how to get the answer

Question

Janice and Donald worked together for 2
hours to build a picnic table, after which Donald continued working for 1
hour without Janice to finish the job. If each is working alone, Janice typically takes 2
less hours than Donald to build a picnic table. Based on this information, how long would it have taken for Janice to build the picnic table alone?

Answer 1

Donald builds the picnic table in d hours ---> 1/d of the picnic table per hour

Janice builds the picnic table in d-2 hours ---> 1/(d-2) of the picnic table per hour

So to complete the picnic table, They work together for 2 hours and then Donald works alone for 1 hour.

2 hours * [ 1/d + 1/(d-2) ] per table per hour + 1 hour * 1/d per table per hour = 1 table

the units cancel:

2/d + 2/(d-2) + 1/d = 1 <-- distributive property: distributes the 2

3/d + 2/(d-2) = 1 <--- combines like terms 2(1/d) + 1/d = 3/d

3(d-2) + 2d = d(d-2) <---- multiplies both sides by d(d-2)

3d - 6 + 2d = d^2 - 2d

5d - 6 = d^2 - 2d

0 = d^2 - 7d + 6

0 = ( d - 6 )(d - 1 )

d=6 ---> Donald takes 6 hours and Janice takes 4 hours

d=1 --> disqualified as it results in negative times.

check:

2(1/6 + 1/4) + 1/6 =

2(2/12 + 3/12) + 1/6 =

2(5/12) + 1/6 =

5/6 + 1/6 = 1

Janice can build the table in 4 hours.

Donald can build the table in 6 hours.

Working together, they build 1/4 + 1/6 = 5/12 of the table per hour...

or the whole table in 12/5 = 2 and 2/5 hours.