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

1.

Diagram is illustrative and is not drawn to scale. Quantity A = Measure of angle 1 + Measure of angle 4 Quantity B = 90^{o} - Measure of angle 2^{o} • 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

In the figure, angles 3 and 4 are vertically opposite angles and are equal. Also the sum of the three angles of a triangle is 180^{o}. Hence angle 1 + angle 2 + angle 4 = 180^{o} or angle 1 + angle 4 = 180^{o} - angle 2.

2. Quantity A = (27 - 13) (296 + 534) Quantity B = (27 + 13) (534 + 296) • 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 y = 296 + 534. Then, Quantity A = 14y and Quantity B = 40y. Now, 14y < 40y. Note that you save time by not calculating the exact values of Quantity A and Quantity B.

3. y^{2} + z^{2} = 34 and yz = 15 Quantity A = y^{2} + 2yz + z^{2} Quantity B = (y + 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 two quantities are equal

It is worth remembering that (y + z)^{2} = (y + z) (y + z) = y^{2} + 2yz + z^{2}

Note that it is not meaningful to solve the equations simultaneously and determine y and z equal +3 or -3 and +5 or -5.

4. Quantity A = (y + 5)^{2} Quantity B = (y - 5)^{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

For y = 0, (y + 5)^{2} = (y - 5)^{2} = 25 For y < 0, (y + 5)^{2} < (y - 5)^{2} For y > 0, (y + 5)^{2} > (y - 5)^{2}

5. Quantity A = (1/25)^{1/2} + (1/144)^{1/2 }Quantity B = [(1/25) + (1/144)]^{1/2} • 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

6. Quantity A = Time to travel 95 miles at 50 miles per hour Quantity B = Time to travel 125 miles at 60 miles per hour • 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

Time = Distance / Speed Time to travel 95 miles at 50 miles per hour = 95/50 < 2 hours. Time to travel 125 miles at 60 miles per hour = 125/60 > 2 hours. Note that you save time by not calculating the exact values of Quantity A and Quantity B.

7. Quantity A = 4 / 100 Quantity B = 0.012 / 3 • 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

4/100 = 0.04 and 0.012/3 = 0.004. Note that 0.04 > 0.004

8. Quantity A = 1.1 Quantity B = 12.1^{1/2} • 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

Now 9^{1/2} = 3 and 16^{1/2} = 4. So, 12.1^{1/2} lies between 3 and 4. Therefore, it is greater than 1.1 Note that you save time by not calculating the exact values of Quantity A and Quantity B. Also, note that 1.1^{2} = 1.1 x 1.1 = 1.21 (and not 12.1).

9. Quantity A = (9/13)^{2} Quantity B = (9/13)^{1/2} • 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

A tricky problem. Let y = 9/13. However, it is not meaningful to try to manually calculate y^{2} or y^{1/2} in this case. The relationship is not indeterminate because the uantities are given numbers. It is also clear that Quantity A does not equal Quantity B here. So, we need to ascertain whether y^{2} is greater than or less than y^{1/2} when 0 < y < 1. Note y^{2} < y when 0 < y < 1. Also y^{1/2} > y when 0 < y < 1. Therefore, y^{1/2} > y^{2} when 0 < y < 1... worth remembering!

10. Quantity A = 0.8^{3} Quantity B = 0.8^{1/3} • 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

A tricky problem. Let y = 0.8 ; however, it is not meaningful to try to manually calculate y^{3} or y^{1/3} in this case. The relationship is not indeterminate because the uantities are given numbers. It is also clear that Quantity A does not equal Quantity B here. So, we need to ascertain whether y^{3} is greater than or less than y^{1/3} when 0 < y < 1. Note y^{3} < y when 0 < y < 1. Also y^{1/3} > y when 0 < y < 1. Therefore, y^{1/3} > y^{3} when 0 < y < 1... worth remembering!

11. Quantity A = (3 x 4 x 17) / (121 x 100) Quantity B = (4 x 5 x 19) / (1000 x 121) • 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 y = 4/121. Then, Quantity A = (3 x 17)/(100 y) = 0.51 y and Quantity B = (5 x 19)/(1000 y) = 0.095 y. Now, 0.51 y > 0.095 y. Note that you save time by not calculating the exact values of Quantity A and Quantity B.

12. 100 < y < 200 and 100 < z < 210 Quantity A = y Quantity B = z • 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

The relationship is clearly indeterminate because y can take any value between 100 and 200, whereas z can take any value between 100 and 210. hus, y = z, y < z or y > z.

13. Consider a triangle PQR. Quantity A = length of PQ + length of QR Quantity B = length of PR • 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

With some visualization, it is clear than PQ + QR can never be less than or equal to PR. In fact, the sum of the lengths of two sides (PQ + QR) is ecessarily greater than the length of the third side PR.

14. Consider a rectangle. The length of its shorter side is 8, and the length of its diagonal is 16. Quantity A = 30^{o} Quantity B = measure of angle formed by diagonal and shorter side • 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

A rough sketch will help in visualizing the solution. The diagonal with the two sides of the rectangle forms a right triangle. The measure of the angle formed by the diagonal and the shorter side is 60^{o} because cos 60^{o} = 1/2 = 8/16. So, Quantity B = 60^{o}

15. Quantity A = Percentage increase from 10 ft to 13 ft Quantity B = Percentage increase from 13 ft to 16 ft • 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

Quantity A = (3/10) x 100 and Quantity B = (3/13) x 100. Now, 3/10 > 3/13. Note that you save time by not calculating the exact values of Quantity A and Quantity B.

16. Quantity A = Time elapsed from 6:47 a.m. to 10.13 the same morning Quantity B = 3 hours 25 minutes • 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

Quantity A = 3 hours 26 minutes (on adding up 13 minutes from 6.47 a.m. to 7 a.m., 3 hours from 7 a.m. to 10 a.m., and again 13 minutes from 10 a.m. to 10.13 a.m.)

17. Quantity A = Volume of a cube in which the length of a side is 5 Quantity B = Volume of a cube in which the length of a diagonal of a face is 6 (2)^{1/2} • 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

If side = 5, then volume of cube = 5^{3} = 125. If side = a, then diagonal of a face = (a^{2} + a^{2})^{1/2} = (2)^{1/2}a. If diagonal of a face = 6 (2)^{1/2}, then side = 6 and volume = 6^{3} = 216. Note that you may save time by not calculating the volumes. If the side of the cube is greater in length, then its volume must be greater.

18. Quantity A = Average of (2y + 3z + 127) and (4y + 5z + 73) Quantity B = Average of (3y + 4z + 173) and (3y + 4z + 73) • 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

19. A task is done by y women in 45 hours. The same task is done by (y + 2) women in z hours. Quantity A = 45 Quantity B = z • 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 the number of women is increased, then the task is done in fewer hours. So, z < 45.

20.

Diagram is illustrative and is not drawn to scale. AB is the diameter of the circle. Angle BAC = 30^{o} Quantity A = Length of side AB Quantity B = 2 (Length of side BC) • 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

Angle subtended by the diameter on the circumference = 90^{o}. So, ABC is a right triangle. sin BAC = opposite side / hypotenuse = BC / AB. sin BAC = sin 30^{o} = 0.5 So, BC / AB = 0.5 or AB = 2 BC.