A and B together can do a piece of work in 12 days; B and C together can do it in 15 days. If A is twice as good a workman as C, in how many days A alone will do the same work
Let the rate at which A works be a; the rate at which B works be b; the rate at which C work be c:
(a + b)*12 = 1
(b + c)*15 = 1
a = 2*c
gives
b + a/2 = 1/15
b = 1/15 - a/2
Substituting into the first equation:
(a + 1/15 - a/2)*12 = 1
a/2 + 1/15 = 1/12
a/2 = 1/12 - 1/15 = (15-12)/(15*12)
a/2 = 3/(15*12) = 1/(5*12) = 1/60
a = 1/30
The rate at which a works is 1/30 of the work per day; so a will complete the task in 30 days.
Let's assign some variables to the work rates of A, B, and C.
Let's say A's work rate is x units per day.
Then B's work rate is also x units per day since A is twice as good as B.
And C's work rate is (1/2)x units per day since A is twice as good as C.
Now let's use these work rates to solve the problem.
According to the given information:
A and B together can complete the work in 12 days.
So, their combined work rate is 1/12 of the work per day.
Therefore, the equation is:
(x + x) / 12 = 1
Similarly, B and C together can complete the work in 15 days.
So, their combined work rate is 1/15 of the work per day.
The equation becomes:
(x + (1/2)x) / 15 = 1
Now let's solve these equations to find the value of x.
(x + x) / 12 = 1
2x / 12 = 1
2x = 12
x = 6
Now that we know x = 6, we can find the work rate of A, B, and C.
A's work rate = 6 units/day
B's work rate = 6 units/day
C's work rate = (1/2) * 6 = 3 units/day
Finally, to find out how many days A alone can complete the work, we can use A's work rate.
Let's assume it takes A alone t days to complete the work.
So, A's work rate multiplied by t should be equal to 1 (the whole work).
6 * t = 1
t = 1/6
Therefore, A alone can complete the same work in 1/6 of a day or approximately 0.1667 days.
To solve this problem, let's break it down step by step:
1. Let's assume that A's work rate is x units per day.
2. According to the problem, A is twice as good a workman as C. Therefore, C's work rate is (1/2)x units per day.
3. A and B together can do the work in 12 days. From this, we can calculate their combined work rate:
A + B = 1/12
=> A = 1/12 - B
4. Similarly, B and C together can do the work in 15 days. Using the same logic, we can calculate their combined work rate:
B + C = 1/15
=> C = 1/15 - B
5. Now, let's substitute the value of C in terms of B into the equation A + B = 1/12:
A = 1/12 - (1/15 - B)
A = 1/12 - 1/15 + B
A = (5 - 4)/60 + B
A = 1/60 + B
6. Since we know that A is twice as good as C, we can write the relationship between their work rates as:
A = 2C
=> 1/60 + B = 2(1/2x)
=> 1/60 + B = 1/x
=> B = 1/x - 1/60
7. Now, we can substitute the value of B back into the equation B + C = 1/15 to solve for C:
1/x - 1/60 + C = 1/15
C = 1/15 - 1/x + 1/60
8. We know that A + B = 1/12, so substituting the values of A and B, we get:
(1/60 + B) + B = 1/12
1/60 + 2B = 1/12
2B = 1/12 - 1/60
2B = 5/60 - 1/60
2B = 4/60
B = 1/30
9. Now, let's substitute the value of B back into the equation A = 1/60 + B to solve for A:
A = 1/60 + 1/30
A = (1 + 2)/60
A = 3/60
A = 1/20
10. Finally, to find how many days A alone can do the work, we can calculate A's work rate:
x = 1/20 units per day
Therefore, A alone can do the work in 20 days.
So, A alone can do the same work in 20 days.