Learn on PengiEureka Math, Grade 5Chapter 23: Multiplication of a Fraction by a Fraction

Lesson 4: Solve word problems using tape diagrams and fraction-by-fraction multiplication.

In this Grade 5 Eureka Math lesson from Chapter 23, students learn to solve word problems involving fraction-by-fraction multiplication by drawing and interpreting tape diagrams. They practice setting up multiplication expressions such as finding a fraction of a fraction and use visual models to find products like 1/3 × 2/5 = 2/15. The lesson builds fluency with multiplying fractions and connects visual representations to real-world problem-solving contexts.

Section 1

Calculate a Fractional Part of a Whole

Property

To find a fractional part of a whole number, you multiply the fraction by the whole. This relationship can be expressed as:

Part=Fraction×Whole \text{Part} = \text{Fraction} \times \text{Whole}

Examples

Section 2

Solving Fraction of a Remainder Problems

Property

To find a fraction of a remainder, you multiply the second fraction by the part that remains from the whole. If an initial fraction ab\frac{a}{b} is taken from a whole, the remainder is (1−ab)(1 - \frac{a}{b}) of the whole. The final part, which is a fraction cd\frac{c}{d} of the remainder, is calculated as:

Part=cd×(1−ab)×Whole \text{Part} = \frac{c}{d} \times \left(1 - \frac{a}{b}\right) \times \text{Whole}

Examples

Section 3

Find the Whole Given a Final Part

Property

To find the original whole from a final part, use the known value of the part to determine the value of a single unit in a tape diagram. The original whole is the value of one unit multiplied by the total number of units that represent the whole.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Calculate a Fractional Part of a Whole

Property

To find a fractional part of a whole number, you multiply the fraction by the whole. This relationship can be expressed as:

Part=Fraction×Whole \text{Part} = \text{Fraction} \times \text{Whole}

Examples

Section 2

Solving Fraction of a Remainder Problems

Property

To find a fraction of a remainder, you multiply the second fraction by the part that remains from the whole. If an initial fraction ab\frac{a}{b} is taken from a whole, the remainder is (1−ab)(1 - \frac{a}{b}) of the whole. The final part, which is a fraction cd\frac{c}{d} of the remainder, is calculated as:

Part=cd×(1−ab)×Whole \text{Part} = \frac{c}{d} \times \left(1 - \frac{a}{b}\right) \times \text{Whole}

Examples

Section 3

Find the Whole Given a Final Part

Property

To find the original whole from a final part, use the known value of the part to determine the value of a single unit in a tape diagram. The original whole is the value of one unit multiplied by the total number of units that represent the whole.

Examples