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

1. Quantity A =(10)^{-1} Quantity B = (100)^{-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 A is greater

(10)^{-1} = 1/10 and (100)^{-1} = 1/100. For fractions with the same numerator, the one with the smaller denominator is the larger one. Hence Quantity A is greater.

2. Quantity A = (0.3)^{3} Quantity B = 3 (0.3) • 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

(0.3)^{3} = 0.3 x 0.3 x 0.3 = 0.027 3 (0.3) = 0.9

3. x = 3 + 5 + 7 + 9 + 11 + 13 y = 2 + 4 + 6 + 8 + 10 + 12 Quantity A = 90 Quantity B = x + 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 two quantities are equal

A quick way of doing this problem is as follows: x = 3 + 5 + 7 + 9 + 11 + 13 y = 12 + 10 + 8 + 6 + 4 + 2 x + y = 15 + 15 + 15 + 15 + 15 + 15 = 15 x 6 = 90.

4. 3x - 9y = 12 Quantity A = 4 - 3y Quantity B = x • 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

3x - 9y = 12 or x - 3y = 4 (on dividing by 3). So x = 4 + 3y. Thus, Quantity A = 4 - 3y and Quantity B = 4 + 3y. Various cases arise depending on whether y is positive, negative or zero. As this information is not given, the relationship is indeterminate.

5.

Quantity A = Measure of angle 3 Quantity B = Measure of angle 4 • 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 3 and 4 are vertically opposite angles and hence always equal.

6.

AB is the diameter of the circle, and the line touching the circle at point D is a tangent to the circle. Quantity A = Measure of angle 3 Quantity B = Measure of angle 4 • 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 3 = angle between the radius and a tangent at the point of contact = 90^{o}. Angle 4 = angle subtended by the diameter on the circumference = 90^{o}.

7.

Given angle 2 = 120^{o} 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 two quantities are equal

The angle subtended by an arc at the center is double the angle subtended by the same arc on the circumference. If angle 2 = 120^{o}, then angle 1 = 60^{o} and angle 1 + angle 2 = 180^{o}.

8.

Given angle 1 = angle 2. Quantity A = AB/BC Quantity B= AC/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

If angle 1 = angle 2, then the opposite sides must be of equal length.So AB = AC or AB/BC = AC/BC.

9.

In the right-angled triangle ABC, right-angled at B, angle 1 = 45^{o}. Quantity A = Length of side AB Quantity B = 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

If one angle in a right-angled triangle is 45^{o}, then the other too must be 45^{o}. Hence it is an isosceles triangle with AB = BC.

10. In an isosceles triangle ABC, right-angled at B: Quantity A = Measure of angle A in degrees Quantity B = 45 degrees • 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

In an isosceles right triangle, two angles will always be 45^{o} each.

11.

In triangle ABC, angle A = 60^{o} and AB = AC = 7. Quantity A = Perimeter of triangle Quantity B = 30 • 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 A = 60^{o} and AB = AC, then angle 1 = angle 2 = 60^{o}. Hence the triangle is equilateral. So perimeter = 7 + 7 + 7 = 21.

12.

From the vertex A of triangle ABD, a line is dropped on the opposite side BD. Quantity A = Length of side AB Quantity B = Length of side AC • 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 anything about the relationship between the lengths of the sides AB and AC. The diagram (is for illustration only and is not drawn to scale) may be misleading!

13.

The triangles ABC and CED are right-angled at A and D respectively. Quantity A = Measure of angle 2 Quantity B = Measure of angle 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

In the two triangles ABC and CED, Angle ACB = Angle DCE as they are vertically opposite angles Angle CAB = Angle CDE = 90^{o} Angle CBA = 180^{o} - angle ACB - angle CAB Angle CED = 180^{o} - angle DCE - angle CDE Hence, Angle CBA = Angle CED or angle 2 = angle 6.

14.

Consider an isosceles triangle ABC right-angled at A, with a perpendicular dropped from A on to BC at point D. 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

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

15. Quantity A = 2^{4} Quantity B = 4^{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

Both 2^{4} and 4^{2} are equal to 16.

16. Consider an isosceles triangle ABC. Quantity A = Length of side AB Quantity B = 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 relationship cannot be determined from the information given

Two sides in an isosceles triangle are of equal length. However, the problem does not state which two sides are equal. Hence the relationship is indeterminate.

17. In rectangle ABCD, AB = π and BC = Radius of circle O. Quantity A =Area of rectangle ABCD Quantity B = Area of circle 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

Area of the rectangle = π r and the area of the circle = π r^{2}. At first glance, the area of the circle may seem larger. But consider the case when radius r is smaller than 1. Then the area of the rectangle becomes larger. Hence relationship is indeterminate.

18. x < y Quantity A =(x + y)/2 Quantity B = y • 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

Quantity A is the mean (average) of x and y. The mean always occurs between the two numbers. Hence Quantity B is greater.

19. x and y are of different signs. Quantity A =|x + y| Quantity B = |x| + |y| • 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

|x| and |y| will both be positive and hence their sum will be greater than |x + y|.

20. The average of 10 numbers is 5. Quantity A = Their sum Quantity B = 50 • 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 (arithmetic mean) of n numbers is their sum divided by n. If the average of 10 numbers is 5, their sum will be 10 x 5 = 50.