Suppose your average, after taking 4 quizzes, is 71 (out of 100). what must your average be on the next 5 quizzes to increase your average to 86 out of 100?

Short 98
Remark
when you have an average of 71, it's like saying that you took 4 quizzes and got 71 on all of them. 
Step One
Find the total point count for the 4 quizzes.
T_4 = 71 * 4 = 284

Step Two
Create an equation that will give you the necessary point count to average 86.

284 + 5*x = 86 * 9
Why did we do this? It is because you have 9 quizzes altogether. All nine must be something that gives a point count of 86 on each quiz.  You have to put together 9 such quizzes with a total count of 774.

Step three 
Solve the equation.
284 + 5x = 774  Subtract 284 from both sides.
5x = 774 - 284
5x = 490   Divide by 5
x = 490/5
x = 98

You must get an average of 98 on the next 5 quizzes

98 <<<<<