Divide a Whole Number by a Unit Fraction
Divide a Whole Number by a Unit Fraction (second entry) is a Grade 5 math skill from Illustrative Mathematics Chapter 3 (Multiplying and Dividing Fractions) where students use the rule a ÷ (1/b) = a × b to solve problems asking how many fractional pieces fit into a whole number. Real-world contexts include portions of cups, lengths cut into fractional pieces, and equal distributions.
Key Concepts
Property To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the fraction. This is because you are finding how many fractional parts fit into the whole number. $$a \div \frac{1}{b} = a \times b$$.
Examples How many $\frac{1}{4}$ cup servings are in 3 cups of sugar? $$3 \div \frac{1}{4} = 3 \times 4 = 12$$ A ribbon is 5 meters long. How many $\frac{1}{3}$ meter pieces can be cut from it? $$5 \div \frac{1}{3} = 5 \times 3 = 15$$.
Explanation Dividing a whole number by a unit fraction asks the question, "How many of these fractional pieces fit into the whole amount?" For example, $2 \div \frac{1}{4}$ is asking how many quarter pieces fit into 2 wholes. Since there are 4 quarters in 1 whole, there must be $2 \times 4 = 8$ quarters in 2 wholes. This concept is the inverse of dividing a fraction by a whole number.
Common Questions
How many 1/4-cup servings are in 3 cups?
Use a ÷ (1/b) = a × b: 3 ÷ (1/4) = 3 × 4 = 12 servings. Dividing by 1/4 asks how many quarter-pieces fit into 3, and since there are 4 quarters in each whole, 3 wholes contain 12 quarters.
Why is dividing by a unit fraction the same as multiplying by the denominator?
Dividing by 1/b asks how many 1/b pieces fit into the whole. Since there are b pieces of size 1/b in each unit, and you have a units, the total count is a × b.
What chapter covers dividing whole numbers by unit fractions in Illustrative Mathematics Grade 5?
Dividing a whole number by a unit fraction is covered in Chapter 3 of Illustrative Mathematics Grade 5, titled Multiplying and Dividing Fractions.
What is an example of dividing a whole number by a unit fraction?
A ribbon is 5 meters long. How many 1/3-meter pieces? 5 ÷ (1/3) = 5 × 3 = 15 pieces. Another: how many 1/2-meter pieces in 7 meters? 7 ÷ (1/2) = 7 × 2 = 14 pieces.
Is the result of dividing a whole number by a unit fraction always larger than the original whole number?
Yes, always. Since the unit fraction is less than 1, many of them fit into each whole unit. So the quotient (number of pieces) is always greater than the whole number you started with.