5.6 Ratios and Rate
Learning Objectives
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
Ratios Involving Decimals
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. | 80 5 |
| Simplify. | 16 1 |
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.
Applications of Ratios
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.
Ratios of Two Measurements in Different Units
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.
Write a Rate as a Fraction
Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use arate. Examples of rates are 120 miles in 2 hours, 160 words in 4 minutes, and $5 dollars per 64 ounces.
Find Unit Price
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.
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.
Notice in Example 5.67 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.
Translate Phrases to Expressions with Fractions
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.
Section 5.6 Exercises
Practice Makes Perfect
Write a Ratio as a Fraction
In the following exercises, write each ratio as a fraction.
- 20 to 36
- 20 to 32
- 42 to 48
- 45 to 54
- 49 to 21
- 56 to 16
- 84 to 36
- 6.4 to 0.8
- 0.56 to 2.8
- 1.26 to 4.2
- 123 to 256
- 134 to 258
- 416 to 313
- 535 to 335
- $18 to $63
- $16 to $72
- $1.21 to $0.44
- $1.38 to $0.69
- 28 ounces to 84 ounces
- 32 ounces to 128 ounces
- 12 feet to 46 feet
- 15 feet to 57 feet
- 246 milligrams to 45 milligrams
- 304 milligrams to 48 milligrams
- total cholesterol of 175 to HDL cholesterol of 45
- total cholesterol of 215 to HDL cholesterol of 55
- 27 inches to 1 foot
- 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
- 180 calories per 16 ounces
- 8.2 pounds per 3 square inches
- 9.5 pounds per 4 square inches
- 488 miles in 7 hours
- 527 miles in 9 hours
- $595 for 40 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
- 180 calories per 16 ounces
- 8.2 pounds per 3 square inches
- 9.5 pounds per 4 square inches
- 488 miles in 7 hours
- 527 miles in 9 hours
- $595 for 40 hours
- $798 for 40 hours
- 576 miles on 18 gallons of gas
- 435 miles on 15 gallons of gas
- 43 pounds in 16 weeks
- 57 pounds in 24 weeks
- 46 beats in 0.5 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?
- 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
- Soap bars at 4 for $3.39
- Women’s sports socks at 6 pairs for $7.99
- Men’s dress socks at 3 pairs for $8.49
- Snack packs of cookies at 12 for $5.79
- Granola bars at 5 for $3.69
- CD-RW discs at 25 for $14.99
- 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?
- 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
- 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
- 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
- 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
- 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
- 78 feet per r seconds
- $3 for 0.5 lbs.
- j beats in 0.5 minutes
- 105 calories in x ounces
- 400 minutes for m dollars
- the ratio of y and 5x
- the ratio of 12x and y
Everyday Math
- One elementary school in Ohio has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate.
- 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.
- 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.
Writing Exercises - Would you prefer the ratio of your income to your friend’s income to be 3/1 or 1/3? Explain your reasoning.
- 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.
- 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.
Self Check
ⓐ 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?