m81303 Ratios and Rate By the end of this section, you will be able to: Write a ratio as a fraction Write a rate as a fraction Find unit rates Find unit price Translate phrases to expressions with fractions c5568dd1-af4e-473d-a0ff-4b7e61e2db69 Before you get started, take this readiness quiz. Simplify: 1624. If you missed this problem, review . 23 Divide: 2.76÷11.5. If you missed this problem, review . 0.24 Simplify: 112234. If you missed this problem, review . 611

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare a and b, the ratio is written as atob,ab,ora:b. A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of a to b is written atob,ab,ora:b. In this section, we will use the fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as 41 instead of simplifying it to 4 so that we can see the two parts of the ratio. Write each ratio as a fraction: ⓐ 15to27ⓑ 45to18. ⓐ 15 to 27 Write as a fraction with the first number in the numerator and the second in the denominator. 1527 Simplify the fraction. 59 ⓑ 45 to 18 Write as a fraction with the first number in the numerator and the second in the denominator. 4518 Simplify. 52 We leave the ratio in ⓑ as an improper fraction. Write each ratio as a fraction: ⓐ 21to56ⓑ 48to32. ⓐ 38 ⓑ 32 Write each ratio as a fraction: ⓐ 27to72ⓑ 51to34. ⓐ 38 ⓑ 32

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator. For example, consider the ratio 0.8to0.05. We can write it as a fraction with decimals and then multiply the numerator and denominator by 100 to eliminate the decimals. Do you see a shortcut to find the equivalent fraction? Notice that 0.8=810 and 0.05=5100. The least common denominator of 810 and 5100 is 100. By multiplying the numerator and denominator of 0.80.05 by 100, we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this: "Move" the decimal 2 places. 805 Simplify. 161 You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same. Write each ratio as a fraction of whole numbers: ⓐ 4.8to11.2 ⓑ 2.7to0.54 ⓐ 4.8 to 11.2 Write as a fraction. 4.811.2 Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right. 48112 Simplify. 37 So 4.8to11.2 is equivalent to 37. ⓑ The numerator has one decimal place and the denominator has 2. To clear both decimals we need to move the decimal 2 places to the right.2.7to0.54 Write as a fraction. 2.70.54 Move both decimals right two places. 27054 Simplify. 51 So 2.7to0.54 is equivalent to 51. Write each ratio as a fraction: ⓐ 4.6to11.5ⓑ 2.3to0.69. ⓐ 25ⓑ 103 Write each ratio as a fraction: ⓐ 3.4to15.3ⓑ 3.4to0.68. ⓐ 29 ⓑ 51 Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions. Write the ratio of 114to238 as a fraction. 114to238 Write as a fraction. 114238 Convert the numerator and denominator to improper fractions. 54198 Rewrite as a division of fractions. 54÷198 Invert the divisor and multiply. 54·819 Simplify. 1019 Write each ratio as a fraction: 134to258. 23 Write each ratio as a fraction: 118to234. 922

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than 5 to 1 is considered good. Hector's total cholesterol is 249 mg/dl and his HDL cholesterol is 39 mg/dl. ⓐ Find the ratio of his total cholesterol to his HDL cholesterol. ⓑ Assuming that a ratio less than 5 to 1 is considered good, what would you suggest to Hector? ⓐ First, write the words that express the ratio. We want to know the ratio of Hector's total cholesterol to his HDL cholesterol. Write as a fraction. total cholesterolHDL cholesterol Substitute the values. 24939 Simplify. 8313 ⓑ Is Hector's cholesterol ratio ok? If we divide 83 by 13 we obtain approximately 6.4, so 8313≈6.41. Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol. Find the patient's ratio of total cholesterol to HDL cholesterol using the given information. Total cholesterol is 185 mg/dL and HDL cholesterol is 40 mg/dL. 378 Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information. Total cholesterol is 204 mg/dL and HDL cholesterol is 38 mg/dL. 10219

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units. We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit. The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of 1 inch for every 1 foot of horizontal run. What is the ratio of the rise to the run? In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem. Write the words that express the ratio. Ratio of the rise to the run Write the ratio as a fraction. riserun Substitute in the given values. 1 inch1 foot Convert 1 foot to inches. 1 inch12 inches Simplify, dividing out common factors and units. 112 So the ratio of rise to run is 1 to 12. This means that the ramp should rise 1 inch for every 12 inches of horizontal run to comply with the guidelines. Find the ratio of the first length to the second length: 32 inches to 1 foot. 83 Find the ratio of the first length to the second length: 1 foot to 54 inches. 29

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are 120 miles in 2 hours, 160 words in 4 minutes, and $5 dollars per 64 ounces. A rate compares two quantities of different units. A rate is usually written as a fraction. When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator. Bob drove his car 525 miles in 9 hours. Write this rate as a fraction. 525 miles in 9 hours Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator. 525 miles9 hours 175 miles3 hours So 525 miles in 9 hours is equivalent to 175 miles3 hours. Write the rate as a fraction: 492 miles in 8 hours. 123 miles2 hours Write the rate as a fraction: 242 miles in 6 hours. 121 miles3 hours

In the last example, we calculated that Bob was driving at a rate of 175 miles3 hours. This tells us that every three hours, Bob will travel 175 miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of 1 unit is referred to as a unit rate. A unit rate is a rate with denominator of 1 unit. Unit rates are very common in our lives. For example, when we say that we are driving at a speed of 68 miles per hour we mean that we travel 68 miles in 1 hour. We would write this rate as 68 miles/hour (read 68 miles per hour). The common abbreviation for this is 68 mph. Note that when no number is written before a unit, it is assumed to be 1. So 68 miles/hour really means 68 miles/1 hour. Two rates we often use when driving can be written in different forms, as shown: Example Rate Write Abbreviate Read 68 miles in 1 hour 68 miles1 hour 68 miles/hour 68 mph 68 miles per hour 36 miles to 1 gallon 36 miles1 gallon 36 miles/gallon 36 mpg 36 miles per gallon Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $12.50 for each hour you work, you could write that your hourly (unit) pay rate is $12.50/hour (read $12.50 per hour.) To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of 1. Anita was paid $384 last week for working 32 hours. What is Anita’s hourly pay rate? Start with a rate of dollars to hours. Then divide. $384 last week for 32 hours Write as a rate. $38432 hours Divide the numerator by the denominator. $121 hour Rewrite as a rate. $12/hour Anita’s hourly pay rate is $12 per hour. Find the unit rate: $630 for 35 hours. $18.00/hour Find the unit rate: $684 for 36 hours. $19.00/hour Sven drives his car 455 miles, using 14 gallons of gasoline. How many miles per gallon does his car get? Start with a rate of miles to gallons. Then divide. 455 miles to 14 gallons of gas Write as a rate. 455 miles14 gallons Divide 455 by 14 to get the unit rate. 32.5 miles1 gallon Sven’s car gets 32.5 miles/gallon, or 32.5 mpg. Find the unit rate: 423 miles to 18 gallons of gas. 23.5 mpg Find the unit rate: 406 miles to 14.5 gallons of gas. 28 mpg

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item. A unit price is a unit rate that gives the price of one item. The grocery store charges $3.99 for a case of 24 bottles of water. What is the unit price? What are we asked to find? We are asked to find the unit price, which is the price per bottle. Write as a rate. $3.9924 bottles Divide to find the unit price. $0.166251 bottle Round the result to the nearest penny. $0.171 bottle The unit price is approximately $0.17 per bottle. Each bottle costs about $0.17. Find the unit price. Round your answer to the nearest cent if necessary. 24-pack of juice boxes for $6.99 $0.29/box Find the unit price. Round your answer to the nearest cent if necessary. 24-pack of bottles of ice tea for $12.72 $0.53/bottle Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves. Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $14.99 for 64 loads of laundry and the same brand of powder detergent is priced at $15.99 for 80 loads. Which detergent has the lowest cost per load? To compare the prices, we first find the unit price for each type of detergent. Liquid Powder Write as a rate. $14.9964 loads $15.9980 loads Find the unit price. $0.234…1 load $0.199…1 load Round to the nearest cent. $0.23/load(23 cents per load.) $0.20/load(20 cents per load) Now we compare the unit prices. The unit price of the liquid detergent is about $0.23 per load and the unit price of the powder detergent is about $0.20 per load. The powder is the better buy. Find each unit price and then determine the better buy. Round to the nearest cent if necessary. Brand A Storage Bags, $4.59 for 40 count, or Brand B Storage Bags, $3.99 for 30 count Brand A costs $0.11 per bag. Brand B costs $0.13 per bag. Brand A is the better buy. Find each unit price and then determine the better buy. Round to the nearest cent if necessary. Brand C Chicken Noodle Soup, $1.89 for 26 ounces, or Brand D Chicken Noodle Soup, $0.95 for 10.75 ounces Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy. Notice in that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction. Translate the word phrase into an algebraic expression: ⓐ 427 miles per h hours ⓑ x students to 3 teachers ⓒ y dollars for 18 hours ⓐ 427 miles perhhours Write as a rate. 427 mileshhours ⓑ xstudents to 3 teachers Write as a rate. xstudents3 teachers ⓒ ydollars for 18 hours Write as a rate. $y18 hours Translate the word phrase into an algebraic expression. ⓐ 689 miles per h hours ⓑ y parents to 22 students ⓒ d dollars for 9 minutes ⓐ 689 mi/h hours ⓑ y parents/22 students ⓒ $d/9 min Translate the word phrase into an algebraic expression. ⓐ m miles per 9 hours ⓑ x students to 8 buses ⓒ y dollars for 40 hours ⓐ m mi/9 h ⓑ x students/8 buses ⓒ $y/40 h Ratios Write Ratios as a Simplified Fractions Involving Decimals and Fractions Write a Ratio as a Simplified Fraction Rates and Unit Rates Unit Rate for Cell Phone Plan

Write a Ratio as a Fraction In the following exercises, write each ratio as a fraction. 20 to 36 59 20 to 32 42 to 48 78 45 to 54 49 to 21 73 56 to 16 84 to 36 73 6.4 to 0.8 0.56 to 2.8 15 1.26 to 4.2 123 to 256 1017 134 to 258 416 to 313 54 535 to 335 $18 to $63 27 $16 to $72 $1.21 to $0.44 114 $1.38 to $0.69 28 ounces to 84 ounces 13 32 ounces to 128 ounces 12 feet to 46 feet 623 15 feet to 57 feet 246 milligrams to 45 milligrams 8215 304 milligrams to 48 milligrams total cholesterol of 175 to HDL cholesterol of 45 359 total cholesterol of 215 to HDL cholesterol of 55 27 inches to 1 foot 94 28 inches to 1 foot Write a Rate as a Fraction In the following exercises, write each rate as a fraction. 140 calories per 12 ounces 35 calories3 ounces 180 calories per 16 ounces 8.2 pounds per 3 square inches 41 lbs15 sq. in. 9.5 pounds per 4 square inches 488 miles in 7 hours 488 miles7 hours 527 miles in 9 hours $595 for 40 hours $1198 hours $798 for 40 hours Find Unit Rates In the following exercises, find the unit rate. Round to two decimal places, if necessary. 140 calories per 12 ounces 11.67 calories/ounce 180 calories per 16 ounces 8.2 pounds per 3 square inches 2.73 lbs./sq. in. 9.5 pounds per 4 square inches 488 miles in 7 hours 69.71 mph 527 miles in 9 hours $595 for 40 hours $14.88/hour $798 for 40 hours 576 miles on 18 gallons of gas 32 mpg 435 miles on 15 gallons of gas 43 pounds in 16 weeks 2.69 lbs./week 57 pounds in 24 weeks 46 beats in 0.5 minute 92 beats/minute 54 beats in 0.5 minute The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour? 8,000 The pressroom at a printing plant prints 540,000 sections in 12 hours. How many sections are printed per hour? Find Unit Price In the following exercises, find the unit price. Round to the nearest cent. Soap bars at 8 for $8.69 $1.09/bar Soap bars at 4 for $3.39 Women’s sports socks at 6 pairs for $7.99 $1.33/pair Men’s dress socks at 3 pairs for $8.49 Snack packs of cookies at 12 for $5.79 $0.48/pack Granola bars at 5 for $3.69 CD-RW discs at 25 for $14.99 $0.60/disc CDs at 50 for $4.49 The grocery store has a special on macaroni and cheese. The price is $3.87 for 3 boxes. How much does each box cost? $1.29/box The pet store has a special on cat food. The price is $4.32 for 12 cans. How much does each can cost? In the following exercises, find each unit price and then identify the better buy. Round to three decimal places. Mouthwash, 50.7-ounce size for $6.99 or 33.8-ounce size for $4.79 The 50.7-ounce size costs $0.138 per ounce. The 33.8-ounce size costs $0.142 per ounce. The 50.7-ounce size is the better buy. Toothpaste, 6 ounce size for $3.19 or 7.8-ounce size for $5.19 Breakfast cereal, 18 ounces for $3.99 or 14 ounces for $3.29 The 18-ounce size costs $0.222 per ounce. The 14-ounce size costs $0.235 per ounce. The 18-ounce size is a better buy. Breakfast Cereal, 10.7 ounces for $2.69 or 14.8 ounces for $3.69 Ketchup, 40-ounce regular bottle for $2.99 or 64-ounce squeeze bottle for $4.39 The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy. Mayonnaise 15-ounce regular bottle for $3.49 or 22-ounce squeeze bottle for $4.99 Cheese $6.49 for 1 lb. block or $3.39 for 12 lb. block The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy. Candy $10.99 for a 1 lb. bag or $2.89 for 14 lb. of loose candy Translate Phrases to Expressions with Fractions In the following exercises, translate the English phrase into an algebraic expression. 793 miles per p hours 793 milesphours 78 feet per r seconds $3 for 0.5 lbs. $30.5 lbs. j beats in 0.5 minutes 105 calories in x ounces 105 caloriesxounces 400 minutes for m dollars the ratio of y and 5x y5x the ratio of 12x and y

One elementary school in Ohio has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate. 15.2 students per teacher The average American produces about 1,600 pounds of paper trash per year (365 days). How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.) A popular fast food burger weighs 7.5 ounces and contains 540 calories, 29 grams of fat, 43 grams of carbohydrates, and 25 grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce. Round to two decimal places. ⓐ 72 calories/ounce ⓑ 3.87 grams of fat/ounce ⓒ 5.73 grams carbs/ounce ⓓ 3.33 grams protein/ounce A 16-ounce chocolate mocha coffee with whipped cream contains 470 calories, 18 grams of fat, 63 grams of carbohydrates, and 15 grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce.

Would you prefer the ratio of your income to your friend’s income to be 3/1 or 1/3? Explain your reasoning. Answers will vary. The parking lot at the airport charges $0.75 for every 15 minutes. ⓐ How much does it cost to park for 1 hour? ⓑ Explain how you got your answer to part ⓐ. Was your reasoning based on the unit cost or did you use another method? Kathryn ate a 4-ounce cup of frozen yogurt and then went for a swim. The frozen yogurt had 115 calories. Swimming burns 422 calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning. Kathryn should swim for approximately 16.35 minutes. Explanations will vary. Mollie had a 16-ounce cappuccino at her neighborhood coffee shop. The cappuccino had 110 calories. If Mollie walks for one hour, she burns 246 calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

ratio A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of a to b is written a to b, ab, or a:b. rate A rate compares two quantities of different units. A rate is usually written as a fraction. unit rate A unit rate is a rate with denominator of 1 unit. unit price A unit price is a unit rate that gives the price of one item.