A bag contains only red, white and blue balls. One third of the balls are red, one fifth of the balls are white. One ball is to be selected at random.

Probability of drawing a white ball

Probability of drawing a blue ball

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 the balls are either red, white or blue, if 1/3 are red and 1/5 are white the rest (1 - 1/3 - 1/5) are blue. But no need to work out the fraction.
If we pick a number that fits in with this distribution it will be easier to see the probability.
So if there are 15 balls in all, 1/3 = 5 will be red, 1/5 = 3 will be white and the rest (7) will be blue.
The probability of drawing white = 3/15
The probability of drawing blue = 7/15 (which is greater)

The cost of hiring props for the school play was to be shared by 8 students. 4 more students decide to share the cost. The new cost per person is $2 less.

Total cost of hiring props

$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

Let the total cost be x.
The cost per student originally = x/8
The cost per students after 4 more students join in = x/12
The difference is 2, so x/8 - x/12 = 2. Add the fractions by putting on acommon denominator (24)
(3x - 2x)/ 24 = 2; x/24 = 2; x = 48

Total savings on 40 gallons of fuel bought at $1.152 per gallon instead of $1.245

$3.70

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

40 gallons at 1.152 = 46.08
40 gallons at 1.245 = 49.8
The savings will be 49.8 - 46.08 = 3.72
(It is slightly quicker to take the difference per gallon [1.245 - 1.152] and multiply by 40)

O is the centre of the circle, and angle POQ is a right angle

MN/PQ

2/1

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

Solution

First notice that MO, ON, OP and QO are all radii. If we assume that the radius =1, then MN = 2
Now we need to find PQ. PQ is the hypotenuse of a right triangle (POQ) with the other sides both = 1. This is a special triangles with side in the ratio x : x : x√2
Since PQ is √2, the ratio MN/PQ = 2/√2 which is less than 2/1

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Points x and y lie on line l

The number of points on line l that are twice as far from x as from y

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

It is easy to see that a point to the left of x could be placed so that it is twice as far tfrom x as from y. But do not forget that there will be a point between x and y, lying one third the way along the segment from x to y, that will also fit the definition. So there are two points.

The line containing point A is rotated 90 degrees anticlockwise about the origin, O.

The y coordinate of point A before rotation.

The y coordinate of point A after rotation.

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

Rotating anticlockwise through 90 degrees is the equivalent of making a mirror image (left-right reversed). The x-coordinates will switch from positive to negative. But the y-coordinate will remain unchanged.

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AC > CB > 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

You need to know that the longest side of a triangle is opposite to the largest angle.
Hence, angle b > a > c

The average of four numbers is 30. The average of three of these numbers is 20.

The value of the fourth number

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 average of four numbers is 30, their total must be 4 x 30 = 120
The average of three of the numbers is 20, and so their total is 3 x 20 = 60
So the fourth number must be 120 - 60 =60

1 < x < 0

x³

x⁵

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 x is a negative fraction, its square will be positive, its cube will again be negative (but closer to zero), the fifth power will also be negative, but even closer to zero, and hence will be further to the right on the number line (i.e. greater).

ab = 0

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 ab = 0, either a or b must be 0. If b = 0, then a could be any number, negative or positive (or even 0). So this is a D answer.

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