173

4.1 INTRODUCTION TO RATIOS

Assess your readiness to complete this activity. Rate how well you understand: Not ready

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Bringit 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

Amber has started a new fi tness 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?

_____________________

Correctly setting up a ratio for a specifi c context

• correct numerator and denominator

• fully reduced, if requested

• units labeled for rates

625

174

Chapter 4 — Ratios and Proportions

Steps in the Methodology Example 1 Example 2

Step 1

Write the ratio in words.

Identify the comparison requested (the ratio) in words and present it in fraction form.

Substitute the fraction bar for the comparison word or phrase, which always precedes the denominator.

T V advertisingradio advertising

heightwidth

Step 2

Insert the quantities.

Insert the corresponding quantities and their units into the ratio.

Note: It may be necessary to calculate a required quantity from the information given (see Model 1).

$40,000$12,000

614

feet feet

Step 3

Reduce the ratio.

Drop the unit labels if they are the same and reduce the ratio to lowest terms.

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).

Both are in dollars; drop the unit labels ($)

40 000 100012 000 1000

401240 412 4103

,,

÷÷

=

=÷÷

=

614

37

÷÷

=22

Step 4

Present the answer.

Present your 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.

For every $10 spent on TV ads, $3 is spent on radio ads.

103

37

Step 5

Validate the reduction.

Validate the reduction by applying the Equality Test for fractions (cross-multiply).

?40 00012 000

103

,,

=

40 000 3 10 12 000120 000 120 000, ,

, ,× = ×

=

?

614

37

6 7 3 1442 42

=

× = ×

=

?

?

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)?

When you want to draw out a specifi c comparison of quantities (a specifi c ratio) from the words that describe a particular context, use the following methodology.

Try It!

175

Activity 4.1 — Introduction to Ratios

Model 1

Penny’s kennel has 55 golden retriever puppies. Thirty-fi ve are females. What is the ratio of female puppies to male puppies? Reduce the ratio to its simplest fraction form.

Step 1

Step 2 The number of females is given; the number of males is not. However, the total – the females = the males 55 – 35 = 20

Step 3

Step 4 The ratio of female puppies to male puppies is 7 to 4.

Step 5 Validate:

female puppiesmale puppies

3520

puppies puppies

35 5

20 574

puppies

puppies

÷

÷=

Answer : 74

3520

74

=? 35 4 7 20

140 140× = ×

=

? 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 fi eld trip to the art museum. What was the ratio of chaperones to children (in simplest fraction form)?

Step 1

Step 2

Step 3 This is a rate. Chaperones and children are different units. Retain the unit labels.

Step 4 “three chaperones for every 13 children”

Step 5 Validate:

chaperoneschildren

1565 chaperones

children

15 565 5

313

chaperones children

chaperones children

÷÷

=

Answer : 3 chaperones13 children

1565

313

=?15 13 3 65

195 195× = ×

=

?

176

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

Step 3 Retain the clearly distinct labels:

Step 4 “The rate of travel is 68 miles per one hour, or 68 mph.”

Notice that this rate reduced to a unit rate.

Step 5 Validate:

2724

miles hours

2724

681

miles 4 hours 4

miles hour

÷÷

=

Answer : 68 miles1 hour

2724

681

=? 272 1 68 4

272 272× = ×

=

?

Make Your Own Model

Problem: _________________________________________________________________________

Step 1

Step 2

Step 3

Step 4

Step 5

Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

Answers will vary.

177

Activity 4.1 — Introduction to Ratios

1. What are three ways you can write a ratio?

2. What words in the English language identify that your comparison of two numbers is a ratio?

3. What characteristics of a ratio defi ne it as a rate?

4. How do you determine the numerator of a ratio? How do you determine the denominator of a ratio?

5. What makes a rate 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?

7. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

a to b a : b ab

The word “to” or “per,” or the phrase “for every” are used to indicate a ratio.

A rate compares two quantities whose units are different. The units must be stated for the numerator and the

The fi rst 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.

Words showing that you are to identify “each, one, unit, single, or per…” makes your answer a unit rate.

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.

Answers will vary.

178

Chapter 4 — Ratios and Proportions

a) ratio of shaded boxes to unshaded boxes: shadedunshaded

= =6

1035

b) ratio of total boxes to shaded boxes: total

shaded= =

166

83

c) ratio of unshaded boxes to total boxes: unshadedtotal

= =1016

58

d) ratio of shaded boxes to total boxes: shadedtotal

= =6

1638

1. Write the ratios. Reduce them to lowest terms.

2. A man, 6-feet tall, casts a shadow 42 inches long. Write the ratio of his height to his shadow as a simplifi ed rate.

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.

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.

b) What is the ratio of orange juice cartons sold to the other juices sold? Reduce fully.

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?

total = 16, shaded = 6, unshaded = 10

his heighthis shadow

feet inches

foot inches

= =6

421

7

selling priceminutes minutes minutes

= =$ $116

200029

500

cartons of orange juicecartons of juice

cartons car

=250400 ttons

=58

orange juiceother juice

=−

= =250

400 250250150

53

malesfemales

males females

male females

= =14

281

2

179

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, fl yers, etc.). In simplest form, what was the ratio of radio ads to print advertising?

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.

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?”

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.

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.

dollards raisedmiles walked miles miles

=×

= =$

.$760

80 0 5760

40$$19

1 mile

radio adsprint ads

= = =18 00033 000

1833

611

,,

radio adsprint ads

611

TV ads+18,000 radio ads

103,000

85 000, 136,000103,000

33,000

−

print ads

518

hits at bats

34

825

pounds potatoes guests

13

31

34

180

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 SolutionWhat is Wrong Here? Identify the Errors Correct Process

a) What was the ratio of footballs to soccer balls?

Wrote the ratio as a whole number.

A ratio must be stated as a comparison of two numbers.

14 footballs

7 soccer balls

14 7

7 7= 2

The ratio of footballs

t

÷

÷

oo soccer balls is 2

1

b) What was the ratio of softballs to jump ropes?

Should identify (with labels) the diff erent types of equipment.

5 softballs2 jump ropes

c) What was the ratio of basketballs to all the balls?

A jump rope is not a ball. 14 footballs

14 basketballs

20 softballs

+7 soccer balls

55 ballls

1455