Divide a Whole Number by a Unit Fraction
Divide a Whole Number by a Unit Fraction is a Grade 5 math skill from Illustrative Mathematics Chapter 3 (Multiplying and Dividing Fractions) where students learn that w ÷ (1/d) = w × d. Dividing a whole number by a unit fraction answers the question of how many fractional pieces fit into the whole, and the answer is always larger than the original whole number. This concept is the inverse of dividing a fraction by a whole number.
Key Concepts
Property To divide a whole number by a unit fraction, you can multiply the whole number by the denominator of the fraction. For a whole number $w$ and a unit fraction $\frac{1}{d}$: $$w \div \frac{1}{d} = w \times d$$.
Examples $6 \div \frac{1}{4} = 6 \times 4 = 24$ $10 \div \frac{1}{2} = 10 \times 2 = 20$ $3 \div \frac{1}{8} = 3 \times 8 = 24$.
Explanation Dividing by a unit fraction is like asking "how many of this fraction fit into the whole number?" For example, $6 \div \frac{1}{4}$ asks how many $\frac{1}{4}$ sized pieces are in 6 wholes. Since there are 4 fourths in 1 whole, there must be $6 \times 4$ fourths in 6 wholes. This shows that dividing by a unit fraction is the same as multiplying by its denominator.
Common Questions
How do you divide a whole number by a unit fraction?
Multiply the whole number by the denominator of the unit fraction. The formula is w ÷ (1/d) = w × d. For example, 6 ÷ (1/4) = 6 × 4 = 24, because there are 4 quarter-pieces in each whole, and 6 wholes contain 24 quarter-pieces.
Why is dividing a whole number by a unit fraction the same as multiplying by the denominator?
Dividing by 1/d asks how many 1/d-sized pieces fit into the whole. Since there are d pieces of size 1/d in each whole unit, and you have w whole units, the total is w × d.
What chapter covers whole number division 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?
10 ÷ (1/2) = 10 × 2 = 20. This means there are 20 half-sized pieces in 10 wholes. Similarly, 3 ÷ (1/8) = 3 × 8 = 24, meaning 24 eighth-sized pieces fit into 3 wholes.
How is dividing by a unit fraction different from dividing by a whole number?
Dividing by a unit fraction makes the quotient larger than the dividend, while dividing by a whole number greater than 1 makes the quotient smaller. This is because unit fractions are less than 1, so more of them fit into any given whole.