rational expressions?

Question

Please help me with these two problems. I need to know quickly if possible.

If Sarah Can do a job in 5hrs and Dick and Sarah working together can do the same job in 2 hours, find how long it takes Dick to do the job alone.

The numerator of a fraction is 4 less than the denominator. If both the numerator and denominator are increased by 2, the resulting fraction is equivalent to 2/3. Find the fraction.

Answer 1

Sarah can do 1/5 of the job in 1hr.
Sarah & Dick together can do 1/2 of the job in 1hr.
Therefore Dick alone can do 1/2 - 1/5 = (5 - 2)/10 = 3/10 of the job in 1hr, and he will take 10/3 or 3 1/3 hr. to do the whole job.

Let the numerator be x.
Then the denominator is x + 4.
Increasing both by 2 gives x + 2 and x + 6.
Therefore:
(x + 2) / (x + 6) = 2 / 3
Multiplying by 3(x + 6):
3(x + 2) = 2(x + 6)
3x + 6 = 2x + 12
x = 6.
The fraction is therefore 6 / 10.

Answer 2

Sarah does 1/5 of the work in an hour.
Dick plus Sarah do 1/2 of the work in an hour.

1/2 - 1/5 = 5/10 - 2/10 = 3/10
Dick does 3/10 of the work in an hour.
Dick can do the work in 10/3 hrs. = 3 1/3 hrs

(x-4)/x = original fraction
(x-4+2)/(x+2) = 2/3
(x-2) / (x+2) = 2/3
3(x-2) = 2(x+2)
3x - 6 = 2x + 4
x = 10
Check 6/10 becomes 8/12 = 2/3 check

Answer 3

3hrs

3/5