Dividing unit fractions and whole numbers Knowledge-Based Study

Study notes

1. What is a Unit Fraction?

   A unit fraction is a fraction where the numerator is 1 and the denominator is a positive whole number, such as $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, or $\frac{1}{8}$. It represents one equal part of a whole divided into several equal shares.

   When dividing with fractions, the relationship between the pieces and the whole depends on whether a whole number is being split into unit fractions, or a unit fraction is being shared among whole parts.

   Whole Number Divided by a Unit Fraction

   Consider the division problem: $$3 \div \frac{1}{4}$$

   This asks: "How many one-fourths are in 3 wholes?"

   • Each whole contains 4 one-fourth pieces: $1 = \frac{4}{4}$.
   • Therefore, 3 wholes contain: $$3 \times 4 = 12$$ pieces of size $\frac{1}{4}$.

   Hence, $$3 \div \frac{1}{4} = 12$$

   General Rule: Dividing a whole number $a$ by a unit fraction $\frac{1}{b}$ results in a larger whole number: $$a \div \frac{1}{b} = a \times b$$

   Unit Fraction Divided by a Whole Number

   Now consider the reverse operation: $$\frac{1}{3} \div 4$$

   This asks: "If $\frac{1}{3}$ of a whole is split equally into 4 parts, what fraction of the original whole is each part?"

   • Imagine 1 whole divided into 3 equal columns. One column represents $\frac{1}{3}$.
   • If that single column is divided into 4 equal pieces, the entire whole would contain $3 \times 4 = 12$ equal pieces in total.
   • Each resulting share is 1 of those 12 equal pieces.

   Hence, $$\frac{1}{3} \div 4 = \frac{1}{12}$$

   General Rule: Dividing a unit fraction $\frac{1}{b}$ by a whole number $c$ (where $c > 0$) yields a smaller unit fraction: $$\frac{1}{b} \div c = \frac{1}{b \times c}$$

   The Multiplication Connection

   Dividing by a number is equivalent to multiplying by its reciprocal (the inverted fraction):

   • The reciprocal of a whole number $n$ (written as $\frac{n}{1}$) is $\frac{1}{n}$.
   • The reciprocal of a unit fraction $\frac{1}{n}$ is $\frac{n}{1} = n$.

   Examples

   Example 1: Evaluate $5 \div \frac{1}{6}$.

   $$5 \div \frac{1}{6} = 5 \times 6 = 30$$

   Example 2: Evaluate $\frac{1}{5} \div 6$.

   $$\frac{1}{5} \div 6 = \frac{1}{5} \times \frac{1}{6} = \frac{1}{30}$$

   Key Takeaways

   • Whole $\div$ Unit Fraction: The answer is a whole number greater than the original dividend because you are counting how many small pieces fit into larger wholes.
   • Unit Fraction $\div$ Whole: The answer is a unit fraction smaller than the original dividend because you are sharing an already small part into even smaller pieces.
   • Order Matters: Division is not commutative. $4 \div \frac{1}{2} = 8$, whereas $\frac{1}{2} \div 4 = \frac{1}{8}$. Always identify whether you are starting with a whole or a fractional part.