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

1.

Quantity A = Measure of angle 3 Quantity B = Measure of angle 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

For the triangles ABC and ABD we have: angle 5 + angle 1 + angle 3 = 180^{o} angle 5 + angle 1 + angle 2 + angle 6 = 180^{o} From the above equations, angle 3 = angle 2 + angle 6. So angle 3 is greater than angle 2.

2. x < 0 ; y < 0 Quantity A = x + y Quantity B = xy • 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

The product of two negative numbers is always positive and their sum is always negative. So Quantity B will always be greater.

3. The sum of three consecutive even numbers is 18. Quantity A = Their average 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

Average = 18/3 = 6. It is not necessary to individually determine the three consecutive even numbers (4, 6, 8).

4.

Quantity A = Measure of angle 2 Quantity B = Measure of angle 3 • 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

Angle 2 + Angle 3 = 180^{o} (supplementary angles). However, the angles 2 and 3 can individually take any value. So the relationship is indeterminate.

5.

Quantity A = Measure of angle 1 + Measure of angle 2 Quantity B = 180^{o} • 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 angle subtended by an arc at the center (angle 2) is double the angle subtended by the same arc on the circumference (angle 1). angle 1 + angle 2 = angle 1 + 2 (angle 1) = 3 (angle 1).Depending on whether angle 1 is less than, greater than or equal to 60^{o}, (angle 1 + angle 2) will be less than, greater than or equal to 180^{o}. Hence the relationship is indeterminate.

6.

In triangle ABC, AB = AC. Quantity A = Measure of angle 1 Quantity B = Measure of angle 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

The angles opposite the equal sides in an isosceles triangle are equal. So angle 1 = angle 2.

7. |x| = 13 , |y| = 4 Quantity A =|x - y| Quantity B = 11 • 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

If x = 13 and y = -4, then |x - y| = 17 which is greater than 11. But if x = 13 and y = 4, then |x - y| = 9 which is less than 11 Hence the relationship is indeterminate.

8. x - y > 10 Quantity A = y - x 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:Quantity B is greater

If x - y > 10, then y - x < -10 (on multiplying by -1). Hence Quantity B is greater.

9. Quantity A = 3/5 + 4/7 Quantity B = 7/12 • 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

3/5 + 4/7 = 41/35, which is greater than 1. On the other hand, 7/12 is less than 1.

10. x = 0, y > 0 Quantity A = x^{y} Quantity B = y^{x} • 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

Zero raised to any positive power is zero, so Quantity A = 0. Any number raised to 0 is unity, so Quantity B = 1.

11.

Quantity A = Measure of angle 1 + Measure of angle 2 + Measure of angle 3 Quantity B = 360^{o} • 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

For the angles, we have the following set of relations: angle 1 = 180^{o} - angle 4 = angle 5 + angle 6 angle 2 = 180^{o} - angle 5 = angle 4 + angle 6 angle 3 = 180^{o} - angle 6 = angle 4 + angle 5 Adding these three equations, we get angle 1 + angle 2 + angle 3 = 360^{o}.

12.

From the vertex A of triangle ABD, a line is dropped on the opposite side BD. Quantity A = Measure of angle 1 Quantity B = Measure of angle 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

There isn't sufficient information here to decide about the relationship between angle 1 and angle 2. The diagram may be misleading... it is for illustration only and is not drawn to scale.

13.

In triangle ABC, AB = AC and angle 1 = 100^{o}. Quantity A = Measure of angle 2 Quantity B = Measure of angle 3 • 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

Angles opposite to equal sides are equal. Hence angle 2 = angle 3.

14.

Consider an isosceles triangle ABC right-angled at A. A perpendicular is dropped from A on to BC at point D. Quantity A = Length of side BD Quantity B = Length of side AD • 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

ABC is an isosceles right triangle. Hence angle 1 = 45^{o}. For the triangle ABD, angle 5 = 90^{o} and angle 1 = 45^{o} and hence angle 2 = 45^{o}. So triangle ABD is also isosceles and hence BD = AD.

15. Jason is 4 times older than Mary and 5 times older than Mark. Quantity A = Mary's Age Quantity B = Mark's Age • 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

Mary's age is 1/4 th Jason's age. Mark's age is 1/5 th Jason's age. So, Mary's age is greater than Mark's.

16. The area of Circle 1 is four times the area of circle 2. Quantity A = Radius of circle 1 Quantity B = Diameter of circle 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

The radius of Circle 1 will be double the radius of Circle 2 if the area of Circle 1 is four times the area of Circle 2. So, the radius of Circle 1 equals the diameter of Circle 2. Mathematically, Area of circle 1 = A_{1} = π r_{1}^{2}and Area of circle 2 = A_{2} = π D_{2}^{2}/4. Since A_{1} = 4 A_{2}, r_{1}^{2} = D_{2}^{2} or r_{1} = D_{2}.

17. xyz = 1; x = 1 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

If xyz = 1 and x = 1, then z is the reciprocal of y (i.e., z = 1/y). For example: y = 10, z = 1/10; y = 1/2, z = 2; y = 1, z = 1. Hence the relationship is indeterminate.

18. x = 0, y > 0, z < 0 Quantity A = (y^{2} + 3x + z^{3})x Quantity B = 0 • 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

Quantity A = (y^{2} + 3x + z^{3}) 0 = 0.

19. 1 mile = 1.6 kilometer Quantity A = 1 kilometer Quantity B = 0.625 mile • 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

20. 1/z < 0 Quantity A = z 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:Quantity B is greater

If 1/z < 0, then z must be negative. So, Quantity B is greater.