Write the mixed number 10 2/3 as an improper fraction.

Step-by-step solution

1. 1

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

   $10\frac{2}{3} = 10 + \frac{2}{3}$

   An improper fraction writes that whole sum over the single denominator $3$, so the job is to convert the $10$ 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 $10$ wholes are

   $10 \times 3 = 30$

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

3. 3

   Add the numerator. Now include the extra $\tfrac{2}{3}$:

   $30 + 2 = 32$

   so there are $32$ 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 $3$:

   $10\frac{2}{3} = \frac{32}{3}$

   Combining as fractions gives the same thing: $\tfrac{30}{3} + \tfrac{2}{3} = \tfrac{32}{3}$.

5. 5

   Sanity-check the size and the form. The same fractional part $\tfrac23$ sits on a larger whole number here, and only the multiplication changes: $10 \times 3 = 30$ instead of $4 \times 3 = 12$. The denominator never changes during the conversion, which is the fastest self-check available.

6. 6

   Reverse the process to confirm. Dividing back, $32 \div 3 = 10$ remainder $2$, which rebuilds $10\frac{2}{3}$ exactly. As a decimal both forms equal $10.666666666666666$, so the conversion is verified in both directions.