SOLUTION: A and B working together can finish the job in 10 days. if A works 4 days and B works 3 days, one-third of the job shall be finished. how many days will it A to finish the job

Algebra ->  Rate-of-work-word-problems -> SOLUTION: A and B working together can finish the job in 10 days. if A works 4 days and B works 3 days, one-third of the job shall be finished. how many days will it A to finish the job      Log On


   

Question 1037379: A and B working together can finish the job in 10 days. if A works 4 days and B works 3 days, one-third of the job shall be finished. how many days will it A to finish the job
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Time for them to do one job each, a and b.
Rates together are additive, and each should be done as JOB/TIME.

system%281%2Fa%2B1%2Fb=1%2F10%2C%281%2Fa%294%2B%281%2Fb%293=1%2F3%29

system%28%28b%2Ba%29%2Fab=1%2F10%2C4%2Fa%2B3%2Fb=1%2F3%29

system%28a%2Bb=ab%2F10%2C%284b%2B3a%29%2Fab=1%2F3%29

system%28a%2Bb=ab%2F10%2C3a%2B4b=ab%2F3%29

system%2810a%2B10b=ab%2C9a%2B12b=ab%29

Isolate either variable in either equation and use for substitution.

10a-ab=-10b
a%2810-b%29=-10b
a%28b-10%29=10b
highlight_green%28a=10b%2F%28b-10%29%29

9%2810b%2F%28b-10%29%29%2B12b=b%2810b%2F%28b-10%29%29

90b%2F%28b-10%29%2B12b=10b%5E2%2F%28b-10%29

%28b-10%29%2890b%2F%28b-10%29%2B12b%29=%28b-10%29%2810b%5E2%2F%28b-10%29%29

90b%2B12b%28b-10%29=10b%5E2

90b%2B12b%5E2-120b=10b%5E2

2b%5E2-30b=0

b%5E2-15b=0

b%28b-15%29=0

Ignore b=0, and take highlight%28b=15%29. Now you can find a for the time A needs to do one whole job.

Return to this one: 10a%2B10b=ab
which became a=10b%2F%28b-10%29;
a=10%2A15%2F%2815-10%29
a=10%2A15%2F5
highlight%28a=30%29


Now you should be able to setup an equation to help answer the question.
Let A finish the job of doing the unfinished two-thirds of a piece of work.
Let x be the number of days for A to do two-thirds of a piece of work.
%281%2F30%29%2Ax=2%2F3

Answer by ikleyn(52792) About Me  (Show Source):
You can put this solution on YOUR website!
.
A and B working together can finish the job in 10 days.
If A works 4 days and B works 3 days, one-third of the job shall be finished.
How many days will it A to finish the job
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There is much easier and more straightforward way to solve the problem, if you correctly choose the unknowns. 

Let "a" be the rate-of-work of A, and
let "b" be the rate-of-work of B.

Then from the condition you immediately get these two equations for two unknowns

10a + 10b = 1,    (1)
 4a +  3b = 1%2F3.   (2)

Multiply equation (2) by 3 (both sides). Then you can rewrite the system in the form

10a + 10b = 1,    (1')
12a +  9b = 1.    (2')

Now, multiply equation (1') by 6, multiply equation (2') by 5 and distract.
In this way you eliminate "a" and obtain a single equation for "b"

60b - 45b = 5 - 4,   or   15b = 1,   b = 1%2F15.

Then from (1)  10a = 1 - 10b = 1 - 10%2F15 = 1 - 2%2F3 = 1%2F3,
Hence, a = 1%2F30.
The rate of "a" is 1%2F30 job-per-day; the rate of "b" is 1%2F15 job-per-day.

Answer. A will complete the job in 30 days.  B will complete the job in 15 days.

The lesson to learn from this solution: choose correctly the unknowns when solving rate-of-work and joint-work problems.