Mobile Math Teachers’ Circle
iClicker Problems - Proportions
June 17, 2016
1. Determine if the following statements do or do not use the term proportional correctly.
Choose one of the following:
A. Correct
B. Incorrect
C. Not sure
(a) The amount of sales tax is proportional to the cost of a purchase.
(b) A person’s weight is proportional to the person’s height.
(c) The amount of weight a person gains is proportional to the increase in calories eaten.
(d) The value of a bar of gold is proportional to its weight.
(e) The distance around a circle is proportional to the distance across the circle.
2. For the given x and y variables, decide whether y is or is not proportional to x. Choose
one of the following:
A. Proportional
B. Not proportional
C. Not sure
(a) y is the cost of x gallons of gasoline.
(b) y is the perimeter of a square whose sides have length x
(c) y is the area of a square whose sides have length x
3. Consider the following students’ methods of solving the proportion
8
20
=
5
x
Richie:
First I divided 20 by 8 and got 2 21 . Then I multiplied 2 12 by 5 and got 12 21 for my answer.
Martha:
I think of it this way. Every time a person takes 8 steps, another person takes 5 steps.
So 16 steps corresponds to 10 steps, and 24 steps corresponds to 15 steps. As 20 is halfway
between 16 and 24, I reasoned that the answer was halfway between 10 and 15. I got 12 12
for the answer.
Which method is valid?
A. Richie
B. Martha
C. Neither
D. Both
4. A tree casts a shadow 45 feet long, compard to a yard stick that casts a shadow 2 feet
long. Which of these equations can be used to determine the height h of the tree?
h
45
=
2
3
45
2
B.
=
h
3
h
2
C.
=
25
3
3
45
D. =
h
2
A.
5. A common method to solving ratio and proportion problems is placing three of the
a
c
given values in the equation = and applying cross-multiplication to solve for the fourth.
b
d
However, students sometimes apply this method in wrong situations. Which of the following
can be solved using this simple application?
For each problem, select: A. Applies
B. Does not apply
C. I’m not sure
(a): Your dog was 4 years old when your cat was 2 years old. When your dog is 9, how old
will your cat be?
(b): A 2-liter bottle of soda contains 100 grams of sugar. How many grams of sugar are in
10 liters?
(c): For every 2 books you read, your friend reads 3 magazines. How many books will you
have read if your friend reads 12 magazines?
(d): Adam and Meg are running around a track. They run at the same speed, but Adam
started late. When Adam has run 8 laps, Meg has run 16 laps. When Ashley has run 24
laps, how many has Megan run?
(e): Sally put 4 white t-shirts out and Bob put 6 white t-shirts out to dry on a clothsline.
Suppose the t-shirts are identical. If it took Sally’s t-shirts 3 hours to dry, how long will it
take Bob’s to dry?
6. Mr. Ed noticed that students often try to solve ratio problems by adding the corresponding
terms:
a : b + c : d = (a + c) : (b + d).
He knows that ratios can be thought of as fractions and it is certainly not appropriate to add
numerators and denominators of fractions.
For which of the following problems is adding the numerators and denominators actually justified?
For each problem, select: A. Applies
B. Does not apply
C. I’m not sure
(a) In Mr. Ed’s class the ratio of boys to girls is 13 : 15. In Ms. Edsel’s class the ratio
of boys to girls is 3 : 4. What is the ratio of boys to girls if the classes are combined?
(b) Jane’s marble collection consists of 13 red marbles and 17 blue marbles. Joe has 23 red
marbles and 15 blue marbles. What is the ratio of red to blue marbles if the two combine their
collection?
(c) At the Dauphin Island Sea Lab two crews were catching and releasing crabs in order
to determine what part of the population was infected with a certain virus. Dr. Robert’s crew
found that 12 of their crabs were infected and 25 were not infected. Doc Martin’s crew reported
that they caught 5 crabs without virus for every 2 infected crabs. What is the ratio of infected
to not infected if the crews’ findings are combined?
(d) Paint A and Paint B come in same size cans. Paint A is made up from red and white
paint in the ratio 1 : 3 and Paint B is made up from red and white paint in the ratio 1 : 7.
What will be the ratio of red to white if one can of A is mixed with one can of B?
7.
Dr. Spock asked his class to solve a percent problem, Julia set up the proportion:
4/5 = x/100.
She then cross-multiplied to solve for x. Which mathematical reason best explains why crossmultiplication works? (Select ONE answer.)
A. In a proportion, the product of the means equals the product of the extremes.
B. You are actually multiplying both sides of the equation by 5 and by 100 and then simplifying.
C. Cross-multiplication is the rule for solving proportions. You multiply the numerator of
one times the denominator of the other, and set them equal.
D. You are actually multiplying 4/5 by the identity for multiplication, 20/20, to get x = 80.
E. 100 divided by 5 equals 20 and 4 times 20 is 80.
F. You do it because Dr. Spock told you so.
8. Dr. Seuss asked his class to find 16% of 25. The Grinch decided to calculate 25% of 16
instead. To Dr. Seuss’ surprise he got 4 as the answer. What can we say about the Grinch’s
method? (Select ONE answer.)
A. The Grinch’s method is not correct and gave the wrong answer to this problem.
B. The Grinch’s method never gives the correct answer.
C. The Grinch was just lucky. The method works because 25% is just a quarter, a special
case. The method would not work in general.
D. The Grinch’s method gives the correct answer and could be used for any numbers.
E. The Grinch stole Christmas.
9. Students in Mr. Bieber’s class were discussing the average speed of two moving objects.
The graph below shows the position of a train and the Batmobile, over ten seconds. The tic
marks on the horizontal axis are located at 1 second intervals, and the tic marks on the vertical
axis are at 20 meter intervals.
distance
in meters
80
60
Train
40
20
Batmobile
1
2
3
4
time in seconds
Which of the following statements is correct? (Select only ONE.)
A. The average speeds of the train and the Batmobile are the same over this interval.
B. The average speed of the train is greater than the average speed of the Batmobile because the graph for the train is always above that for the Batmobile.
C. There isn’t enough information to determine the average speed of either vehicle.
D. You can determine the average speed of the train because it is a straight line, but cannot determine the average speed of the Batmobile from this graph.
E. The Batmobile is always faster.
10. Al is 5 feet tall and has a shadow that is 18 inches long. At the same time, a tree has a
shadow that is 15 feet long. Al sets up and solves the proportion as follows:
5 ft
18 in
=
.
15 ft
x in
Al claims the solution is, x = 54 in.
Which of the following statements is correct? (Select only ONE.)
A. The units work out correctly, that indicates that Al’s solution is correct.
B. Al’s solution is wrong because he did not convert all units to feet.
C. Al set up the ratios correctly but he did not properly solve for x.
D. Shadow problems are tricky and cannot be solved with ratios.
E. Al’s answer is incorrect. A more appropriate ratio would have been
18 in
5 ft
=
.
x ft
15 ft