Annette has 33 hours to spend training for an upcoming race. She completes her training by running full speed the distance of the race and walking back the same distance to cool down. If she runs at a speed of 9mph9⁢mph and walks back at a speed of 3mph3⁢mph, how long should she plan to spend walking back?

To solve this problem, we first need to understand that the total time Annette has for training is divided into two parts: the time she spends running and the time she spends walking back.

Let's denote the distance of the race as D (in miles), the running speed as R (in mph), the walking speed as W (in mph), and the total time as T (in hours).

From the problem, we know that: R = 9 mph W = 3 mph T = 33 hours

The time spent on running and walking can be calculated by the formula: time = distance/speed.

Since Annette runs and walks the same distance (D), we can write the total time as the sum of the time spent running and walking:

T = D/R + D/W

Substituting the given values into the equation, we get:

33 = D/9 + D/3

To solve for D, we first need to find a common denominator for the fractions, which is 9 in this case. So, we rewrite the equation as:

33 = D/9 + 3D/9

Combine like terms:

33 = 4D/9

To isolate D, multiply both sides of the equation by 9/4:

D = 33 * (9/4) = 74.25 miles

Now that we know the distance, we can find out how long Annette should plan to spend walking back. Using the formula time = distance/speed, we get:

Time walking = D/W = 74.25/3 = 24.75 hours

So, Annette should plan to spend approximately 24.75 hours walking back.