4.1 INTRODUCTION TO RATIOS
Amber has started a new fitness program, one aspect of which is that she will consume no more than 2000
calories per day. She got up this morning, went for a walk, then had breakfast. The calorie total for her
breakfast was 480.
What is the fully reduced ratio
of calories she consumed at
breakfast, to her daily calorie
allowance?
6
25
_____________________
Assess your readiness to complete this activity. Rate how well you understand:
Not
ready
Almost
ready
Bring
it on!
• the terminology and notation associated with ratios
• how to translate from English language to the mathematical representation
of a ratio
• the characteristics of a rate which make it a special type of ratio
• the characteristic of a unit rate which makes it a special type of rate
Correctly setting up a ratio for a specific context
• correct numerator and denominator
• fully reduced, if requested
• units labeled for rates
173
Chapter 4 — Ratios and Proportions
When you want to draw out a specific comparison of quantities (a specific ratio) from the words that describe
a particular context, use the following methodology.
Example 1:
A car dealership spends $40,000 a year on TV ads and $12,000 on radio ads. What is the
ratio of its TV advertising to radio advertising (in simplest or reduced fraction form)?
Example 2:
The largest landscape painting on display at the museum
is 14 feet wide by 6 feet high. What is the ratio of its height
to its width (in simplest or reduced fraction form)?
Steps in the Methodology
Step 1
Write the
ratio in
words.
Step 2
Insert the
quantities.
Step 3
Reduce the
ratio.
Identify the comparison requested
(the ratio) in words and present it
in fraction form.
T V advertising
radio advertising
Insert the corresponding quantities
and their units into the ratio.
$40,000
$12,000
Note: It may be necessary to
calculate a required quantity from
the information given (see Model 1).
Drop the unit labels if they are
the same and reduce the ratio to
lowest terms.
Step 4
Present your answer.
Present the
answer.
Note that the instructions ask for the
reduced fraction form. If they did
not, we would represent the ratio as
“x to y” and “x:y”, as well as fraction
form.
Step 5
Validate the reduction by applying
the Equality Test for fractions
(cross-multiply).
Both are in dollars;
drop the unit labels ($)
40, 000 ÷ 1000
40
=
12, 000 ÷ 1000
12
40 ÷ 4
=
12 ÷ 4
10
=
3
10
3
height
width
6 feet
14 feet
6 ÷2
3
=
14 ÷ 2 7
3
7
For every $10 spent
on TV ads, $3 is spent
on radio ads.
40, 000 ? 10
=
12, 000
3
?
40, 000 × 3 = 10 × 12, 000
120, 000 = 120, 000
174
Example 2
Substitute the fraction bar for the
comparison word or phrase, which
always precedes the denominator.
If the unit labels are clearly
distinct, retain them (because
the ratio is a rate) and reduce to
lowest terms (see Models 2 and 3).
Validate the
reduction.
Example 1
Try It!
6
14
6 ×7
? 3
=
7
?
= 3 ×14
42 = 42
Activity 4.1 — Introduction to Ratios
Model 1
Penny’s kennel has 55 golden retriever puppies. Thirty-five are females. What is the ratio of female
puppies to male puppies? Reduce the ratio to its simplest fraction form.
Step 1
female puppies
male puppies
Step 2
The number of females is given; the number of males is not.
However, the total – the females = the males
55
–
35
=
20
35 puppies
20 puppies
Step 3
35 puppies ÷ 5
20 puppies ÷ 5
=
7
4
Step 4
The ratio of female puppies to male puppies is 7 to 4.
Step 5
Validate:
35 ? 7
=
20
4
?
35 × 4 = 7 × 20
140 = 140
Answer :
7
4
This only validates the reduction.
It does not validate that the ratio
was set up properly.
Model 2
Fifteen parent chaperones and 65 children went on a class field trip to the art museum. What was the
ratio of chaperones to children (in simplest fraction form)?
Step 1
Step 2
chaperones
children
15 chaperones
65 children
Step 3
This is a rate. Chaperones and children are different units.
Retain the unit labels.
15 chaperones ÷ 5 3 chaperones
=
65 children ÷ 5
13 children
Step 4
Step 5
Answer :
3 chaperones
13 children
Validate:
15 ? 3
=
65
13
“three chaperones for every 13 children”
?
15 × 13 = 3 × 65
195 = 195
175
Chapter 4 — Ratios and Proportions
Model 3
It takes 4 hours to travel 272 miles. What is the rate of travel in miles per hours? Reduce to lowest
terms.
Steps 1 & 2
272 miles
4 hours
Step 3
Retain the clearly distinct labels:
Step 4
Answer :
Step 5
Validate:
Make Your Own Model
68 miles
1 hour
272 miles ÷ 4
68 miles
=
4 hours ÷ 4
1 hour
“The rate of travel is 68 miles per one hour, or 68 mph.”
Notice that this rate reduced to a unit rate.
272 ? 68
=
4
1
?
272 × 1 = 68 × 4
272 = 272
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
Step 1
Step 2
Step 3
Step 4
Step 5
176
Activity 4.1 — Introduction to Ratios
1. What are three ways you can write a ratio?
a to b
a:b
a
b
2. What words in the English language identify that your comparison of two numbers is a ratio?
The word “to” or “per,” or the phrase “for every” are used to indicate a ratio.
3. What characteristics of a ratio define it as a rate?
A rate compares two quantities whose units are different. The units must be stated for the numerator and the
4. How do you determine the numerator of a ratio? How do you determine the denominator of a ratio?
The first number in a comparison statement is the numerator. The comparison number is on the bottom (the
denominator) and is the second number in a statement of the relationship.
5. What makes a rate a unit rate?
Words showing that you are to identify “each, one, unit, single, or per…” makes your answer a unit rate.
6. In what circumstances might the same ratio be interpreted as a rate by one person and not as a rate by
another?
These circumstances apply only when there is a common unit (between the numerator and denominator of the
ratio).
The interpretation has to do with the presentation of the units.
For example, “female puppies/male puppies” can be construed as a rate. However, if you remove the adjectives
describing the common unit, you now have “puppies/puppies” which is not a rate.
7. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
177
Chapter 4 — Ratios and Proportions
1. Write the ratios. Reduce them to lowest terms.
total = 16, shaded = 6,
unshaded = 10
a) ratio of shaded boxes to unshaded boxes:
shaded
6
3
=
=
unshaded 10
5
b) ratio of total boxes to shaded boxes:
total
16
8
=
=
shaded
6
3
c) ratio of unshaded boxes to total boxes:
unshaded 10
5
=
=
total
16
8
d) ratio of shaded boxes to total boxes:
shaded
6
3
=
=
total
16
8
2. A man, 6-feet tall, casts a shadow 42 inches long. Write the ratio of his height to his shadow as a simplified
rate.
his height
6 feet
1 foot
=
=
his shadow
42 inches
7 inches
3. A long distance provider sells pre-paid phone cards with 2000 minutes of calling time for $116. What is
the ratio of the selling price to the minutes purchased? Reduce fully.
selling price
$116
$29
=
=
minutes
2000 minutes
500 minutes
4. Tamika places food orders for a market. She noticed that in one month the market sold 250 cartons of
orange juice out of the 400 total cartons of juice that were sold.
a) Compare, as a ratio in its simplest form, the cartons of orange juice sold to the total cartons of juice
sold.
5
cartons of orange juice
250 cartons
=
=
8
cartons of juice
400 cartons
b) What is the ratio of orange juice cartons sold to the other juices sold? Reduce fully.
orange juice
250
25 0
5
=
=
=
other juice
400 − 250 15 0
3
5. The Humane Society is looking for new homes for 42 kittens. Twenty-eight of them are females. In
simplest form, what is the ratio of male kittens to female kittens?
males
14 males
1 male
=
=
females
28 females
2 females
178
Activity 4.1 — Introduction to Ratios
6. A retailer with a $136,000 advertising budget spent $85,000 last year on TV ads and $18,000 on radio ads.
The remainder of the budget was spent on print advertising (newspapers, flyers, etc.). In simplest form,
what was the ratio of radio ads to print advertising?
radio ads 18, 000
18
6 radio ads
=
=
=
print ads
33, 000
33 11 print ads
85, 000 TV ads
+18,000 radio ads
103,000
6
11
136,000
−103,000
33,000 print ads
7. Mary and her teammates walked a total of 80 laps on a 0.5 mile track for a walk-a-thon fundraiser. They
raised $760 in pledges. Write the ratio of dollars raised to miles walked. Reduce fully.
dollards raised
$760
$760
$19
=
=
=
miles walked
80 × 0.5 miles
40 miles 1 mile
1. In a baseball season, a major league player got 125 hits in his 450 “at bats.” What was
his ratio of hits to “at bats?”
5 hits
18 at bats
2. In a neighborhood elementary school, 120 students walk to school and 160 are driven to school by
car or bus. What is the ratio of walkers to non-walkers (in simplest fraction form)?
3. As a general guideline, a caterer prepares 16 pounds of potatoes for every 50
dinner guests. Write in reduced fraction form the ratio of pounds of potatoes
used to the number of guests.
4. In a class of 60 students, 45 of the students are women.
a) What is the ratio of men students to women students? Simplify the ratio.
8 pounds potatoes
25 guests
1
3
b) What is the ratio of women students to men students in the class? Simplify.
c) What is the ratio of women students to the entire class? Simplify.
3
4
3
1
3
4
179
Chapter 4 — Ratios and Proportions
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column.
1. In the P.E. equipment locker at the end of the school year, there were 14 footballs, 14 basketballs,
20 softballs, 7 soccer balls, and 8 jump ropes.
Worked Solution
What is Wrong Here?
a) What was the ratio of
footballs to soccer balls?
Identify the Errors
Wrote the ratio as a whole
number.
A ratio must be stated as a
comparison of two numbers.
Correct Process
14 footballs
7 soccer balls
14 ÷ 7
=2
7÷7
The ratio of footballs
2
to soccer balls is
1
b) What was the ratio of
softballs to jump ropes?
Should identify (with
labels) the different types
of equipment.
c) What was the ratio of
basketballs to all the balls?
A jump rope is not a ball.
5 softballs
2 jump ropes
14 footballs
14 basketballs
20 softballs
+7 soccer balls
55 ballls
14
55
180