Let "a", "b" and "c" be the rate-of-work of each of the persons A, B, and C respectively.
We are given that
a + b = , (1)
b + c = , (2)
a + c = . (3)
To solve the system (1), (2), (3), let us start adding the equations (1), (2) and (3).
You will get
2a + 2b + 2c = = = = 1.
Hence,
a + b + c = 1/2. (4)
Thus we just found the combined rate-of-work of the three persons working together. It is of job per day.
Now we have to find individual rate-of-work for each person. For it, let us first subtract the equation (1) from (4). You will get
c = = .
Next, subtract the equation (2) from (4). You will get
a = = .
Finally, subract the equation (3) from (4). You will get
b = = = = .
Answer. The individual rates of work are for C, for A and for B (in job-per-day units).
So, A will complete the job in 6 days; B will complete the job in 12 days; and C will complete the job in 4 days.
For many other similar solved problems See the lesson
- Joint-work problems for 3 participants
in this site.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this textbook under the topic
"Rate of work and joint work problems" of the section "Word problems".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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Notice. The way on how I solved the system of equations is THE STANDARD WAY of dealing with such special systems.
The way that @gosgarithmetic proposes in his solution is the way to NOWHERE.