Predict the result
Convert three and one sixth into an improper fraction.
How many sixths make its three wholes?

Set up the model
Here a positive mixed number means three plus one sixth.
Three wholes contain eighteen sixths. The denominator remains six.

Work the example
Multiply three by six, then add one. The numerator is nineteen.
Keep denominator six: the exact improper fraction is nineteen sixths.
For the reverse direction, divide nineteen by six: quotient three and remainder one.
So nineteen sixths equals three plus one sixth. Keep the fraction exact.

Check the result
Now convert two and three fourths into an improper fraction.
Keep denominator four. Take eight seconds to think.
[Eight-second silent pause to work; the answer remains hidden.]
Two times four plus three equals eleven.
Keep denominator four: the exact improper fraction is eleven fourths.

Interpret the result
For these positive mixed numbers, add the whole part and the proper fraction.
Multiply the whole by the denominator, then add the numerator; keep that denominator.
To reverse, use the quotient and remainder, and keep the fraction exact.