Section 7: Ration As Measure
Gia Elise Barboza
Gia Barboza
MATH 201
Department of Mathematics, Michigan State University
April 12, 2005
1
Ratio As Measure
We have talked in the past about how we measure certain quantities. A quantity is any attribute of an object
that can be measured.
Consider the following problem:
An new housing subdivision offers rectangular lots of three different sizes:
a. 75 feet by 114 feet
b. 455 feet by 508 feet
c. 185 feet by 245 feet
If you were to view these lots from above, which would appear most square? Which would appear least square?
Explain your answers.
The closer the ratio of the sides is to 1, the more square the object is. Therefore, looking at the ratio we can
obtain the answer:
75
= .66
114
455
= .90
508
185
= .76
245
(1)
(2)
(3)
This means that the lot that measures 455 feet by 508 feet would appear to be the most square.
What attribute or characteristic of the lots are we interested in? In what ways can this attribute by quantified?
We are interested in the squareness of the lots. We can quantify this by looking at the ratio of sides.
1.1
Activity: Downhill
In Japan, indoor skiing has become a big thing. Ski slopes are built in very large indoor arenas and covered
with plastic fiber that simulates packed snow. Suppose you have measurements on each of the three slopes
that tell you the length, the width of the base, and the height. How could you decide which is the most steep
or the least steep?
1
1.2
1.2
Exercises for Section 7
1
RATIO AS MEASURE
Exercises for Section 7
1. You are running a contest to see if people can tell which of three different sized batches of coffee (from the
same type of coffee bean) is strongest. As the manager of the contest, how would you measure the strength of
each batch?
I would look at the ratio of coffee to the amount of water
2. A conference on population growth has the following data for a recent year. (page 46). Devise a way of
quantifying population growth to enable comparisons among different countries.
Country
1=China
2=Mexico
3=Philippines
4=United Kingdom
5=United States
population
214
248
185
75
391
100
150
deaths
83
46
48
63
234
out-migration
24
45
23
34
45
200
in-migration
12
23
10
22
67
25
30
35
40
45
300 350 400
50
births
83
46
48
63
234
100 150 200 250
births
|
|
|
|
|
||
|
|
|
60
50
100
150
200
deaths
||
|
45
||
10
20
30
40
50
im
40
om
C hina
35
Mex ico
Philippines
30
U nited Kingdom
25
U nited States
||
100
150
200
250
300
350
400
10
20
30
40
50
|
60
Figure 1: Scatter Plots of Birth, Death, In-migration, Out-migration
2
|
2
COMPARING RATIOS
Population size at t = 2
500
0.05
1000
0.10
1500
p2
natpoprate
2000
0.15
2500
0.20
Natural Growth Population Rate
1
2
3
4
5
1
Country
B-D/Pop
2
3
4
5
Index
p2=population+(births-deaths)+(im-om)
Figure 2: Natural Population Growth vs. Projected Population Growth
3. Discuss how each of the following attributes might be quantified:
c. the steepness of a line on a coordinate plane
→ The ratio of the change in the y-coordinate to the change in the x-coordinate
e. population density
→ The ratio of the number of people to the number of square miles they live in
f. the likelihood of drawing a red ball from a bag containing a mixture of red and blue balls
→ The ratio of the number of red balls to the total number of balls
g. quality of performance of an undergraduate in all her course work
→ the ratio of difficulty of major or course load to the number of credits taken
2
Comparing Ratios
2.1
Orange Juicy
The orange drink in pitcher A is made by mixing one can of orange concentrate with 3 cans of water. The
mixture in pitcher B is made by mixing 2 cans of orange concentrate with 6 cans of water. Which will taste
more “orangey,” the mixture in pitcher A or the mixture in pitcher B?
1. Give an argument to support your answer to the above problem. How do you know?
3
2.2
Activity: Pepperoni to Go!
2
COMPARING RATIOS
→ Even though the pitcher in B has twice as much orange mixture, it also has twice as much water.
Therefore, when we compare the two ratios we see they are proportional. 2 cans of orange concentrate
with 4 cans of water is proportional to 1 can of orange concentrate with 2 cans of wanter.
2. Some students reasoned that the mixture in pitcher A is more orangey since less water went into making
it while others believed that the mixture in B is more orangey because it has more orange concentrate.
How would you settle the argument? What is wrong with the reasoning of the students in each group?
→ In this situation, the students on either side of the argument are focusing on only one of the two
quantities but one has to focus on both quantities in order to appropriately assess the relative strength
of the mixtures.
3. There was another student who argued as follows: ... How would you deal with this student’s thinking?
→ This student is reasoning additively not multiplicatively. Again, we must consider both quantities
and he considers only the water differential between the two pitchers. A ratio is the appropriate
measure since both pitchers have three times the amount of water.
Think of another situation in which a ratio would be needed in order to make a comparison.
Allan Anderson made 11 free throws out of 19 attemps while Kelvin Torbert made 17 out of 35. Who has been
more effective in making free throws? Kelvin made 6 more than Al but Al has been more effective. We must
17
consider the number of free throws relative to the number of attempts. Here, 11
19 = .58 while 35 = .49
2.2
Activity: Pepperoni to Go!
Luigi’s is a pizza parlor that caters to the local college crowd. The twenty-four members of the chess club
come in to celebrate. Eighteen pizzas had been ordered in advance. None of the tables will hold 24 persons,
so they sit at two tables, each with 9 pizzas and 12 members.
1b. Here is a seating arrangement where pizzas and members were moved from one table to three different
tables in such a way that all got a fair share of pizza:
In how many ways could 18 pizzas and 24 members have been arranged at each table so they all got a fair
share of pizza? (three ways). How much pizza does each person get? ( 34 of a pizza). Is there more than
one way to distribute pizzas among members? (Of course).
2. Consider some new situations. Keep in mind that pizzas are shared fairly in each situation.
10p
8p
a. Would the distribution 14m
and 10m
be fair for each member?
It is possible to compare the two ratios through some rote procedure like dividing the numerator
by the denominator. However, more insight is provided by explaining these situation as one would
5p
8p
10p
expect from a child. As above, we could show that 14m
is equivalent to 7m
and 10m
is equivalent to
4p
.
Then
we
could
draw
out
how
much
each
member
would
get...
5m
4
2
COMPARING RATIOS
2.3
Exercises for section 7.1: 2ab,5,6,7,9,10,12,15
b. Who would get more pizza, a person sitting at a table of
7p
3p
since 9m
= .78 while 5m
= .6
3p
5m
or of
7p
9m .
The person sitting at the
7p
9m
c. One person’s plate has 18 + 16 + 16 + 16 pizza. How were the pizzas served and distributed? How many
5p
people were at the table? 81 + 16 + 61 + 16 = 8m
d. Another person has 1 23 pizzas. At what table might she be sitting?
might be sitting at a table that has 5 pizzas.
e. Can someone who is served
table gets 25 pizza and 25 <
1
2
1
2
5
3
+
5
3
=
10
3
+
5
3
=
15
3
= 5. She
2p
2p
+ 31 pizzas have sat at the table 5m
? No, since a person sitting at a 5m
1
1
1
1
< 2 + 3 , so a person sitting at such a table must get less than 2 + 3 .
f. If someone is given a serving of
1p 2p 4p
following: 2m
, 4m , 8m , etc.
1
2
at which tables might he be sitting? He can get any one of the
4p
. At which table would she get only half as much? At a
g. Someone is sitting at a table designated 5m
2p
4p
table with either half the pizza or twice the people, so either 5m
or 10m
.
2.3
Exercises for section 7.1: 2ab,5,6,7,9,10,12,15
2a. Three scoops of coffee are used with 4 cups of water in one coffee machine; 4 scoops of coffee are used
with 6 cups of water in another machine. Which brew will be stronger? How do you know?
→ In the first coffee machine, the ratio of coffee to water is 3:4. In the second it is 2:3. If we put both
ratios in terms of a common denominator, we get 8:12 vs. 9:12. This means there is less coffee in the
first machine, so it must be weaker.
2b. How about 4 scoops for 8 cups of water and 7 scoops for 15 cups of water?
→ 4:8 is stronger, compare ratio’s. The first has 50% coffee and the second has less than 50%.
5. A manufacturer recommends 10 tablespoons of cocoa be mixed with 4 cups of milk to make hot cocoa. A
school cafeteria is fixing hot cocoa for 50 first-graders. How many tablespoons of cocoa should they use if
they are to mix enough to give every first grader a cup? Given that 2 tablespoons of cocoa are equivalent
to 81 cup of cocoa, how many cups of cocoa should be used?
→ We start off with a ratio of 10 tablespoons for every 4 cups of hot cocoa made. Note, the problem
does not state that we are to assume each child gets 1 cup. If we reduce this, we find that we need
2.5 tablespoons for every one cup. There are 50 first-graders, so we multiply by 50 to get a ratio of
125:50, so to accommodate 50 children we need 125 tablespoons of cocoa.
→ Given that 2 tablespoons of cocoa are equivalent to
cocoa.
1
8
cup, we need (125 ÷ 2) ×
1
8
= 7.8125 cups of
7. Car A can travel a greater distance in 3 hours than car B can travel in 2 hours. If possible, find which car
will travel a greater distance: car A in 5 hours or car B in 6 hours.
→ Suppose car A travels 12 miles in 3 hours and B travels 10 miles in 2 hours. The ratios for A and B
are 4:1 and 5:1 respectively. This means that in 5 hours, car A travels 20 miles and in 6 hours car B
travels 30 miles. Now suppose that Car A travels 30 miles in 3 hours and B stays the same. In this
situation, A travels more in 5 ours. Therefore, it is not possible to arrive at a conclusion (thanks
Amber!).
10. Two workers working 9 hours made 243 parts. Worker A makes 13 parts in 1 hour. If the workers work
at a steady rate throughout the day, who is more productive, Worker A or Worker B?
→ For worker A, the ratio of parts to hours is 13:1. This means that in 9 hours, the ratio of parts to
hours is 13 × 9 : 9 × 1 = 117 : 9. We know their total ratio of parts to hours, which is 243:9. From
this, we can deduce the ratio of parts to hours for Worker B, which is 243 − 117 : 9 = 126 : 9. From
this, we see that worker B is more productive.
15. In Boogleville,
2
3
of the men are married to
3
4
of the women. What is the ratio of men to women?
5
2.3
Exercises for section 7.1: 2ab,5,6,7,9,10,12,15
2
→ Try a simpler question first, to get the hang of this. In Boogleville,
the women. What is the ratio of men to women?
2
5
COMPARING RATIOS
of the men are married to
1
3
of
→ For every married male, there must be 2 unmarried females. How can we tell? Quite simply, draw
the table,
2
5
of the men
M
M
M
M
M
1
3
of the women
W
W
W
If this were the case, there would be an extra woman left. Try another
2
of the men
6 of the women
M
W
M
W
M
W
M
W
M
W
W
This means that for every married male there are two unmarried females. Why
2
5
→ Because there are 2 married males for every two married females. This also means that for every 2
married males, there are four unmarried females, or 2:4. This is the same ratio as 1:2. What is the
ratio of unmarried men to unmarried women? (Ans. 3:4).
Back to the original question
2
3
of the men
M
M
M
3
4
of the women
W
W
W
W
If this were the case, there would be an extra woman left. Try another
6
4
of
the
men
6
8 of the women
M
W
M
W
M
W
M
W
M
W
M
W
W
W
If this were the case, there would be extra women for the married men, we cannot have that!
6
6
9 of the men
8 of the women
M
W
M
W
M
W
M
W
M
W
M
W
M
W
M
W
M
6
2
COMPARING RATIOS
2.4
2.4
Section 7.2: Percents in Comparisons
Section 7.2: Percents in Comparisons
Read on your own, if you can do 7.1, this section is not a problem.
2.5
Section 7.3: Practice Multiplicative Reasoning
Do problems 1-8. Anything on the exam will be included in the review on Tues. during class.
7