How do i solve a real world problem that involves finding the product of a fraction and a mixed number?

You solve it the same way you solve a textbook or classroom problem.

a) convert the mixed number to an improper fraction, then multiply the fractions in the usual way, then convert the result to a mixed number.

or

b) multiply the fraction by the integer part of the mixed number, multiply the fraction by the fractional part of the mixed number, and add the result. Sometimes, this is easier to do when you're doing it in your head.

or

c) divide the mixed number into an integer and another mixed number and do it in a fashion similar to b).

Example:
.. I have a recipe that calls for 5 1/3 cups of flour. I want to make 1/4 of the recipe. How much flour do I need?

a) (1/4)*(5 1/3)
.. = (1/4)*(16/3)
.. = (1*16)/(4*3)
.. = (4*4)/(4*3)
.. = 4/3
.. = 1 1/3 . . . . cups of flour

b) (1/4)*(5 1/3)
.. = (1/4)*5 +(1/4)*(1/3)
.. = 5/4 +1/12
.. = 64/48 . . . or . . . (1 1/4) +1/12
.. = 4*16/(3*16) . . . or . . . 1 +(3/12 +1/12)
.. = 4/3 . . . or . . . 1 +4/12
.. = 1 1/3 . . . . cups of flour

c) (1/4)*(5 1/3)
.. = (1/4)*(4 + 4/3)
.. = (1/4)*4 +(1/4)*(4/3)
.. = 1 1/3 . . . . cups of flour