Learn on PengiSaxon Math, Course 2Chapter 5: Lessons 41-50, Investigation 5

Lesson 46: Rates

In this Grade 7 Saxon Math Course 2 lesson, students learn what a rate is and how to calculate common rates including speed using the formula r = d/t, mileage rates, pay rates, and unit prices by dividing two related measures. Through worked examples, students practice expressing rates as fractions with a denominator of 1 and applying rates by multiplying or dividing to solve real-world problems.

Section 1

📘 Rates

New Concept

A rate is a ratio of two measures. Unit Price is the cost for a single unit of a product.

rate=distancetimer=dt \text{rate} = \frac{\text{distance}}{\text{time}} \quad r = \frac{d}{t}
distance=rate×timed=rt \text{distance} = \text{rate} \times \text{time} \quad d = rt

What’s next

Now you will apply these formulas in worked examples. These problems will cover calculating speed, fuel efficiency, pay rates, and travel distance to build your problem-solving skills.

Section 2

Rate

Property

A rate is a ratio of two measures. The formula for a rate of speed is:

rate=distancetime \text{rate} = \frac{\text{distance}}{\text{time}}

Examples

An awesome driver traveled 420 miles in 7 hours. Their average speed was:

420 miles7 hours=60 miles1 hour \frac{420 \text{ miles}}{7 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}}

A car used 12 gallons of gas to go 360 miles. Its mileage rate was:
360 miles12 gallons=30 miles1 gallon \frac{360 \text{ miles}}{12 \text{ gallons}} = \frac{30 \text{ miles}}{1 \text{ gallon}}

You earned 510 dollars for 30 hours of work. Your pay rate was:
510 dollars30 hours=17 dollars1 hour \frac{510 \text{ dollars}}{30 \text{ hours}} = \frac{17 \text{ dollars}}{1 \text{ hour}}

Explanation

Think of a rate as a 'how much for one' deal! The word 'per' is your secret code for division. Whether it is miles per hour or cookies per minute, you are just simplifying a fraction to find the amount for a single unit, like one hour or one gallon.

Section 3

Unit Price

Property

Unit Price is the cost for a single unit of a product. It is a ratio of price to quantity that is often posted in supermarkets to help customers identify the better buy.

Examples

Find the unit price of a 20-ounce box of cereal that costs 4.00 dollars:

4.00 dollars20 ounces=0.20 dollars1 ounce \frac{4.00 \text{ dollars}}{20 \text{ ounces}} = \frac{0.20 \text{ dollars}}{1 \text{ ounce}}

Find the unit price of a 16-ounce bottle of juice that costs 4.00 dollars:
4.00 dollars16 ounces=0.25 dollars1 ounce \frac{4.00 \text{ dollars}}{16 \text{ ounces}} = \frac{0.25 \text{ dollars}}{1 \text{ ounce}}

Explanation

Unit price is your superpower for smart shopping! It is a special rate that reveals the cost for just one item, like one ounce of chips or one can of soda. By dividing the total price by the number of items, you can easily compare different packages and find the true best deal.

Section 4

The Distance Formula

Property

An important formula that relates distance to rate and time is the following:

distance=rate×timed=rt \text{distance} = \text{rate} \times \text{time} \quad d = rt

Examples

You plan to ride your bike for 6 hours at an average speed of 14 miles per hour:

d=14 miles1 hour⋅6 hours=84 miles d = \frac{14 \text{ miles}}{1 \text{ hour}} \cdot 6 \text{ hours} = 84 \text{ miles}

Your car gets 30 miles per gallon and has a 10-gallon tank. The total distance it can go is:
d=30 miles1 gallon⋅10 gallons=300 miles d = \frac{30 \text{ miles}}{1 \text{ gallon}} \cdot 10 \text{ gallons} = 300 \text{ miles}

Explanation

This handy formula lets you predict the future of your trip! If you know how fast you are going (rate) and for how long you will be traveling (time), just multiply them together. This tells you the total distance you will cover, making it perfect for planning epic road trips or bike rides.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Lessons 41-50, Investigation 5

  1. Lesson 1

    Lesson 41: Using Formulas, Distributive Property

    LearnPractice
  2. Lesson 2

    Lesson 42: Repeating Decimals

    LearnPractice
  3. Lesson 3

    Lesson 43: Converting Decimals to Fractions, Converting Fractions to Decimals, Converting Percents to Decimals

    LearnPractice
  4. Lesson 4

    Lesson 44: Division Answers

    LearnPractice
  5. Lesson 5

    Lesson 45: Dividing by a Decimal Number

    LearnPractice
  6. Lesson 6Current

    Lesson 46: Rates

    LearnPractice
  7. Lesson 7

    Lesson 47: Powers of 10

    LearnPractice
  8. Lesson 8

    Lesson 48: Fraction-Decimal-Percent Equivalents

    LearnPractice
  9. Lesson 9

    Lesson 49: Adding and Subtracting Mixed Measures

    LearnPractice
  10. Lesson 10

    Lesson 50: Unit Multipliers and Unit Conversion

    LearnPractice
  11. Lesson 11

    Investigation 5: Creating Graphs

    LearnPractice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Rates

New Concept

A rate is a ratio of two measures. Unit Price is the cost for a single unit of a product.

rate=distancetimer=dt \text{rate} = \frac{\text{distance}}{\text{time}} \quad r = \frac{d}{t}
distance=rate×timed=rt \text{distance} = \text{rate} \times \text{time} \quad d = rt

What’s next

Now you will apply these formulas in worked examples. These problems will cover calculating speed, fuel efficiency, pay rates, and travel distance to build your problem-solving skills.

Section 2

Rate

Property

A rate is a ratio of two measures. The formula for a rate of speed is:

rate=distancetime \text{rate} = \frac{\text{distance}}{\text{time}}

Examples

An awesome driver traveled 420 miles in 7 hours. Their average speed was:

420 miles7 hours=60 miles1 hour \frac{420 \text{ miles}}{7 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}}

A car used 12 gallons of gas to go 360 miles. Its mileage rate was:
360 miles12 gallons=30 miles1 gallon \frac{360 \text{ miles}}{12 \text{ gallons}} = \frac{30 \text{ miles}}{1 \text{ gallon}}

You earned 510 dollars for 30 hours of work. Your pay rate was:
510 dollars30 hours=17 dollars1 hour \frac{510 \text{ dollars}}{30 \text{ hours}} = \frac{17 \text{ dollars}}{1 \text{ hour}}

Explanation

Think of a rate as a 'how much for one' deal! The word 'per' is your secret code for division. Whether it is miles per hour or cookies per minute, you are just simplifying a fraction to find the amount for a single unit, like one hour or one gallon.

Section 3

Unit Price

Property

Unit Price is the cost for a single unit of a product. It is a ratio of price to quantity that is often posted in supermarkets to help customers identify the better buy.

Examples

Find the unit price of a 20-ounce box of cereal that costs 4.00 dollars:

4.00 dollars20 ounces=0.20 dollars1 ounce \frac{4.00 \text{ dollars}}{20 \text{ ounces}} = \frac{0.20 \text{ dollars}}{1 \text{ ounce}}

Find the unit price of a 16-ounce bottle of juice that costs 4.00 dollars:
4.00 dollars16 ounces=0.25 dollars1 ounce \frac{4.00 \text{ dollars}}{16 \text{ ounces}} = \frac{0.25 \text{ dollars}}{1 \text{ ounce}}

Explanation

Unit price is your superpower for smart shopping! It is a special rate that reveals the cost for just one item, like one ounce of chips or one can of soda. By dividing the total price by the number of items, you can easily compare different packages and find the true best deal.

Section 4

The Distance Formula

Property

An important formula that relates distance to rate and time is the following:

distance=rate×timed=rt \text{distance} = \text{rate} \times \text{time} \quad d = rt

Examples

You plan to ride your bike for 6 hours at an average speed of 14 miles per hour:

d=14 miles1 hour⋅6 hours=84 miles d = \frac{14 \text{ miles}}{1 \text{ hour}} \cdot 6 \text{ hours} = 84 \text{ miles}

Your car gets 30 miles per gallon and has a 10-gallon tank. The total distance it can go is:
d=30 miles1 gallon⋅10 gallons=300 miles d = \frac{30 \text{ miles}}{1 \text{ gallon}} \cdot 10 \text{ gallons} = 300 \text{ miles}

Explanation

This handy formula lets you predict the future of your trip! If you know how fast you are going (rate) and for how long you will be traveling (time), just multiply them together. This tells you the total distance you will cover, making it perfect for planning epic road trips or bike rides.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Lessons 41-50, Investigation 5

  1. Lesson 1

    Lesson 41: Using Formulas, Distributive Property

    LearnPractice
  2. Lesson 2

    Lesson 42: Repeating Decimals

    LearnPractice
  3. Lesson 3

    Lesson 43: Converting Decimals to Fractions, Converting Fractions to Decimals, Converting Percents to Decimals

    LearnPractice
  4. Lesson 4

    Lesson 44: Division Answers

    LearnPractice
  5. Lesson 5

    Lesson 45: Dividing by a Decimal Number

    LearnPractice
  6. Lesson 6Current

    Lesson 46: Rates

    LearnPractice
  7. Lesson 7

    Lesson 47: Powers of 10

    LearnPractice
  8. Lesson 8

    Lesson 48: Fraction-Decimal-Percent Equivalents

    LearnPractice
  9. Lesson 9

    Lesson 49: Adding and Subtracting Mixed Measures

    LearnPractice
  10. Lesson 10

    Lesson 50: Unit Multipliers and Unit Conversion

    LearnPractice
  11. Lesson 11

    Investigation 5: Creating Graphs

    LearnPractice