The sum of the even numbers from 2 to 100 inclusive

52 x 50

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

The sum of the even numbers will be the middle value in the series (the average) multiplied by the number of terms. There are 50 even numbers, and the average will lie between the 25^th and 26^th term (i.e. 51). The sum will be 51 x 50 which is less than the right column.

x⁴ / 4

(x / 4)⁴

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

There is no limit set on x, which could, therefore, be 0, a fraction, 1, or anything else!
If x is 1, then the left column is � and the right column is 1/256 which is smaller. But if you take x as 0, both columns are zero. Therefore, this is a D answer.

Fred is now 10 years older than George was 5 years ago

Fred’s age 5 years ago

George’s age now

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

We can express the information in an equation. Let Fred's age now be F and Georges's age now be G.
F = (G-5) +10 ; F = G + 5
From this we can get the difference in their ages (5). Now if we take 5 years off Fred's age now it will be the same as George's age.

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y/x

q/p

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

Imagine lines drawn from each of the points to the origin. The slope of the line will then be the y-coordinate / x-coordinate. So one line willslope y/x and the other q/p. The greatest (steepest) slope is the one that makes the greatest angle with the x-axis. Hence, y/x is greater. (No need to calculate anything. Just judge the slope by eye).

The rate of a certain chemical reaction doubles for every 10 degree rise in temperature.

Twice the rate at 10°

Half the rate at 30°

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

If the rate at 10 degrees is x, the rate will be 2x at 20 degrees and 4x at 30 degrees.
Hence twice the rate at 10 degrees (2x) and half the rate at 30 degrees (1/2 of 4x) will be the same.

Twice an odd integer greater than 6

Three times an odd integer greater than 2

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

No limits are placed on the upper size of the odd integers, so it is easy to pick cases where columns A or B can be greater. Hence it is a D answer.

Discount in dollars on $60 at the rate of 4 percent

Discount in dollars on $40 at the rate of 6 percent

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

Four percent of 60 = 0.04 x 60 = 2.4
Six percent of 40 = 0.06 x 40 = 2.4

The beach resort is 2 kilometers from the city and the sports complex is 10 kilometers from the city.
The city, resort and sports complex all lie at sea level.

The distance from the beach resort to the sports complex.

7 kilometers

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

Draw a sketch. Put a point for the city. Draw a circle radius 2 centered on the city on which the beach resort can lie. Draw a circle radius 10 centered on the city, on which the sports center can lie. Wherever you put the resort and the sports center on these circles, they can never be closer than 8 kilometers, which is the shortest distance between the circles.

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AB is a diameter of the circle. All triangles above the diameter in the diagram are equal in area. All triangles below the diameter are equal in area.

Total area of the triangles above AB.

Total area of the triangles below AB.

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

The total area of the upper triangles is less than the area of the lower triangles. The more triangles that you cut the semicircle into, the more of the circle that is occupied.

The greatest prime factor of 36

The greatest prime factor of 48

A. The quantity on the left is greater
B. The quantity on the right is greater
C. Both are equal
D. The relationship cannot be determined without further information

Solution

Prime factors of 36 are 3 and 2. Prime factors of 48 are 3 and 2 also. So the greatest prime factor in both cases is 3.

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