Compare the two quantities and choose the best answer from four choices given.

1. Amy, Beth and Charlie divided a pizza amongst themselves. Amy took 30% of the pizza and ate (3/4) of what she took. Beth took 20% of the pizza. Charlie ate (2/5) of what he took. Quantity A = The amount Amy ate Quantity B = The amount Charlie ate • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

If Amy took 30% and Beth took 20%, then Charlie took 50% of the pizza. Let P denote the size of the total pizza. Amount Amy ate = (3/4) (0.3 P) = 0.225 P. Amount Charlie ate = (2/5) (0.5 P) = 0.2 P.

2. Quantity A = (55 + 59)(61 - 79) Quantity B = (55 - 59)(61 - 79) • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity B is greater

Be careful! Note that (61 - 79) is a negative number. Quantity A is negative because it is the product of a positive number and a negative number. Quantity B is positive because it is the product of two negative numbers.

3. a, b, c and d are four consecutive integers. Quantity A = The arithmetic mean of a and d Quantity B = The arithmetic mean of b and c • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The two quantities are equal

If a, b, c and d are four consecutive integers, then b = a + 1 ; c = a + 2 ; and d = a + 3. Quantity A = (a + d)/2 = (a + a + 3)/2 = (2a + 3)/2. Quantity B = (b + c)/2 = (a + 1 + a + 2)/2 = (2a + 3)/2.

4. 2 < z < 4 Quantity A = π^{2}z^{3} Quantity B = π^{3}z^{2} • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

You could substitute values for z between 2 and 4 to compare. An alternative simple strategy is to divide both Quantity A and B by π^{2}z^{2}. Then Quantity A = z and Quantity B = π. We note that Quantity A lies between 2 and 4, whereas Quantity B equals about 3.14

5. yz < 0 Quantity A = (y − z)^{2} Quantity B = y^{2} + z^{2} • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

(y − z)^{2} = (y − z)(y − z) = y^{2} + z^{2} − 2yz. Since yz is negative, −2yz is positive. So, Quantity A > Quantity B.

6. 11y^{4}/(3y^{2}) = 11/3 Quantity A = y Quantity B = 1 • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

Dividing both sides by 11/3 gives y^{4}/y^{2} = 1 or y^{4} = y^{2}. Now, dividing both sides by y^{2} gives y^{2} = 1. So, y can equal 1 or -1.

7. y > 0 Quantity A = 100 y Quantity B = 100 / y • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

When y = 1, Quantity A equals Quantity B. When y > 1, Quantity A is greater. When y < 1, Quantity B is greater.

8. Quantity A = The perimeter of a rectangle whose area is 33. Quantity B = 28 • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

Area of rectangle = Length x Breadth Perimeter of rectangle = 2 (Length + Breadth) If the area of the rectangle is 33, then there are many possible values for the length and breadth as given below. Length = 11, Width = 3 and Perimeter = 28. Length = 33, Width = 1 and Perimeter = 68. Length = 6, Width = 5.5 and Perimeter = 23.

9. n(n + 2)(n + 4) = 480 Quantity A = n + 2 Quantity B = 6 • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

Expanding the left-hand side will give a cubic polynomial and the resulting cubic equation will be difficult to solve. The following strategy may be adopted. If n + 2 = 6, then n = 4, n + 4 = 8 and 4 x 6 x 8 = 192. Since the product must be 480, n + 2 must be larger than 6.

10. Andrew stitches shirts thrice as fast as Bill. Bill charges 40% more per shirt than Andrew. Quantity A = The amount Andrew earns in 8 days Quantity B = The amount Bill earns in 15 days • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

Let Bill stitch b shirts per day. Then, Andrew stitches 3b shirts per day. Let Andrew charge a dollars per shirt. Then, Bill charges 1.4a dollars per shirt. Amount Andrew earns in 8 days = 8 (3b) a = 24 ab. Amount Bill earns in 15 days = 15 b (1.4a) = 21 ab.

11. p = 3z/4, q = 4r/5, and r = 5z/6 Quantity A = 12p Quantity B = 12q • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

12p = 12 (3z/4) = 9z. q = 4r/5 = 4z/6. 12q = 12 (4z/6) = 8z. If z = 0, then 9z = 8z = 0. If z > 0, then 9z > 8z. If z < 0, then 9z < 8z.

12. For any positive integer n, n! is the product of all positive integers less than or equal to n. Quantity A = 20! / 17! Quantity B = 80! / 78! • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

20! / 17! = (20 x 19 x 18 x 17 x 16 x ...)/(17 x 16 x 15 x ...) = 20 x 19 x 18. 80! / 78! = (80 x 79 x 78 x 77 x ...)/(78 x 77 x 76 x ...) = 80 x 79 = 20 x 4 x 79. Now, 19 x 18 > 4 x 79.

13. 0 < a < 1 Quantity A = a^{3} Quantity B = a^{2} • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity B is greater

When a > 1, then a^{3} > a^{2}. When a = 1, then a^{3} = a^{2} = 1. When 0 < a < 1, then a^{3} < a^{2}. For example, (1/2)^{3} = 1/8, (1/2)^{2} = 1/4, and 1/8 < 1/4.

14. m is a negative number. Quantity A = m^{3} Quantity B = -m^{2} • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

On dividing by m^{2} (a positive quantity), Quantity A = m and Quantity B = -1. When m = -1, then Quantity A equals Quantity B. When -1 < m < 0, then Quantity A is greater. When m < -1, then Quantity B is greater. For example, (-1/2)^{3} = -1/8, -(-1/2)^{2} = -1/4, and -1/8 > -1/4.

15. y is a positive integer less than 200. Quantity A = The number of multiples of 3 between y and 200. Quantity B = The number of multiples of 5 between y and 200. • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

Every third integer is a multiple of 3 and every fifth integer is a multiple of 5. The number of multiples of 3 will typically be greater than the number of multiples of 5 in a large enough interval (if y is small). However, this will not be the case if y is large, e.g., if y = 199, then there is no multiple of 3 between 199 and 200, and there is only one multiple of 5.

16. In the correctly worked out addition problem given below, each letter represents a different digit. BC + BC = CAA Quantity A = A Quantity B = 6 • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The two quantities are equal

BC is a two-digit number, so it is less than 100. BC + BC gives a three-digit number which is therefore between 100 and 198. So, C in CAA must represent 1. Adding the digits in the units place gives 1 + 1 = 2. So, A must represent 2. Finally, C + C = 12 and therefore C must represent 6.

17. Quantity A = The average of the measures of the angles of a scalene triangle Quantity B = The average of the measures of the angles of an isosceles triangle • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The two quantities are equal

The average of the three angles of a triangle is the sum divided by 3. The sum of the angles of any triangle is always 180^{o}. So, the average is 60^{o} irrespective of the kind of triangle.

18. Quantity A = Twice the area of an equilateral triangle whose sides are a Quantity B = The area of a square whose sides are a • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity B is greater

Let h be the height of the equilateral triangle. By Pythagoras' Theorem, h^{2} = a^{2} - (a/2)^{2} = 3a^{2}/4. So, the height h is less than a. Area of triangle = (base x height)/2. 2 (Area of triangle) = base x height < a^{2} = Area of square.

19. A small circle is drawn inside a large circle, whose radius is 40% more than the radius of the small circle. Quantity A = The area of the small circle Quantity B = The area between the two circles • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:Quantity A is greater

If r is the radius of the small circle, then 1.4r is the radius of the large circle. Area of small circle = π r^{2}. Area between circles = π (1.4r)^{2} − π r^{2} = (1.4^{2} − 1) π r^{2} = 0.96 π r^{2}.

20. m and n are prime numbers. m < n and m + n = 16 Quantity A = n Quantity B = 12 • Quantity A is greater • Quantity B is greater • The two quantities are equal • The relationship cannot be determined from the information given Answer:The relationship cannot be determined from the information given

3 + 13 = 16 and 5 + 11 = 16 are the two possibilities. So, n can be either 11 or 13.