On a winter morning the temperature before sunrise was -11 f. the temperature then rose by 1/2 degree each hour for 7 hours before dropping by 2 1/4 degrees each hour for 3 hours. what was the temperature in degrees fahrenheit after 10 hours

Your starting temperature is -11.

Over the next 7 hrs, the temperature rose by 1/2°F for 7 hours, meaning it rose by (1/2)*7=(7/2)°F over the 7 hours.

Then, in the next 3 hours, it dropped by 2 1/4°F each hour. To do this by hand, convert 2 1/4 into an improper fraction. Multiply the denominator of the fraction part by the whole number: 2*4=8. Then add the numerator of the fraction to that value: 8+1=9. Put that final number of the denominator to get the improper fraction form of 2 1/4: 9/4. Now multiply 9/4 by 3: (9/4)*3=27/4. The temperature dropped by (27/4)°F over those 3 hours. 

Finally do the math. Add (7/2)°F to 11°F, then subtract (27/4)°F. You should get 7.75 or 7 3/4°F as the final temperature.