Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 7: Ratios, Conversions, and Rates

Lesson 6: Other Rates

In this Grade 4 AMC Math lesson from The Art of Problem Solving: Prealgebra, students explore rates beyond speed, learning how to apply unit cancellation and conversion factors to solve problems involving typing rates, fill rates, and work rates. Drawing from Chapter 7 on Ratios, Conversions, and Rates, the lesson teaches students to recognize "per" as a signal for a rate and to set up expressions where unwanted units cancel to reveal the desired answer. Practice problems guide students through multi-step rate calculations, such as determining how long it takes to fill a pool or how many pages to prepare for a timed speech.

Section 1

Rate Definition and Structure

Property

A rate is a ratio that compares two different quantities, expressed as rate=quantitytime\text{rate} = \frac{\text{quantity}}{\text{time}}. The word "per" indicates a rate relationship, as in "miles per hour" or "words per minute".

Examples

Section 2

Multi-Step Rate Conversions

Property

For multi-step rate problems, chain conversion factors to cancel intermediate units:

Starting quantity×unit1unit2×unit3unit1×unit4unit3=Final quantity in unit4\text{Starting quantity} \times \frac{\text{unit}_1}{\text{unit}_2} \times \frac{\text{unit}_3}{\text{unit}_1} \times \frac{\text{unit}_4}{\text{unit}_3} = \text{Final quantity in unit}_4

Examples

Book overview

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Chapter 7: Ratios, Conversions, and Rates

  1. Lesson 1

    Lesson 1: What is a Ratio?

  2. Lesson 2

    Lesson 2: Multi-way Ratios

  3. Lesson 3

    Lesson 3: Proportions

  4. Lesson 4

    Lesson 4: Conversions

  5. Lesson 5

    Lesson 5: Speed

  6. Lesson 6Current

    Lesson 6: Other Rates

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Rate Definition and Structure

Property

A rate is a ratio that compares two different quantities, expressed as rate=quantitytime\text{rate} = \frac{\text{quantity}}{\text{time}}. The word "per" indicates a rate relationship, as in "miles per hour" or "words per minute".

Examples

Section 2

Multi-Step Rate Conversions

Property

For multi-step rate problems, chain conversion factors to cancel intermediate units:

Starting quantity×unit1unit2×unit3unit1×unit4unit3=Final quantity in unit4\text{Starting quantity} \times \frac{\text{unit}_1}{\text{unit}_2} \times \frac{\text{unit}_3}{\text{unit}_1} \times \frac{\text{unit}_4}{\text{unit}_3} = \text{Final quantity in unit}_4

Examples

Book overview

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

Continue this chapter

Chapter 7: Ratios, Conversions, and Rates

  1. Lesson 1

    Lesson 1: What is a Ratio?

  2. Lesson 2

    Lesson 2: Multi-way Ratios

  3. Lesson 3

    Lesson 3: Proportions

  4. Lesson 4

    Lesson 4: Conversions

  5. Lesson 5

    Lesson 5: Speed

  6. Lesson 6Current

    Lesson 6: Other Rates