Write the mixed number 20 1/2 as an improper fraction.

Step-by-step solution

1. 1

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

   $20\frac{1}{2} = 20 + \frac{1}{2}$

   An improper fraction writes that whole sum over the single denominator $2$, so the job is to convert the $20$ wholes into halves and add on the halves you already have.

2. 2

   Multiply the whole number by the denominator. Each whole equals $\tfrac{2}{2}$, i.e. $2$ halves, so $20$ wholes are

   $20 \times 2 = 40$

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

3. 3

   Add the numerator. Now include the extra $\tfrac{1}{2}$:

   $40 + 1 = 41$

   so there are $41$ halves altogether.

4. 4

   Keep the original denominator. The size of each piece never changed — only how many you are counting — so the denominator stays $2$:

   $20\frac{1}{2} = \frac{41}{2}$

   Combining as fractions gives the same thing: $\tfrac{40}{2} + \tfrac{1}{2} = \tfrac{41}{2}$.

5. 5

   Sanity-check the size and the form. Halves are the easiest case to check mentally: $20\tfrac12 = 20.5$, and $\tfrac{41}{2} = 20.5$. Notice the numerator is odd — it must be, because an even numerator over $2$ would reduce to a whole number.

6. 6

   Reverse the process to confirm. Dividing back, $41 \div 2 = 20$ remainder $1$, which rebuilds $20\frac{1}{2}$ exactly. As a decimal both forms equal $20.5$, so the conversion is verified in both directions.