Learn on PengiSaxon Math, Course 2Chapter 6: Lessons 51-60, Investigation 6

Lesson 55: Average and Rate Problems with Multiple Steps

In this Grade 7 Saxon Math Course 2 lesson, students learn how to use the relationship between average, sum, and quantity to solve multi-step average and rate problems. Key skills include finding a missing number in a data set when the average is known, calculating unit price to compare costs, and determining average rate of pay across multiple pay rates. The lesson emphasizes organizing multi-step solutions clearly to reach a final answer efficiently.

Section 1

📘 Average and Rate Problems with Multiple Steps

New Concept

If we know the average of a group of numbers and how many numbers are in the group, we can determine the sum of the numbers.

Why it matters

Mastering averages allows you to condense vast amounts of data into a single, meaningful value, a foundational skill in statistics and data science. This tool enables you to analyze trends, compare performance, and make predictions, turning raw numbers into strategic insights.

What’s next

Next, you’ll apply this by using the total sum to find a missing number in a set and solve multi-step rate problems.

Section 2

Finding the Total from an Average

Property

To find the sum of a group of numbers, multiply their average by the quantity of numbers in the group.

Sum=Average×Quantity \text{Sum} = \text{Average} \times \text{Quantity}

Examples

The average of five numbers is 20. Their total sum is 5×20=1005 \times 20 = 100.
Tisha averaged 15 points in 3 basketball games. She scored a total of 3×15=453 \times 15 = 45 points.

Explanation

Think of the average as the 'fair share' amount. If you know the fair share and how many people are sharing, you can find the total amount they have altogether by multiplying! It's like everyone temporarily had the exact same number.

Section 3

Finding a Missing Number

Property

First, find the total sum using the average and quantity. Then, subtract the sum of the known numbers from the total to find the missing number.

Examples

The average of four numbers is 10. Three numbers are 5, 8, and 12. Total sum: 4×10=404 \times 10 = 40. Fourth number: 40−(5+8+12)=1540 - (5+8+12) = 15.
The average of three exam scores is 90. Two scores are 85 and 95. Total needed: 3×90=2703 \times 90 = 270. Third score: 270−(85+95)=90270 - (85+95) = 90.

Explanation

This is like a math treasure hunt! You know the total value of all the items (the sum). You're told the value of all but one. To find the last one's value, just subtract what you already know from the grand total!

Section 4

Reaching a New Average

Property

To find the value needed to achieve a new average, calculate the required total for the new average and subtract the current total.

New Value=(New Average×New Quantity)−(Old Sum) \text{New Value} = (\text{New Average} \times \text{New Quantity}) - (\text{Old Sum})

Examples

Your average for 3 tests is 80. To get an average of 82 after 4 tests: New total needed: 4×82=3284 \times 82 = 328. Old total: 3×80=2403 \times 80 = 240. Score needed on the 4th test: 328−240=88328 - 240 = 88.
After 4 bowling games, your average is 100. To get a 102 average after 5 games: New total: 5×102=5105 \times 102 = 510. Old total: 4×100=4004 \times 100 = 400. Score needed in game 5: 510−400=110510 - 400 = 110.

Explanation

Time to level up your average! Figure out the total score you need for your new level (new average × new count). Then, subtract the score you already have (your current total) to see what you need on the next try to hit your goal.

Book overview

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Chapter 6: Lessons 51-60, Investigation 6

  1. Lesson 1

    Lesson 51: Scientific Notation for Large Numbers

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  2. Lesson 2

    Lesson 52: Order of Operations

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  3. Lesson 3

    Lesson 53: Ratio Word Problems

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  4. Lesson 4

    Lesson 54: Rate Word Problems

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  5. Lesson 5Current

    Lesson 55: Average and Rate Problems with Multiple Steps

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  6. Lesson 6

    Lesson 56: Plotting Functions

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  7. Lesson 7

    Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

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  8. Lesson 8

    Lesson 58: Symmetry

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  9. Lesson 9

    Lesson 59: Adding Integers on the Number Line

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  10. Lesson 10

    Lesson 60: Fractional Part of a Number, Part 1, Percent of a Number, Part 1

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  11. Lesson 11

    Investigation 6: Classifying Quadrilaterals

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Average and Rate Problems with Multiple Steps

New Concept

If we know the average of a group of numbers and how many numbers are in the group, we can determine the sum of the numbers.

Why it matters

Mastering averages allows you to condense vast amounts of data into a single, meaningful value, a foundational skill in statistics and data science. This tool enables you to analyze trends, compare performance, and make predictions, turning raw numbers into strategic insights.

What’s next

Next, you’ll apply this by using the total sum to find a missing number in a set and solve multi-step rate problems.

Section 2

Finding the Total from an Average

Property

To find the sum of a group of numbers, multiply their average by the quantity of numbers in the group.

Sum=Average×Quantity \text{Sum} = \text{Average} \times \text{Quantity}

Examples

The average of five numbers is 20. Their total sum is 5×20=1005 \times 20 = 100.
Tisha averaged 15 points in 3 basketball games. She scored a total of 3×15=453 \times 15 = 45 points.

Explanation

Think of the average as the 'fair share' amount. If you know the fair share and how many people are sharing, you can find the total amount they have altogether by multiplying! It's like everyone temporarily had the exact same number.

Section 3

Finding a Missing Number

Property

First, find the total sum using the average and quantity. Then, subtract the sum of the known numbers from the total to find the missing number.

Examples

The average of four numbers is 10. Three numbers are 5, 8, and 12. Total sum: 4×10=404 \times 10 = 40. Fourth number: 40−(5+8+12)=1540 - (5+8+12) = 15.
The average of three exam scores is 90. Two scores are 85 and 95. Total needed: 3×90=2703 \times 90 = 270. Third score: 270−(85+95)=90270 - (85+95) = 90.

Explanation

This is like a math treasure hunt! You know the total value of all the items (the sum). You're told the value of all but one. To find the last one's value, just subtract what you already know from the grand total!

Section 4

Reaching a New Average

Property

To find the value needed to achieve a new average, calculate the required total for the new average and subtract the current total.

New Value=(New Average×New Quantity)−(Old Sum) \text{New Value} = (\text{New Average} \times \text{New Quantity}) - (\text{Old Sum})

Examples

Your average for 3 tests is 80. To get an average of 82 after 4 tests: New total needed: 4×82=3284 \times 82 = 328. Old total: 3×80=2403 \times 80 = 240. Score needed on the 4th test: 328−240=88328 - 240 = 88.
After 4 bowling games, your average is 100. To get a 102 average after 5 games: New total: 5×102=5105 \times 102 = 510. Old total: 4×100=4004 \times 100 = 400. Score needed in game 5: 510−400=110510 - 400 = 110.

Explanation

Time to level up your average! Figure out the total score you need for your new level (new average × new count). Then, subtract the score you already have (your current total) to see what you need on the next try to hit your goal.

Book overview

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

Continue this chapter

Chapter 6: Lessons 51-60, Investigation 6

  1. Lesson 1

    Lesson 51: Scientific Notation for Large Numbers

    LearnPractice
  2. Lesson 2

    Lesson 52: Order of Operations

    LearnPractice
  3. Lesson 3

    Lesson 53: Ratio Word Problems

    LearnPractice
  4. Lesson 4

    Lesson 54: Rate Word Problems

    LearnPractice
  5. Lesson 5Current

    Lesson 55: Average and Rate Problems with Multiple Steps

    LearnPractice
  6. Lesson 6

    Lesson 56: Plotting Functions

    LearnPractice
  7. Lesson 7

    Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

    LearnPractice
  8. Lesson 8

    Lesson 58: Symmetry

    LearnPractice
  9. Lesson 9

    Lesson 59: Adding Integers on the Number Line

    LearnPractice
  10. Lesson 10

    Lesson 60: Fractional Part of a Number, Part 1, Percent of a Number, Part 1

    LearnPractice
  11. Lesson 11

    Investigation 6: Classifying Quadrilaterals

    LearnPractice