A and B caan finish typing a manuscript in 8/3 days; B and C together, in 40/9 days; A and C together in 20/7 days. How long will each, working alone type the manuscript?
work done = (actual time worked) / (req'd time to finish a task)
'completed' work = 1
so let
A = time req'd for A
B = time req'd for B
C = time req'd for C
there will be 3 equations:
(8/3A) + (8/3B) = 1
(40/9B) + (40/9C) = 1
(20/7A) + (20/7C) = 1
use elimination method to solve the problem.
So if i eliminate C first would the resulting two equations be:
(8/3A) + (8/3B) = 1
(1/A) + (1/B) = (-8/20)
???????
To solve this problem, we'll need to use the concept of work rates. Let's assume that A, B, and C each have individual work rates, represented by the variables R(A), R(B), and R(C) respectively.
We can start by expressing the work completed by A and B working together. We are given that they can finish typing the manuscript in 8/3 days. This means their combined work rate is:
R(A+B) = 1 / (8/3) = 3/8
Similarly, we can find the combined work rates for B and C:
R(B+C) = 1 / (40/9) = 9/40
And for A and C:
R(A+C) = 1 / (20/7) = 7/20
Now, let's set up a system of equations using these work rates. Since work rates are additive, we have:
R(A+B) = R(A) + R(B)
R(B+C) = R(B) + R(C)
R(A+C) = R(A) + R(C)
Substituting the previously calculated values, we get:
3/8 = R(A) + R(B)
9/40 = R(B) + R(C)
7/20 = R(A) + R(C)
We now have a system of three equations that we can solve simultaneously to find the individual work rates of A, B, and C.
By rearranging and solving the equations, we find:
R(A) = 1/5 (or 0.2), which means A can complete the manuscript in 5 days.
R(B) = 2/15 (or 0.1333...), which means B can complete the manuscript in approximately 7.5 days.
R(C) = 3/40 (or 0.075), which means C can complete the manuscript in 40/3 (or approximately 13.33) days.
Therefore, A can type the manuscript in 5 days, B in approximately 7.5 days, and C in approximately 13.33 days.