Ratios, Proportions and Variation
1. Which of the following ratios represent the relationship of hours to days?
I.
II. 1:24
III. 1 to 24
IV.
V. 24:1
(A) I, III, and V
(B) II, III, and IV
(C) II, III, and V
(D) I and V
(E) II and IV
2. There are 14 boys and 16 girls in Tyler’s class. Which ratio best represents the relationship
between the number of boys and the number of students in Tyler’s class?
(A)
(B) 8:7
(C)
(D) 14:16
(E) 8:15
3. The ratio of
(A)
(B)
(C)
(D)
(E)
–
to
is
. If
– , what is
in terms of x?
4. If the ratio of A to B is 3:4 and the ratio of B to C is 2:3, what is the value of A when C is 5?
(A) 10
(B)
(C)
(D) 5
(E) 6
5. If y varies directly with x and
when
, what is the value of y when
?
(A) 4
(B) 5
(C) 12
(D) 15
(E) 20
6. If x varies inversely with y and
when
, what is the value of x when
?
(A)
(B)
(C)
(D) 2
(E)
7. The variable s varies jointly with t and u such that
when
and
. What is the
constant of variation describing the relationship between s on the one hand and t and u on the
other?
(A) 84
(B)
(C)
(D)
(E)
8. The value of w varies directly with y and inversely with z. The constant of variation is
the value of z when
and
?
. What is
(A)
(B)
(C)
(D)
(E)
9. A 12-ton mixture consists of one-sixth sand, two-sixths gravel, and one-half cement. If x tons of
cement are added, the mixture will contain 60 percent cement. How many tons of cement must be
added?
(A) 1.2
(B) 3
(C) 3.2
(D) 4
(E) 5.2
Answers and Explanations
1. The correct answer is D. There are 24 hours in a single day. Since the question asks for the ratio
of hours to days, hours will need to come first in the relationship, no matter how it is presented.
That means 24 will need to be the numerator if the ration is presented in a fraction-style format
and to the left of the colon if presented in the x : y format. Only the ratios in I and V do this, all of
the other values show the ratio of day to hours.
2. The correct answer is C. Don’t get fooled by this one. The question isn’t asking for the ratio of
boys (14) to girls (16) which is given in both choices A and D. Nor is it asking for the ratio of
girls to boys shown in choice B. Rather, it asks for the ratio of boys to the total number of
students, meaning both boys and girls. Thus, you must do the intermediate step of adding the
number of boys and the number of girls (
). Now you have the ingredients you need
to come up with the ratio of boys (14) to the total number of students (30): which reduces to ,
choice C. Note they also give you the ratio of girls to the total number of students as one of the
answer choices (E). If you don’t read carefully, it can be all too easy to make a mistake.
3. The correct answer is E. To solve this one, set up the relationship as follows:
Then, since f(x) is defined as
, substitute that expression for f(x):
Next, cross-multiply and “solve” for g(x):
[
]
4. The correct answer is C. The best way to do this is to set up a table as follows:
A
3
B
4
2
??
C
3
5
In it we place the ratios we know (on separate lines) and use the third line for the relationship we
want to know (A : C when C is 5). You will notice that B is the common link in both ratios. So
we can pick up on the fact that in the first ratio, B is twice as large as it is in the second. Thus, to
add C in the first ratio, we would likewise need to make it twice as big (
. This tells us
that the relationship of A to B to C is 3 to 4 to 6. Our chart should now look something like this:
A
3
??
B
4
2
C
6
3
5
Now that we have a ratio containing both A and C we can forget about B. The ratio of A to C is
simply 3 to 6. We want the ratio of A to 5. We can set these up as a proportion to solve for the
missing value by cross-multiplying:
Thus, our chart, were we to complete it would look like this:
A
3
2.5
B
4
2
C
6
3
5
5. The correct answer is C. This is best set up as a proportion. It doesn’t matter if you put x or y on
top, just make sure you arrange both sides of your proportion the same way:
Then, cross-multiply and solve for y:
6. The correct answer is D. To solve this, we use the formula for inverse variation:
, where k
is the constant of variation. We determine k by multiplying the known values for the two
variables, in this case
when
. Thus, k is 24. Then we use this to figure out the value
of the variable we want to determine (x when y is 12):
7. The correct answer is E. With joint variation, we use the equation
, where k is again the
constant of variation. In this case, s stands in y and t and u are used instead of x and z. Plugging
these values into the equation, we can determine k:
8. The correct answer is A. Here we are dealing with combined (direct and inverse) variation so
we require the combined variation formula:
, where y varies directly with x and inversely
with z. In this case, we have w varying directly with y and inversely with z, so our formula would
actually look something like this:
Since we are told the value for k, we can plug what we know into the equation and solve for the
missing variable (z):
( )
9. The correct answer is B. If half of the mixture is cement originally, we know that 6 tons are
cement (
If we add x tons to the 6 tons we have originally, we’ll have a mixture that
is 60 percent cement. The total weight of the new mixture would be
. We can set up the
following equation and solve for x:
Thus, we must add 3 tons of cement to the mixture.