Solve Word Problems: Fraction of a Fraction
Solve Word Problems: Fraction of a Fraction is a Grade 5 math skill from Illustrative Mathematics Chapter 3 (Multiplying and Dividing Fractions) where students identify real-world situations requiring fraction-of-fraction multiplication and set up the expression (1/c) × (1/b) = 1/(c × b). Common contexts include dividing remaining fractions of food or adjusting partial recipe quantities. The word "of" signals multiplication.
Key Concepts
Property To find a fraction of a fraction, you multiply the two fractions. This is equivalent to dividing the first fraction by the reciprocal of the second fraction. For a unit fraction $\frac{1}{b}$ and a whole number $c$, taking $\frac{1}{c}$ of $\frac{1}{b}$ is calculated as: $$\frac{1}{c} \times \frac{1}{b} = \frac{1}{c \times b}$$.
Examples Maria has $\frac{1}{2}$ of a chocolate bar left. She eats $\frac{1}{3}$ of the leftover chocolate. The fraction of the original chocolate bar she ate is $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$. A recipe calls for $\frac{1}{4}$ cup of flour. If you only want to make half of the recipe, you would need $\frac{1}{2}$ of the flour. The amount of flour needed is $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$ of a cup.
Explanation This skill applies the concept of fraction division to real world scenarios. Often, a word problem will ask for a "fraction of a fraction," such as "one half of the remaining one third." The word "of" in this context signifies multiplication. Solving these problems is equivalent to dividing the initial fraction by a whole number, connecting the application directly to the visual models of partitioning a fractional piece.
Common Questions
How do you solve a fraction of a fraction word problem?
Identify the two fractions in the problem. The word of signals multiplication. Multiply: (1/c) × (1/b) = 1/(c × b). For example, Maria has 1/2 of a chocolate bar and eats 1/3 of what remains: (1/3) × (1/2) = 1/6 of the original bar.
What real-world situations involve fraction of a fraction?
Situations like: eating 1/3 of the remaining 1/2 pizza, making half of a recipe that calls for 1/4 cup of flour, or using 1/2 of a 1/3-acre plot. The word of between two fractions always signals multiplication.
What chapter covers fraction of a fraction word problems in Illustrative Mathematics Grade 5?
Solving word problems: fraction of a fraction is covered in Chapter 3 of Illustrative Mathematics Grade 5, titled Multiplying and Dividing Fractions.
What is an example of a fraction of a fraction problem?
A recipe calls for 1/4 cup of flour. Making 1/2 of the recipe: (1/2) × (1/4) = 1/8 cup. The expression connects "1/2 of the amount" to multiplication by 1/2.
How is fraction of a fraction division related to unit fraction multiplication?
Finding 1/c of a unit fraction 1/b is the same as dividing 1/b by c (partitive division). Both give 1/(b × c). The word problem context determines which interpretation applies, but the calculation is the same.