Ratios, Rates & Proportions

Topics Covered Writing Ratios Writing Ratios of Mixed Numbers and Decimals Applications of Ratios

Writing Ratios A ratio is a comparison of two quantities. There are three different ways to write a ratio We will primarily use the fraction form. Examples: Write the ratios in lowest terms. 10 lb to 14 lb 14 oz to 7 oz

Writing Ratios of Mixed Numbers and Decimals Examples: 10.15 hr to 8.12 hr 11.55 km to 6.6 km

Writing Ratios from word statements For a recent year, 250 million persons were covered by health insurance in the United States and 45 million were not covered. Write a ratio of the number of insured persons to the number of uninsured persons. Write a ratio of the number of uninsured persons to the number of insured persons. Write a ratio of the number of uninsured persons to the total number of persons.

Application #1 The temperature at 8:00AM in Los Angeles was 66 degrees Fahrenheit. By 2:00PM the temperature had risen to 90 degrees. Find the increase in temperature from 8:00AM to 2:00PM Write a ratio representing the increase in temperature to the temperature at 8:00AM

Application #2 In 1981, a giant sinkhole in Winter Park Florida, “swallowed” a home, a public pool, and a car dealership (including five Porsches). One witness said that in a single day, the hole widened from 5ft in diameter to 320 ft. Write a ratio representing the increase in diameter to the original diameter of 5 ft. 1/64 64/1 1/63 63/1

Application #3 A construction company needs 2 weeks to construct a family room and 3 days to add a porch. Find the ratio of the time it takes for constructing the porch to the time constructing the family room, with all units in weeks. Find the ratio of the time it takes for constructing the porch to the time constructing the family room, with all units in days.

Application #4 A window is 2ft wide and 3yd in length (2 ft is 2/3 yd). Find the ratio of the width to length with all units in yards.

Application #5 Find the ratio of the shortest side to the longest side

Definition of a Rate A rate is a type of ratio used to compare different types of quantities. Several key words imply rates; per a miles per b hours For a dollars for b pounds In a meters in b seconds On a miles on b gallons of gas

Examples There are 130 calories in 8 snack crackers During a bad storm there was 2 inches of rain in 6 hours Jill’s portfolio changed by -$2600 in 6 months

Unit Rates A rate having a denominator of 1 unit is called a unit rate. Examples: 45 mph is a unit rate (45 miles per 1 hour) $0.23/ounce is a unit cost (23 cents per 1 ounce) 21 mpg is a unit rate (21 miles per 1 gallon of gasoline) Note: Any rate can be turned into a unit rate by dividing the two numbers. EX. He traveled 534 miles on 20 gallons of gas (26.7mpg).

Application #1 Compute the amount of carbohydrate per fluid ounce for each soda. Then determine which has the greatest amount of carbohydrate per fluid ounce. soda amount carbohydrates Coca-Cola 20 fl oz 65 g Mello Yello 12 fl oz 47 g Canada Dry Ginger Ale 8 fl oz 25 g

Application #2 According to the National Institutes of Health, a platelet count below 20,000 per microliter of blood is considered a life-threatening condition. Suppose a patient’s test results yield a platelet count of 13,000,000 for 100 microliters. Write this as a unit rate. Does the patient have a life-threatening condition?

Definition of a Proportion A proportion states that two ratios are equal. Examples:

Reading Proportions This proportion is read, “a is to b as c is to d” Write a proportion from each statement. 3 is to 18 as 4 is to 24. The numbers 2 and 1 are proportional to the numbers 26 and 13. Try: $115 per week is proportional to $460 per 4 weeks.

Determining whether Two Ratios Form a Proportion. We determine whether two ratios are proportional by using the cross product. Determine whether the given ratios are proportional. Are the numbers -7.1 and 2.4 proportional to the numbers -35.5 and 10?

Try These: Are the numbers 1 2/3 and 5/6 proportional to the numbers 5 and 2 1/2 ? Are the numbers -6.3 and 9 proportional to the numbers -12.6 and 16?

Solving Proportions Process: Set the cross products equal to each other Solve the equation Check the solution in the original proportion.

More examples The numbers -4.75 and 8 are proportional to the numbers -9.5 and k.

Application #1 Didi takes her pulse for 10 sec and counts 13 beats. How many beats per minute is that?

Application #2 Suppose two adults produce 63.4 lb of garbage in one week. At this rate, how many pounds will 50 adults produce in one week?

Application #3 On a map, the distance from Nashville, Tennessee, to Atlanta, Georgia, is 3.5 in. and the actual distance is 210 mi. If the map distance between Dallas, Texas, and Little Rock, Arkansas, is 4.75 in., what is the actual distance?

Application #4 Pam drives her Toyota Prius 244 mi in city driving on 4 gal. of gas. At this rate how many miles can she drive on 10 gal. of gas?

Application #5 At Central Community College, the ratio of female students to male students is 31 to 19. If there are 6200 female students, how many male students are there?

Application #6 Evelyn won an election by a ratio of 6 to 5. If she received 7230 votes, how many votes did her opponent receive?

Application #7 Each gram of fat consumed has 9 calories. If a ½ cup serving of gelato has 81 calories from fat, how many grams of fat are in this serving?

Application #8 Yellowstone National Park in Wyoming has the largest population of free-roaming bison. To approximate the number of bison, 200 are captured and tagged and then let free to roam. Later, a sample of 120 bison is observed and 6 have tags. Approximate the population of bison in the park.

Application #9 A 32-ft tree casts a shadow of 18 ft. How long will the shadow be for a 22-ft tree?