3.1 Ratios and Rate
Learning Objectives
By the end of this section, you will be able to:
- Write a ratio as a fraction
- Find unit rates
- Find unit price
- Translate phrases to expressions with fractions
Write a Ratio as a Fraction
Ratios
A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of to is written
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 instead of simplifying it to so that we can see the two parts of the ratio.
EXAMPLE 1
Write each ratio as a fraction: a) b).
a)
| Write as a fraction with the first number in the numerator and the second in the denominator. | |
| Simplify the fraction. |
We leave the ratio in b) as an improper fraction.
b)
| Write as a fraction with the first number in the numerator and the second in the denominator. | |
| Simplify. |
TRY IT 1.1
Write each ratio as a fraction: a) b) .
Answer
TRY IT 1.2
Write each ratio as a fraction: a) b) .
Answer
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 . We can write it as a fraction with decimals and then multiply the numerator and denominator by to eliminate the decimals.
Do you see a shortcut to find the equivalent fraction? Notice that and . The least common denominator of and is . By multiplying the numerator and denominator of by , 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. | = |
| Simplify. |
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.
EXAMPLE 2
Write each ratio as a fraction of whole numbers:
a)
b)
| a) | |
| Write as a fraction. | |
| Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right. | |
| Simplify. |
So is equivalent to .
| b) The numerator has one decimal place and the denominator has . To clear both decimals we need to move the decimal places to the right. |
|
| Write as a fraction. | |
| Move both decimals right two places. | |
| Simplify. |
So is equivalent to .
TRY IT 2.1
Write each ratio as a fraction: a) b) .
Answer
TRY IT 2.2
Write each ratio as a fraction: a) b) .
Answer
Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.
EXAMPLE 3
Write the ratio of as a fraction.
| Write ” ” as a fraction. | |
| Convert the numerator and denominator to improper fractions. | |
| Rewrite as a division of fractions. | |
| Invert the divisor and multiply. | |
| Simplify. |
TRY IT 3.1
Write each ratio as a fraction: .
Answer
TRY IT 3.2
Write each ratio as a fraction: .
Answer
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 to 1 is considered good.
EXAMPLE 4
Hector’s total cholesterol is mg/dl and his HDL cholesterol is mg/dl. a) Find the ratio of his total cholesterol to his HDL cholesterol. b) Assuming that a ratio less than to is considered good, what would you suggest to Hector?
a) 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. | |
| Substitute the values. | |
| Simplify. |
b) Is Hector’s cholesterol ratio ok? If we divide by we obtain approximately , so . Hector’s cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.
TRY IT 4.1
Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is mg/dL and HDL cholesterol is mg/dL.
Answer
TRY IT 4.2
Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is mg/dL and HDL cholesterol is mg/dL.
Answer
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.
EXAMPLE 5
The Canadian National Building Code (CNBC) Guidelines for wheel chair ramps require a maximum vertical rise of inch for every 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. | |
| Substitute in the given values. | |
| Convert 1 foot to inches. | |
| Simplify, dividing out common factors and units. |
So the ratio of rise to run is to . This means that the ramp should rise inch for every inches of horizontal run to comply with the guidelines.
TRY IT 5.1
Find the ratio of the first length to the second length: inches to foot.
Answer
TRY IT 5.2
Find the ratio of the first length to the second length: foot to inches.
Answer
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 a rate. Examples of rates are miles in hours, words in minutes, and dollars per ounces.
Rate
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.
EXAMPLE 6
Bob drove his car miles in hours. Write this rate as a fraction.
| Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator. | |
So miles in hours is equivalent to .
TRY IT 6.1
Write the rate as a fraction: miles in hours.
Answer
TRY IT 6.2
Write the rate as a fraction: miles in hours.
Answer
Find Unit Rates
In the last example, we calculated that Bob was driving at a rate of . This tells us that every three hours, Bob will travel 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 unit is referred to as a unit rate.
Unit Rate
A unit rate is a rate with denominator of unit.
Unit rates are very common in our lives. For example, when we say that we are driving at a speed of miles per hour we mean that we travel miles in hour. We would write this rate as miles/hour (read miles per hour). The common abbreviation for this is mph. Note that when no number is written before a unit, it is assumed to be .
So miles/hour really means
Two rates we often use when driving can be written in different forms, as shown:
| Example | Rate | Write | Abbreviate | Read |
|---|---|---|---|---|
| miles in hour | miles/hour | mph | ||
| miles to gallon | miles/gallon | mpg |
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 for each hour you work, you could write that your hourly (unit) pay rate is (read per hour.)
To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of .
EXAMPLE 7
Anita was paid last week for working . What is Anita’s hourly pay rate?
| Start with a rate of dollars to hours. Then divide. | |
| Write as a rate. | |
| Divide the numerator by the denominator. | |
| Rewrite as a rate. |
Anita’s hourly pay rate is per hour.
TRY IT 7.1
Find the unit rate: for hours.
Answer
$18.00/hour
TRY IT 7.2
Find the unit rate: for hours.
Answer
$19.00/hour
EXAMPLE 8
Sven drives his car miles, using gallons of gasoline. How many miles per gallon does his car get?
Start with a rate of miles to gallons. Then divide.
| Write as a rate. | |
| Divide 455 by 14 to get the unit rate. |
Sven’s car gets miles/gallon, or mpg.
TRY IT 8.1
Find the unit rate: miles to gallons of gas.
Answer
23.5 mpg
TRY IT 8.2
Find the unit rate: miles to gallons of gas.
Answer
28 mpg
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 price
A unit price is a unit rate that gives the price of one item.
EXAMPLE 9
The grocery store charges for a case of 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. | |
| Divide to find the unit price. | |
| Round the result to the nearest penny. |
The unit price is approximately per bottle. Each bottle costs about .
TRY IT 9.1
Find the unit price. Round your answer to the nearest cent if necessary.
of juice boxes for
Answer
$0.29/box
TRY IT 9.2
Find the unit price. Round your answer to the nearest cent if necessary.
of bottles of ice tea for
Answer
$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.
EXAMPLE 10
Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at for loads of laundry and the same brand of powder detergent is priced at for loads.
Which is the better buy, the liquid or the powder detergent?
To compare the prices, we first find the unit price for each type of detergent.
| Liquid | Powder | |
| Write as a rate. | ||
| Find the unit price. | ||
| Round to the nearest cent. |
Now we compare the unit prices. The unit price of the liquid detergent is about per load and the unit price of the powder detergent is about per load. The powder is the better buy.
TRY IT 10.1
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand A Storage Bags, for count, or Brand B Storage Bags, for count
Answer
Brand A costs $0.11 per bag. Brand B costs $0.13 per bag. Brand A is the better buy.
TRY IT 10.2
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand C Chicken Noodle Soup, for ounces, or Brand D Chicken Noodle Soup, for ounces
Answer
Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy.
Notice in the above example 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.
EXAMPLE 11
Translate the word phrase into an algebraic expression:
a) miles per hours
b) students to teachers
c) dollars for hours
| a) | |
| Write as a rate. |
| b) | |
| Write as a rate. |
| c) | |
| Write as a rate. |
TRY IT 11.1
Translate the word phrase into an algebraic expression.
a) miles per hours b) parents to students c) dollars for minutes
Answer
- 689 mi/h hours
- y parents/22 students
- $d/9 min
TRY IT 11.2
Translate the word phrase into an algebraic expression.
a) miles per hours b) students to buses c) dollars for hours
Answer
- m mi/9 h
- x students/8 buses
- $y/40 h
Access to Additional Online R
Glossary
- ratio
- A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of to is written to , , or .
- 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.
Practice Makes Perfect
Write a Ratio as a Fraction
In the following exercises, write each ratio as a fraction.
| 1. to | 2. to |
| 3. to | 4. to |
| 5. to | 6. to |
| 7. to | 8. to |
| 9. to | 10. to |
| 11. to | 12. to |
| 13. to | 14. to |
| 15. to | 16. to |
| 17. to | 18. to |
| 19. ounces to ounces | 20. ounces to ounces |
| 21. feet to feet | 22. feet to feet |
| 23. milligrams to milligrams | 24. milligrams to milligrams |
| 25. total cholesterol of to HDL cholesterol of | 26. total cholesterol of to HDL cholesterol of |
| 27. inches to foot | 28. inches to foot |
Write a Rate as a Fraction
In the following exercises, write each rate as a fraction.
| 29. calories per ounces | 30. calories per ounces |
| 31. pounds per square inches | 32. pounds per square inches |
| 33. miles in hours | 34. miles in hours |
| 35. for hours | 36. for hours |
Find Unit Rates
In the following exercises, find the unit rate. Round to two decimal places, if necessary.
| 37. calories per ounces | 38. calories per ounces |
| 39. pounds per square inches | 40. pounds per square inches |
| 41. miles in hours | 42. miles in hours |
| 43. for hours | 44. for hours |
| 45. miles on gallons of gas | 46. miles on gallons of gas |
| 47. pounds in weeks | 48. pounds in weeks |
| 49. beats in minute | 50. beats in minute |
| 51. The bindery at a printing plant assembles magazines in hours. How many magazines are assembled in one hour? | 52. The pressroom at a printing plant prints sections in hours. How many sections are printed per hour? |
Find Unit Price
In the following exercises, find the unit price. Round to the nearest cent.
| 53. Soap bars at for | 54. Soap bars at for |
| 55. Women’s sports socks at pairs for | 56. Men’s dress socks at pairs for |
| 57. Snack packs of cookies at for | 58. Granola bars at for |
| 59. CD-RW discs at for | 60. CDs at for |
| 61. The grocery store has a special on macaroni and cheese. The price is for boxes. How much does each box cost? | 62. The pet store has a special on cat food. The price is for 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.
| 63. Mouthwash, size for or size for | 64. Toothpaste, ounce size for or size for |
| 65. Breakfast cereal, ounces for or ounces for | 66. Breakfast Cereal, ounces for or ounces for |
| 67. Ketchup, regular bottle for or squeeze bottle for | 68. Mayonnaise regular bottle for or squeeze bottle for |
| 69. Cheese for lb. block or for lb. block | 70. Candy for a lb. bag or for lb. of loose candy |
Translate Phrases to Expressions with Fractions
In the following exercises, translate the English phrase into an algebraic expression.
| 71. miles per hours | 72. feet per seconds |
| 73. for lbs. | 74. beats in minutes |
| 75. 105 calories in ounces | 76. minutes for dollars |
| 77. the ratio of and | 78. the ratio of and |
Everyday Math
| 79. One elementary school in Saskatchewan has students and teachers. Write the student-to-teacher ratio as a unit rate. | 80. The average Canadian produces about pounds of paper trash per year How many pounds of paper trash does the average Canadian produce each day? (Round to the nearest tenth of a pound.) |
| 81. A popular fast food burger weighs ounces and contains calories, grams of fat, grams of carbohydrates, and grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce. Round to two decimal places. | 82. A chocolate mocha coffee with whipped cream contains calories, grams of fat, grams of carbohydrates, and grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce. |
Writing Exercises
| 83. Would you prefer the ratio of your income to your friend’s income to be or Explain your reasoning. | 84. The parking lot at the airport charges for every minutes. a) How much does it cost to park for hour? b) Explain how you got your answer to part a). Was your reasoning based on the unit cost or did you use another method? |
| 85. Kathryn ate a cup of frozen yogurt and then went for a swim. The frozen yogurt had calories. Swimming burns calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning. | 86. Arjun had a cappuccino at his neighbourhood coffee shop. The cappuccino had calories. If Arjun walks for one hour, he burns calories. For how many minutes must Arjun walk to burn off the calories in the cappuccino? Explain your reasoning. |
Answers
| 1. | 3. | 5. |
| 7. | 9. | 11. |
| 13. | 15. | 17. |
| 19. | 21. | 23. |
| 25. | 27. | 29. |
| 31. | 33. | 35. |
| 37. 11.67 calories/ounce | 39. 2.73 lbs./sq. in. | 41. 69.71 mph |
| 43. $14.88/hour | 45. 32 mpg | 47. 2.69 lbs./week |
| 49. 92 beats/minute | 51. 8,000 | 53. $1.09/bar |
| 55. $1.33/pair | 57. $0.48/pack | 59. $0.60/disc |
| 61. $1.29/box | 63. 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. | 65. 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. |
| 67. The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy. | 69. The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy. | 71. |
| 73. | 75. | 77. |
| 79. 15.2 students per teacher | 81. a) 72 calories/ounce
b) 3.87 grams of fat/ounce c) 5.73 grams carbs/once d) 3.33 grams protein/ounce |
83. Answers will vary. |
| 85. Answers will vary. |
Attributions
This chapter has been adapted from “Ratios and Rate” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.