Write the mixed number 6 1/3 as an improper fraction.

Step-by-step solution

1. 1

   Understand what the mixed number means. A mixed number is a sum in disguise:

   $6\frac{1}{3} = 6 + \frac{1}{3}$

   An improper fraction writes that whole sum over the single denominator $3$, so the job is to convert the $6$ wholes into thirds and add on the thirds you already have.

2. 2

   Multiply the whole number by the denominator. Each whole equals $\tfrac{3}{3}$, i.e. $3$ thirds, so $6$ wholes are

   $6 \times 3 = 18$

   thirds in total. This is the step the shortcut “multiply, add, keep the bottom” compresses.

3. 3

   Add the numerator. Now include the extra $\tfrac{1}{3}$:

   $18 + 1 = 19$

   so there are $19$ thirds altogether.

4. 4

   Keep the original denominator. The size of each piece never changed — only how many you are counting — so the denominator stays $3$:

   $6\frac{1}{3} = \frac{19}{3}$

   Combining as fractions gives the same thing: $\tfrac{18}{3} + \tfrac{1}{3} = \tfrac{19}{3}$.

5. 5

   Sanity-check the size and the form. Compare this with $6\tfrac{7}{12}$: same whole number, but a unit fraction on top. $6 \times 3 = 18$ and $18 + 1 = 19$, so the numerator is only one more than a multiple of the denominator — the signature of a mixed number whose fractional part is $\tfrac1d$.

6. 6

   Reverse the process to confirm. Dividing back, $19 \div 3 = 6$ remainder $1$, which rebuilds $6\frac{1}{3}$ exactly. As a decimal both forms equal $6.333333333333333$, so the conversion is verified in both directions.