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

Lesson 2: Multi-way Ratios

In this Grade 4 AMC Math lesson from The Art of Problem Solving: Prealgebra, students learn how to extend two-way ratios to multi-way ratios that compare three or more quantities simultaneously. They practice simplifying multi-part ratios by finding the greatest common factor of all terms, including ratios involving fractions and mixed numbers. The lesson also teaches students how to interpret a multi-way ratio as parts of a whole and how to extract simpler two-way ratios from a larger multi-part ratio.

Section 1

Simplify multi-way ratios using GCF

Property

To simplify a multi-way ratio a:b:c:...a:b:c:..., find the greatest common factor (GCF) of all terms and divide each term by the GCF: aGCF:bGCF:cGCF:...\frac{a}{\text{GCF}}:\frac{b}{\text{GCF}}:\frac{c}{\text{GCF}}:...

Examples

Section 2

Simplify fractional multi-way ratios

Property

To simplify a multi-way ratio with fractions, multiply all terms by the least common denominator (LCD) to eliminate fractions, then simplify the resulting integer ratio if possible: ab:cd:ef=a⋅LCD:c⋅LCD:e⋅LCD\frac{a}{b} : \frac{c}{d} : \frac{e}{f} = a \cdot \text{LCD} : c \cdot \text{LCD} : e \cdot \text{LCD}

Examples

Section 3

Parts-of-the-whole method for multi-way ratios

Property

For a multi-way ratio a:b:ca:b:c, each part represents a fraction of the total where:

First quantity=aa+b+c×total\text{First quantity} = \frac{a}{a+b+c} \times \text{total}
Second quantity=ba+b+c×total\text{Second quantity} = \frac{b}{a+b+c} \times \text{total}
Third quantity=ca+b+c×total\text{Third quantity} = \frac{c}{a+b+c} \times \text{total}

Examples

Section 4

Combine two-part ratios into multi-way ratios

Property

To combine separate two-part ratios with a common term, scale each ratio so the common term has the same value, then write as a multi-way ratio: If a:b=m:na:b = m:n and b:c=p:qb:c = p:q, then a:b:c=mp:nq:nq⋅qpa:b:c = mp:nq:nq \cdot \frac{q}{p}

Examples

Book overview

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Chapter 7: Ratios, Conversions, and Rates

  1. Lesson 1

    Lesson 1: What is a Ratio?

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  2. Lesson 2Current

    Lesson 2: Multi-way Ratios

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  3. Lesson 3

    Lesson 3: Proportions

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  4. Lesson 4

    Lesson 4: Conversions

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  5. Lesson 5

    Lesson 5: Speed

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  6. Lesson 6

    Lesson 6: Other Rates

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Lesson overview

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Section 1

Simplify multi-way ratios using GCF

Property

To simplify a multi-way ratio a:b:c:...a:b:c:..., find the greatest common factor (GCF) of all terms and divide each term by the GCF: aGCF:bGCF:cGCF:...\frac{a}{\text{GCF}}:\frac{b}{\text{GCF}}:\frac{c}{\text{GCF}}:...

Examples

Section 2

Simplify fractional multi-way ratios

Property

To simplify a multi-way ratio with fractions, multiply all terms by the least common denominator (LCD) to eliminate fractions, then simplify the resulting integer ratio if possible: ab:cd:ef=a⋅LCD:c⋅LCD:e⋅LCD\frac{a}{b} : \frac{c}{d} : \frac{e}{f} = a \cdot \text{LCD} : c \cdot \text{LCD} : e \cdot \text{LCD}

Examples

Section 3

Parts-of-the-whole method for multi-way ratios

Property

For a multi-way ratio a:b:ca:b:c, each part represents a fraction of the total where:

First quantity=aa+b+c×total\text{First quantity} = \frac{a}{a+b+c} \times \text{total}
Second quantity=ba+b+c×total\text{Second quantity} = \frac{b}{a+b+c} \times \text{total}
Third quantity=ca+b+c×total\text{Third quantity} = \frac{c}{a+b+c} \times \text{total}

Examples

Section 4

Combine two-part ratios into multi-way ratios

Property

To combine separate two-part ratios with a common term, scale each ratio so the common term has the same value, then write as a multi-way ratio: If a:b=m:na:b = m:n and b:c=p:qb:c = p:q, then a:b:c=mp:nq:nq⋅qpa:b:c = mp:nq:nq \cdot \frac{q}{p}

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?

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  2. Lesson 2Current

    Lesson 2: Multi-way Ratios

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  3. Lesson 3

    Lesson 3: Proportions

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  4. Lesson 4

    Lesson 4: Conversions

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  5. Lesson 5

    Lesson 5: Speed

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  6. Lesson 6

    Lesson 6: Other Rates

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