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## GRE Test Prep : Quantitative Comparison III

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 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 = 90o.
Angle 4 = angle subtended by the diameter on the circumference = 90o.

7. Given angle 2 = 120o
Quantity A = Measure of angle 1 + Measure of angle 2
Quantity B = 180o
• 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 = 120o, then angle 1 = 60o and angle 1 + angle 2 = 180o.

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 = 45o.
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 45o, then the other too must be 45o. 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 45o each.

11. In triangle ABC, angle A = 60o 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 = 60o and AB = AC, then angle 1 = angle 2 = 60o.
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 = 90o
Angle CBA = 180o - angle ACB - angle CAB
Angle CED = 180o - 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 = 45o.
For the triangle ABD, angle 5 = 90o and angle 1 = 45o and hence angle 2 = 45o.

15. Quantity A = 24 Quantity B = 42
• 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 24 and 42 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 r2.
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.

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