUF Together) We do not know the smallest number INSUF E

Given that x and y are real numbers, what is the value of x + y ? I) (x² - y²) / (x-y) = 7 II) (x + y)² = 49 Data Sufficiency

I) Factor (x² - y²) (x+y)(x-y)/(x-y)=7 (x+y)=7 SUFF II) Since (x+y) is squared, it could either equal 7 or -7 and still result in 49 in this equation. This answer is INSUF A

If x and y are integers, and 3x + 2y = 13, which of the following could be the value of y ? A. 0 B. 1 C. 2 D. 3 E. 4

Start by substituting the values for y. Note that x and y are integers, so x must equal a whole number for the value of y to be a possibility y=0, 3x=13, not whole number y=1, 3x+2=13, 3x=11, not whole number y=2, 3x+4=13, 3x=9, x=3 whole number- CORRECT C

A machine puts c caps on bottles in m minutes. How many hours will it take to put caps on b bottles? A. 60bm/c B. bm/60c C. bc/60m D. 60b/cm E. b/60cm

Substitute numbers. If the machine adds 6 caps in 2 minutes, it will bottle 6/2=3 caps per minute, or (6/2)*60 caps per hour. Caps/hour= (6/2)*60 or (c/m)*60 For the number of bottles b, set this equal to b b=(c/m)*60, or b=60c/m Divide the number of bottles b by the caps per hour b*(m/60c), or bm/60c B

(Figure of a circle and square where two sides of the square are tangent to the circle, and one side of the square is the circle's diameter) In the figure above the square has two sides which are tangent to the circle. If the area of the circle is 4a²π, what is the area of the square? A. 2a² B. 4a C. 4a² D. 16a² E. 64a²

Tangent means the edge of the circle touch the edges of the square If the area of the circle is 4a²π , the radius will be the square root of 4a² (i.e. 2a). (Check- 2a^2 * π = 4a²π) If r=2a, d=4a The diameter is the side of the square Area of the square is (4a)² = 16a² D

One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle? I 24 II 20 III 5 A. I only B. II only C. III only D. II and III only E. I, II and III

The maximum area of the triangle will come when the given sides are placed at right angles. WE CAN take 8 as the base and 5 as the height so the area = ½ x 8 x 5 = 20 We can alter the angle between the sides to make it less or more than 90, but this will only reduce the area. (Draw it out for yourself). Hence the area can be anything less than or equal to 20. My first instinct was to calculate the max and minimum possible sides for the triangle, but this is unnecessary since we just need to calculate area with a base and a height.

The slope of the line passing through the point (5,5) is 5/6. All of the following points could be on the line except A. (2.5, 2) B. (11, 10) C. (8, 7.5) D. (-1, 0) E. (-7, -5)

x intercept = (-1,0) Slope equation = (y2-y1)/(x2-x1), use the x intercept for y1 and y2 C) (7.5-0)/8+1)=7.5/9 aka 5/6 A) (2-0)/(2.5+1)=2/3.5 which could not have a point (5,5) with a slope 5/6 A

If V = 12R / (r + R) , then R = A. Vr / (12 - V) B. Vr + V /12 C. Vr - 12 D. V / r - 12 E. V (r + 1) /12

Rearrange the equation to make R the subject. Cross multiply by (r+R). V(r+R)=12R Distribute Vr+VR=12R, Vr=12R-VR Factor Vr=R(12-V), R=Vr/(12-V) A Important to identify that 12R-VR can be factored to pull out R on its own.

If n ≠ 0, which of the following must be greater than n? I 2n II n² III 2 - n A. I only B. II only C. I and II only D. II and III only E. None

Remember that n could be positive, negative, or a fraction. Try out a few cases: In case I, if n is -1, then 2n is less than n. In case II, if n is a fraction such as ½ then n2 will be less than n. In case III, if n is 2, then 2-n = 0, which is less than n. Therefore, none of the choices must be greater than n

2^30 + 2^30 + 2^30 + 2^30 = A. 8^120 B. 8^30 C. 2^32 D. 2^30 E. 2^26

Rewrite as 4(2^30) 4 can be rewritten as 2^2 2^2 * 2^30 = 2^(2+30) =2^23 C

Is the area of circle C greater than the area of square S? I) The diameter of C is equal to the diagonal of square S. II) The perimeter of S is less than the circumference of C. Data Sufficiency

I) If the diagonal of the square equals the diameter of the circle, the square will lie inside the circle with the corners inscribed on the circumference, making the area of the square smaller. SUFF II) If the perimeter of the square is less than the circle, the square will be smaller since it is a less efficient distribution of perimeter space. SUFF D

The arithmetic mean (average) of a set of 10 numbers is 10. Is the median value of the same set also equal to 10? I) Exactly half of the numbers are less than 10. II) The mode of the set of numbers is 10. Data Sufficiency

I) Numbers 1-5 are less than 10 and the median is between 5 and 6, but 5 and 6 could be different distances from 10. Ex. number 5=8 and number 6=11, with a median of 9.5 INSUF II) Mode on its own is INSUF Together) From I) we know that number 5 is less than 10 From II) we know that number 6 is 10, because if the mode is 10 and number 5 is less than 10, then number 6 and at least one more have to be 10. Number 5 is less than 10 and Number 6 is 10, so the median is less than 10 C

What is the ratio of male to female officers in the police force in town T? I) The number of female officers is 250 less than half the number of male officers. I) The number of female officers is 1/7 the number of male officers.

I) We do not know the actual number of males or females, so we cannot determine a ration INSUF II) Since the ratio is the same as a fraction, this is SUFF B

Is w an integer? I) 3w is an odd number. II) 2w is an even number.

I) w could be a whole number like 1, or a fraction like 5/3 which still satisfies the condition. INSUF II) All even numbers divided by 2 give a whole number (ex. 1/2 *2=1 is odd, so the answer must be an integer) SUFF B

If the average age of three people of different ages is 21 years, is the youngest older than 13? I) The oldest is 25. II) One person is 24. Data Sufficiency

If the average age of three people is 21, the sum of their ages is 63 I) If the oldest is 25, than the other two are 63-25=38 collectively. Neither of the remaining two is older than 24 (the oldest), so at most 38-24=14. The youngest is at least 14. SUFF II) 63-24=39, but we do not know if 24 is the oldest, youngest, or middle, so the remaining two people could be any two ages that add up to 39. INSUF A

What digit appears in the units place in the number obtained when 2^320 is multiplied out? A. 0 B. 2 C. 4 D. 6 E. 8

Look for patterns 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32 Pattern is 2,4,8,6 320/4= 80 (divisible by 4), so the unit's place will be 6 D

What is the average of x, y, and z? I) 2x + y + 4z = 23 II) 3x + 4y + z = 22 Data Sufficiency

Need the total of (x+y+Z) Neither statement alone can give this value Together) Add the equations to get 5x+5y+5z=45 Divide by 5= x+y+z=9 Average= 9/3=3 C

If x, y and z are different integers, is x divisible by 11? I) xyz is divisible by 22 and 33 II) yz is divisible by 72 Data Sufficiency

Need to determine if x specifically has 11 as a factor I) The factors of 22 are 2, 11, and the factors of 33 are 3, 11, so xyz has factors 2,3,11, but we do not know if 11 is specifically a factor of x INSUF II) The factors of 72 are 2,2,2,3,3, but this does not answer anything for x INSUF T) Together these do not tell us if x has 11 as a factor INSUF E

What is the value of 3x² + 2x - 1 ? I) x² + 2x = 0 II) x = -2 Data Sufficiency

Need- value of x I) Rewrite x² + 2x = 0 as x(x+2)=0. This means x can either be 0 or -2 and satisfy the condition. INSUF. II) This tells us exactly what x is, so we can solve 3x² + 2x - 1 for 3(-2)² + 2(-2) - 1, 3(4) -4 -1, 7 SUFF

Rectangle ABCD with a shaded semicircle with AD serving as its radius/one length of the square. Rectangle ABCD has a perimeter of 26. The half circle with diameter AD has an area of 8π. What is the perimeter of the part of the figure that is not shaded? A. 26 + 4π B. 18 + 8π C. 18 + 4π D. 14 + 4π E. 14 + 2π

Perimeter of unshaded section will be l+W+W+semicircle arc Half circle has area 8π=1/2πr^2, or 16π=πr^2. r=4, so diameter/AD=8, l=8 Perimeter of the half circle = 1/2*2πr, or πr, 4π Perimeter of rectangle without AD is 26-8=18 Total perimeter of unshaded is 18+4π C

6 people meet for a business lunch. Each person shakes hands once with each other person present. How many handshakes take place? A. 30 B. 21 C. 18 D. 15 E. 10

Person 1 shakes hands with 5 people Person 2 shakes hands with 4 people besides person 1 Total shakes= 5+4+3+2+1=15 D

In a certain village, m litres of water are required per household per month. At this rate, if there are n households in the village, how long (in months) will p litres of water last? A. p /mn B. mn / p C. mp / n D. np / m E. npm

Pick numbers m=100 liters/household/month n=20 households p=1000 liters 1000 liters will last 1000/20*100 months time=p/mn A

A triangle has a perimeter 13. The two shorter sides have integer lengths equal to x and x + 1. Which of the following could be the length of the other side? A. 2 B. 4 C. 6 D. 8 E. 10

The measure of the third side of a triangle must lie between the sum and the difference of the other two sides. Using this fact along with the answer choices, we can eliminate the wrong answers. (A) cannot be correct because 2 would not be the longest side. (B) cannot be correct because again it would not be the longest side (C) could be correct because the other two sides would be 3 and 4. (D) and (E) cannot be correct because this third side would be greater than the sum of the other two sides. C

n and p are integers greater than 1 5n is the square of a number 75np is the cube of a number ?The smallest value for n + p is ? A. 14 B. 18 C. 20 D. 30 E. 50

The smallest value for n such that 5n is a square is 5. 75np can now be written as 75 x 5 x p. This gives prime factors.... 3 x 5 x 5 x 5 x p To make the expression a perfect cube, p will have to have factors 3 x 3 , and hence p =9 n + p = 5 + 9 = 14

Large triangle ABC with smaller triangle ADE forming the tip of ABC. DE and BC are horizontal In triangle ABC, AD = DB , DE is parallel to BC, and the area of triangle ABC is 40. What is the area of triangle ADE ? A. 10 B. 15 C. 20 D. 30 E. it cannot be determined from the information given

The triangles are SIMILAR because their bases are parallel. If we know the ratio of a corresponding side of the two triangles, we can find the ratios for the areas. We know from the prompt that the left side of ABC is twice as long as the left side of ADE, so the ratio of the sides is 1:2. A1=1/2(b*h), and A2=1/2*(2b*2h), so the ratio of the areas will be (1)^2 : (2)^2, or 1:4 Since the big triangle has area 40, the small one has area 10.

Picture of a rectangle inscribed inside an arc which represents 1/4 of a circle. ASB is a quarter circle. PQRS is a rectangle with sides PQ = 8 and PS = 6. What is the length of the arc AQB ? A. 5π B. 10π C. 25 D. 14 E. 28

To find the radius we need to see that the diagonal SQ of the rectangle= radius. Now use Pythagoras to find the diagonal; 82 + 62 = r2 So r = 10 (You could have spotted that this is a 3-4-5 triangle and so saved this part of the calculation) The length of the arc = 1/4 * 20 π = 5π A

3x + y = 19 , and x + 3y = 1. Find the value of 2x + 2y A. 20 B. 18 C. 11 D. 10 E. 5

To solve a pair of simultaneous equations such as those given we can add or subtract them. Adding we get 4x + 4y = 20 Therefore 2x + 2y = 10 D

If a and b are both positive, what percent of b is a? I) a = 3/11 II) b/a = 20 Data Sufficiency

We need to be able to solve (a/b)*100 A) does not indicate the value for B so it is INSUF B) shows us that b/a=20 or a/b=1/20 SUF B

Does x³ - x give a whole number when divided by 3? I) x is a positive integer greater than 1. II) |x|> 0

x³ - x can be expressed x(x²-1) x²-1 is a difference of two squares (a squared number minus another squared number), so it can be factored a^2-b^2=(a+b)(a-b), for x²-1=(x+1)(x-1) x(x+1)(x-1) This represents three continuous integers x-1, x, x+1, so IF x is an integer, one of these values will be a multiple of three, meaning that x³ - x is divisible by 3 as long as one of the values is not 0 (which multiplied by the others would make the whole numerator 0). As a result, x cannot be 0, 1, or -1. I) This confirms that x³ - x will be divisible by 3 (check- if x=2, then 2(2-1)(2+1)=2*1*3=6 which is divisible by 3) SUFF II) This does not confirm if x is an integer or if it is 1 INSUF A