4.1 INTRODUCTION TO RATIOS

Transcription

1 .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 80. 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 17

2 Chapter 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 $0,000 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 1 feet wide by 6 feet high. What is the ratio of its height to its width (in simplest or reduced fraction form)? Try It! 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 advertising radio advertising height width 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). $0,000 $12,000 6 feet 1 feet Step Reduce the ratio. Drop the unit labels if they are the same and reduce the ratio to lowest terms. If the unit labels are clearly distinct, retain them (because the ratio is a rate) and reduce to lowest terms (see Models 2 and ). Both are in dollars; drop the unit labels ($) 0, , Step 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. 10 For every $10 spent on TV ads, $ is spent on radio ads. 7 Validate the reduction. Validate the reduction by applying the Equality Test for fractions (cross-multiply). 0, 000? 10 12, 000? 0, , , , 000 6? ?

3 Activity.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 5 puppies 20 puppies The number of females is given; the number of males is not. However, the total the females the males Step 5 puppies 5 20 puppies 5 7 Step The ratio of female puppies to male puppies is 7 to. Answer : 7 Validate: 5 20?? 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 Step chaperones children 15 chaperones 65 children This is a rate. Chaperones and children are different units. Retain the unit labels. 15 chaperones 5 65 children 5 1 chaperones children Step Answer : chaperones 1 children three chaperones for every 1 children Validate: 15 65??

4 Chapter Ratios and Proportions Model It takes hours to travel 272 miles. What is the rate of travel in miles per hours? Reduce to lowest terms. Steps 1 & 2 Step 272 miles hours Retain the clearly distinct labels: 272 miles hours 68 miles 1 hour Step Answer : 68 miles 1 hour The rate of travel is 68 miles per one hour, or 68 mph. Notice that this rate reduced to a unit rate. Validate: 272?? Make Your Own Model 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. Problem: Step 1 Step 2 Step Step 176

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

6 Chapter Ratios and Proportions 1. Write the ratios. Reduce them to lowest terms. a) ratio of shaded boxes to unshaded boxes: b) ratio of total boxes to shaded boxes: shaded unshaded total shaded total 16, shaded 6, unshaded 10 c) ratio of unshaded boxes to total boxes: d) ratio of shaded boxes to total boxes: unshaded total shaded total A man, 6-feet tall, casts a shadow 2 inches long. Write the ratio of his height to his shadow as a simplified rate. his height his shadow 6 feet 1 foot 2 inches 7 inches 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. Tamika places food orders for a market. She noticed that in one month the market sold 250 cartons of orange juice out of the 00 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. cartons of orange juice cartons 250 cartons of juice 00 cartons 5 8 b) What is the ratio of orange juice cartons sold to the other juices sold? Reduce fully. orange juice other juice The Humane Society is looking for new homes for 2 kittens. Twenty-eight of them are females. In simplest form, what is the ratio of male kittens to female kittens? males females 1 males 1 male 28 females 2 females 178

7 Activity.1 Introduction to Ratios 6. A retailer with a $16,000 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 print ads 18, , 11 radio ads print ads , 000 TV ads 16, ,000 radio ads 10,000 10,000,000 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 miles 0 miles 1 mile 1. In a baseball season, a major league player got 125 hits in his 50 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)?. 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. 8 pounds potatoes 25 guests. In a class of 60 students, 5 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

8 Chapter 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 1 footballs, 1 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 1 footballs 7 soccer balls The ratio of footballs to soccer balls is 2 1 b) What was the ratio of softballs to jump ropes? Should identify (with labels) the different types of equipment. 5 softballs 2 jump ropes c) What was the ratio of basketballs to all the balls? A jump rope is not a ball. 1 footballs 1 basketballs 20 softballs +7 soccer balls 55 balls