Rs 150 will be given to cultivate one hectare of land.A and B can do this work 12 hours.B and C can do it in 15 hours and C and A ca do it in 10 hours.How much each will get if all the three work together?
Sri Chaitanya public school
school;:chaitanya vidyalaya
To solve this problem, let's assign variables to each person's rate of work. Let's say A's rate of work is a hectares per hour, B's rate of work is b hectares per hour, and C's rate of work is c hectares per hour.
We know that A and B working together can finish the job in 12 hours, so their combined rate of work is 1/12 hectares per hour.
Similarly, B and C working together can finish the job in 15 hours, so their combined rate of work is 1/15 hectares per hour.
C and A working together can finish the job in 10 hours, so their combined rate of work is 1/10 hectares per hour.
Now, let's set up a system of equations based on the given information:
a + b = 1/12 (Equation 1 - A and B's combined rate of work)
b + c = 1/15 (Equation 2 - B and C's combined rate of work)
c + a = 1/10 (Equation 3 - C and A's combined rate of work)
To solve this system of equations, we can use the method of substitution:
From Equation 1, we can rearrange it to express a in terms of b: a = 1/12 - b.
Substituting a into Equation 3, we get: c + (1/12 - b) = 1/10.
Simplifying this equation, we get: c - b = 1/10 - 1/12.
Combining the fractions on the right side, we get: c - b = 6/60 - 5/60 = 1/60.
Therefore, c - b = 1/60. This equation (Equation 4) represents the relationship between C and B's individual rates of work.
Now, let's solve for b using Equation 2:
b + c = 1/15.
Since we already have c - b = 1/60 from Equation 4, we can substitute c - b for c in Equation 2:
b + (c - b) = 1/15.
Simplifying this equation, we get: c = 1/15.
Now, we can solve for b: b + (1/15 - b) = 1/15.
Simplifying further, we get: b - b/15 = 1/15.
Combining like terms, we get: (15b - b)/15 = 1/15.
Simplifying, we get: 14b/15 = 1/15.
Cross-multiplying, we get: 14b = 15.
Finally, dividing both sides by 14, we find: b = 15/14.
Now that we know the value of b, we can substitute it back into Equation 1 to find a:
a + b = 1/12.
Substituting b = 15/14, we get: a + 15/14 = 1/12.
Simplifying this equation, we get: a = (1/12) - (15/14).
Combining the fractions on the right side, we get: a = 14/168 - 180/168 = -166/168 = -83/84.
Since the rate of work cannot be negative, we discard this negative value of a. We are left with b = 15/14 and c = 1/15.
To find how much each person will get if all three work together, we need to find their respective proportions of work.
A's proportion of work = a / (a + b + c) = (-83/84) / (-83/84 + 15/14 + 1/15)
= (-83/84) / (-83/84 + 90/84 + 28/420)
= (-83/84) / (28/420)
= (-83/84) / (1/15)
= -83/56.
B's proportion of work = b / (a + b + c) = (15/14) / (-83/84 + 15/14 + 1/15)
= (15/14) / (-83/84 + 90/84 + 28/420)
= (15/14) / (28/420 + 7/84)
= (15/14) / (7/84 + 1/12)
= (15/14) / (1/12)
= 180/14.
C's proportion of work = c / (a + b + c) = (1/15) / (-83/84 + 15/14 + 1/15)
= (1/15) / (-83/84 + 90/84 + 28/420)
= (1/15) / (28/420 + 7/84)
= (1/15) / (7/84 + 1/12)
= (1/15) / (1/12)
= 4/15.
Now, we divide the total amount of money, Rs 150, proportionally among the three people:
Amount A will get = (A's proportion of work) * (Total amount) = (-83/56) * 150 = -282.59.
Amount B will get = (B's proportion of work) * (Total amount) = (180/14) * 150 = 1928.57.
Amount C will get = (C's proportion of work) * (Total amount) = (4/15) * 150 = 40.
Therefore, A will get - Rs 282.59, B will get Rs 1928.57, and C will get Rs 40 if all three of them work together.