You know that whole numbers can be written as fractions ($1 = \dfrac{3}{3}$, $2 = \dfrac{6}{3}$) and how to add like fractions. Put those together and an amount bigger than 1 can be written two ways.
Key idea
$2\dfrac{1}{3}$ (a mixed number) and $\dfrac{7}{3}$ (a fraction greater than 1) name the same amount. Mixed to fraction: wholes × denominator, plus the numerator: $2 \times 3 + 1 = 7$, so $\dfrac{7}{3}$. Fraction to mixed: divide, $7 \div 3 = 2$ R 1, so $2\dfrac{1}{3}$.
Why it works
Each whole is 3 thirds. Two wholes are $3 + 3 = 6$ thirds, and the extra $\dfrac{1}{3}$ makes 7 thirds. Going back, every 3 thirds you can group make one whole; 7 thirds make 2 groups with 1 third left over.
Worked example: Mixed number to fraction
Write $3\dfrac{2}{5}$ as a fraction.
- Each whole is 5 fifths, so 3 wholes are $3 \times 5 = 15$ fifths.
- Add the 2 fifths: $15 + 2 = 17$.
Answer: $\dfrac{17}{5}$
Worked example: Fraction to mixed number
Write $\dfrac{11}{4}$ as a mixed number.
- How many groups of 4 fourths are in 11 fourths? $11 \div 4 = 2$ R 3.
- 2 wholes and 3 fourths are left.
Answer: $2\dfrac{3}{4}$
Common mistake
Multiplying only the whole number ($3 \times 2 = 6$, so "$\dfrac{6}{5}$") or just adding $3 + 2$. The wholes must be turned into fifths first: 3 wholes are 15 fifths, then add the 2 fifths. Check: $\dfrac{17}{5}$ is a bit more than $\dfrac{15}{5} = 3$.