# Using one variable, what time did they finish the float in the problem below? Don, Ric and Mike were in a committee to build a float for the homecoming parade. Each of them thought that he could build it alone in 16 hours. Don started work at 8:00 a.m., Ric at 8:30 a.m. and Mike at 9:00 a.m. What time did they finish the float? (use one variable) Given:

One guy can finish the boat alone in 16 hours.

Don start work at 8.00 am

Ric start work at 8.30 am

Mike at 9.00 am

Required:

The time at when they finished the boat.

Solution:

Let's calculate the amount of work to be done to finish the boat.

The amount of man-hours needed (A man hour is the amount of work a single man can do in an hour)

This work can be done by a single person in 16 hours. Therefore the amount of work, W in man-hours is,

W = 1 x 16 man-hours

W = 16.

Now if we assume that the total time consumed by the given working pattern is t, we can derive expressions for each person's work.
The sum of these individual works must be equal to 16.

Amount of work Don did = 1 x time consumed.

At the end of work, Don would have worked all the t amount of hours.

Therefore amount of work Don did = 1 x t = t

In he same manner,

Amount of work Ric did = 1 x (t-0.5) = (t-0.5)

Amount of work Mike did = 1 x (t-1) = (t-1)

The sum of above three must be 16.

t+(t-0.5)+(t-1) = 16

3t-1.5 = 16

t = 17.5/3 hours.

t = 5 hours and 50 minutes

Therefore the time taken to finish the boat is 5 hours and 50 minutes.

Checking:

Now this time must be greater than the time to finish the boat if all three people worked same t hours together from the beginning. That time would be t'.

t' = 16/3 = 5 hours and 20 minutes.

t = 5 hours and 50 minutes.

So t>t'

Therefore this answer must be correct. To be more precise, we can calculate each person's work and add them together and then check whether they sum up to 16.

The amount of work Don did = 1 x 35/6 = 35/6

The amount of work Ric did = 1 x (35/6 -1/2) = 16/3

The amount of work Mike did = 1 x (35/6 -1) = 29/6

The sum of work = 35/6 +16/3+29/6 = 96/6 = 16.

Therefore the answer is correct. It is the same amount of work.