Question

Two workers in a holiday boutique are filling stockings with small gifts and candy. Kate has already filled 5 stockings and will continue to fill them at a rate of 4 stocking per hour. Todd, who just arrived to help, can fill 5 stockings per hour. At some point, Todd will catch up with Kate and they will have completed the same number of stockings. How log will it take for Todd to catch up?

Answer 1

This question is stated in a complicated way, but all the information we need is right there waiting to be untangled.

We'll start the clock when Todd arrives.  At that time:

-- Kate has 5 done.  Todd has none yet.  Todd is 5 units behind.

From then on:

-- The clock is running.  Kate adds 4 an hour to her total.  Todd adds 5 an hour.

-- She started out 5 ahead of Todd when he arrived, but Todd does 1 more than Kate every hour.

-- So Kate's 'lead' shrinks by 1 every hour.

-- So Todd will catch up with Kate in 5 hours.

That's the answer to the question ... How long ?  It doesn't ask us how many stockings have been filled, but that's easy for us to figure out:

-- Kate had 5 done when the clock started.  She fills 4 every hour.  After 5 hours, she has (5 x 4) = 20 more filled, and a total of 25 ready to sell.

-- Todd started out with none done.  He fills 5 every hour.  After 5 hours, he has (5 x 5) = 25 filled and ready to sell.  He has caught up with Kate in 5 hours.