Lesson 12: Represent Multiplication and Division Situations

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:

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.

Property

To find a fractional part of a quantity, you multiply the fraction by the quantity. This applies to situations involving scaling recipes, calculating distances, or finding a portion of a given amount. The operation is represented as:

Examples

• A recipe calls for $\frac{3}{4}$ cup of sugar. If you are making $\frac{1}{2}$ of the recipe, how much sugar do you need?

• A runner wants to complete $\frac{2}{3}$ of a race that is $\frac{9}{10}$ of a mile long. What distance did the runner cover?

Explanation

This skill involves translating real-world scenarios into fraction multiplication problems. Key phrases like "fraction of a quantity" or "part of a total" often indicate that multiplication is needed. To solve, you multiply the numerators together and the denominators together to find the resulting fraction. This skill extends the concept of "fraction of a fraction" to a wider variety of practical applications.

Property

A real-world situation involving sharing or splitting a fractional amount ($\frac{1}{b}$) into a number of equal groups ($c$) can be solved using division. The equation is:

Examples

• There is $\frac{1}{2}$ of a pizza left. If 3 friends share it equally, what fraction of the whole pizza does each friend get?

• A runner completes $\frac{1}{4}$ of a race in 5 minutes, running at a constant speed. What fraction of the total race distance does the runner complete each minute?

Explanation

These problems require you to divide a unit fraction by a whole number. To solve, identify the initial fractional amount and the number of equal groups it is being divided into. Dividing a unit fraction by a whole number results in a smaller unit fraction. This is because you are splitting an existing part into even smaller pieces.

Property

Dividing a whole number $c$ by a unit fraction $\frac{1}{b}$ is equivalent to multiplying the whole number by the denominator $b$. This answers the question: "How many pieces of size $\frac{1}{b}$ fit into $c$ wholes?"

Examples

• A baker has 4 pounds of flour. If each cake recipe requires $\frac{1}{3}$ of a pound of flour, how many cakes can the baker make?

• A relay race is 5 miles long. If each runner runs for $\frac{1}{2}$ of a mile, how many runners are needed for the race?

Explanation

This skill involves situations where you need to find out how many fractional parts fit into a whole number. For example, if you have 2 pizzas and you want to know how many $\frac{1}{4}$-pizza slices there are, you are solving $2 \div \frac{1}{4}$. Since each whole pizza has 4 quarter slices, 2 pizzas would have $2 \times 4 = 8$ slices. Dividing a whole number by a unit fraction is the same as multiplying the whole number by the denominator of the fraction.