1) The fraction 1/8
is between the numbers listed in which of the following pairs?
a) 1/10 and 2/17
b) 0.7 and 0.9
c) 0.07 and 0.09
d) 1/9 and 2/15
Solution: The decimal form of 1/8 is found by dividing 1
by 8. It is 0.125. This would eliminate choices b) and c). Note that
choice b) and c) are constructed so as to snare the person who mistakenly
thinks 1/8 is 0.8 or 0.08. Now we must decide between choice a) and d).
The obvious way of picking the right choice would be to convert the
fractions into decimals so that they can be compared with 0.125. This,
however, would use up a lot of precious time. Fortunately, with a little
number sense it is possible to solve the problem another way. We discern
that 1/9 is less than 1/8 and 2/15 is greater than 2/16 which is equal to
1/8. Thus, 1/8 lies between 1/9 and 2/15. Therefore the right answer is
d).
2) A man dies and leaves his estate to his wife, three
children, and a certain charity. The will designates that the estate be
divide so that for every five parts of the estate his wife receives, three
parts goes to each child, and a single part goes to the charity. If at the
time of his death his estate is worth $60,000, how much money did his wife
receive?
a) $5,000
b) $12,000
c) $15,000
d) $20,000
Solution: It is necessary to find out what portion of
the estate the wife is to receive. This portion will be the ratio of the
parts received by the wife to the total number of parts dividing the
estate. The wife is designated to receive 5 parts of the estate. The total
number of parts dividing the estate will be the five parts designated to
the wife plus the 9 parts designated to the children and one part to the
charity. Thus the portion the wife is designated to receive is 5/15 or 1/3
of the estate. The estate is worth $60,000, and so the wife will receive
$20,000. The right answer is d).
3) Consider the table below:
Which of the following formulas expresses
the relationship between x and y given in the table.
a) y = 3x + 2
b) y = x + 3
c) y = 3x – 1
d) y = 4x - 2
Solution: The values for x and y given in the first row
of the table are 1 and 2 respectively. If we place these values in for x
and y in any of the formulas it will either make the formula true or
false. When placed in formula a), for example, we get 2 = 3(1) + 2 which
is false. This tells us that formula a) does not express the relationship
between x and y. Formula b) is also false, while formulas c) and d) are
true. The correct formula, however, will hold true for all the values of x
and y in the table. Placing in other values for x and y reveal that only
formula c) holds true for all the values of x and y. Thus formula c)
expresses the relationship between x and y.
4) Two places leave the same airport, one headed due
west at 200 miles per hour and the other due south at 150 miles per hour.
How far apart are the planes after four hours.
a) 250 miles
b) 1000 miles
c) 350 miles
d) 1,400 miles
Solution: Draw a picture to see what’s going on. From
the picture we see that the distance
we want is the hypotenuse of a right triangle. We
can find its length by
using the Pythagorean Theorem. Be careful! The lengths of the other two
sides of the triangle are not 200 and 150. The distances must be founding
by calculating the rate 
4 hours. The sides of the triangle will be:
and 
Using the Pythagorean Theorem the desired distance will
be:
5) Consider the figure below:
In the figure above, segment AB equals segment AC and
angle ABC equals 40 degrees. What is the measure of angle x?
a) 40 degrees
b) 100 degrees
c) 60 degrees
d) 90 degrees
Solution: First it is important to note that triangle
ABC is an isosceles triangle. One of the properties of isosceles triangles
is that the angles opposite the equal sides are also equal. This means
angle ABC and angle ACB have equal measure. Thus angle ACB measures 40
degrees. Now using the fact that the angles of any triangle add up to 180
degrees we can deduce that angle BAC equals 100 degrees. Finally, since
angle BAC and the angle whose measure we want to find are vertical angles,
they both have the same measure. Thus angle x is 100 degrees.
6) A micro-millimeter is defined as one millionth of a
millimeter. A length of 150 micro-millimeters may be represented by
a)
centimeters
b)
centimeters
c)
centimeters
d) 
centimeters
Solution: Looking over the possible answers it looks
like we want to convert micro-millimeters to centimeters and then write
the answer in scientific notation. To make the conversion we use the fact
that 1 micro-millimeter is one millionth of a millimeter. One millionth
can be written 0.000001. Thus
1 micro-mm = 0.000001 mm
Hence
150
1
micro-mm = 150
0.000001
mm
Thus 150 micro mm equals 0.000150 mm. Now to convert to
centimeters we need to divide the result by 10, which amounts to moving
the decimal place over one place to the left giving 0.0000150 cm. To place
this into scientific notation we must move the decimal point over five
places to the right, and the answer is 
.
7)
Suppose a survey is taken of 90 men between the ages of
20 and 70 selected randomly from the men in America who have a full-time
job with benefits. The following histogram displays the distribution of
the men grouped by their age. If a man between the age of 20 and 70 is
randomly selected from the men in America possessing a full-time job with
benefits, then use the above data to estimate the probability that the man
will be between the ages of 30 and 50.
a) 50
b) 0.56
c) 0.50
d) 0.40
Solution: When randomly selecting a person from the
population of men between the ages of 20 and 70 who have full time jobs,
the probability that the individual is between the age of 30 and 50 is the
ratio:
There is no way of finding the actual numbers for this
ratio as far as all men in America are concerned, but we can use the data
from our survey to get an estimate of these numbers. From the histogram,
we see that out of the 90 men who were surveyed, 50 of them were between
the ages of 30 and 50. Thus, an estimate of the desired probability would
be 50/90, which is 0.56 when rounded to two decimal places.
8) If the truth of A is sufficient to guarantee the
truth of B, then it must follow that:
a) The truth of A is necessary for the truth of B.
b) If A is false, then B is false.
c) If B is false, the A is false.
d) If B is true, then A is true.
B,
which means the truth of A implies the truth of B. This means the truth of
B must follow whenever A is true. Thus, if B was false, then it can only
mean that A was not true in the first place because the truth of A would
have implied that B was true, which is a contradiction. Hence, the right
answer is c).