n is a positive integer
p = 8 x 9 x 11 x n

The remainder when p is divided by 6

The remainder when p is divided by 33

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

Since p has factors 8 and 9, it will be divisible by 6.
Since p has factors 11 and 9, it will be divisible by 33. There will be no remainder in either case.

Refer to the previous question

The mean (average) of the 5 annual sales figures

The median of the 5 annual sales figures

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

To find the mean, add all the values and divide by 5. (1000 + 2000 + 3000 + 2000 + 1000)/5 = 9000/5 = 1800
To find the median, arrange the numbers in order of magnitude and find the middle term
1000,1000,2000,2000,3000. The median = 2000

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This question and the next refer to the following:
The graph shows the sales figures for a certain company in five consecutive years.

Percentage increase in sales from 1989 to 1991

Percentage fall in sales from 1991 to 1993

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

From 89 to 91 the rise is 2000. From 91 to 93 the fall is 2000.
But the percent increase is (2000/1000) x 100; and the percent fall is (2000/3000) x 100
Therefore, the percent rise is greater.

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C is the midpoint of segment AE and AB < DE

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

BC = AC - AB
CD = CE - DE, but AC and CE are both equal. Since DE is greater than AB, CD must be less than BC. (We are taking off a smaller amount from half the segment, when we subtract AB from AC.)

There are 28 students in a class all born in 1990

The probability that all their birthdays are in February

The probability that all their birthdays fall on a Friday

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 are 12 months in a year so each student has a 1/12 chance of being born in a particular month. There are only 7 days in a week, so each student has a 1/7 chance of being born on a particular day of the week The chances that all will be born on a particular day of the week is thus higher than the chances of them all being born in the same month.

The sum of all the integers from -10 to 12 inclusive.

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 integers from -10 to 10 inclusive is 0. (-10 + 10, -9 + 9, -8 + 8 etc.) This leaves only 11 and 12 to be added = 23

Three boxes contain 20, 25 and 27 sweets respectively.

The least number of sweets that need to be transferred so that there is an equal number in each box.

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 number of sweets = 20 + 25 + 27 = 72
If there are an equal number in each, there will be 72/3 = 24
To make the numbers equal we will need to remove one from the second box, and three from the third (and put them in the first). This means we must move 4 sweets.

A circle and a line lie in the same plane.

The greatet possible number of points that are on both the circle and the line.

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 line can cut through the circle at maximum two points.

p + q = 3, and p-q = 6

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

These are a pair of simultaneous equations. To solve, we can add them to get
2p = 9. p = 4.5
Substituting in the first equation 4.5 + q = 3; q = -1.5
So 0 is greater than q

x is 3 less than y

x + 5

y + 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

x is 3 less than y can be written x = y - 3
If we add 5 to both sides we have x + 5 = y + 2

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