Time and work questions using LCM method(shortcut method)
Method for calculating Quickly:
L.C.M Method:
If A, B, C complete a piece of work in X, Y, Z Days receptively
Then total work is equal to the L.C.M of X, Y, Z
Then work per day :
For,
A = L.C.M / X
B = L.C.M /Y
C= L.C.M /Z
Total work per day = work per day of (A+B+C)
Total work is done = L.C.M / Total work per day
Ex. If A, B, C completes a piece of work in 12 days, 15 days, 20 days respectively. In how many days the work is completed if they work together?
Sol. L.C.M of 12,15,20 is 60
Work per day of A= 60/12 = 5
B= 60/15 = 4
C= 60/20 = 3
Total work is done per day (A+B+C) = 12
Total work done by A+B+C in = 60/12 = 5 days
hence they complete work in 5 days if they work together.
Solve it like This:
Time and Work solved examples:
Ex.1 If A can complete a work in 10 days, B can complete work in 20 days, and C can complete work in 30 days then in how many days they will complete the work together?
Sol.LCM of 10,20 and 30 is 60.
One day work of
A= 60/10 = 6
B = 60/20 = 3
C = 60/30 = 2
One day work of A+B+C = 11
Together they will complete the work = 60/11 days
Ex.2 If A+B can do work in 30 days, B+C can do work in 24 days and C+A can do work in 20 days. Then in how many days A+B+C can complete the work?
Sol.LCM of 30,24 and 20 is 120
One day work of
A+B = 120/30 = 4
B+C = 120/24 = 5
C+A = 120/20 = 6
2(A+B+C) = 15
A+B+C = 7.5
Together they will finish work in 120/7.5 = 16 days
Ex.3 A can finish work in 15 days and B can finish work in 10 days. if A starts work and after 5 days B joins then in how many days the whole work will be completed?
Sol.LCM of 15 and 10 is 30
One day work
A = 30/15 = 2
B = 30/10 = 3
A+B = 5
Now 30 is the total work
A worked for 5 days completed 10 work
The remaining work is 20, A+B together can complete the work in 20/5 = 4 days
Total work is done in 9 days
Ex.4 If A can complete work in 18 days and B can complete work for 30 days. If they both started the work and after 6 days A left the work. Then in how many days B will finish the remaining work?
Sol.LCM of 18 and 30 is 90. Thus 90 is total work
One day work
A = 90/18 = 5
B = 90/30 = 3
A+B = 8
They worked for 6 days together
Total work done is 48
The remaining work is 90-48 = 42
B will finish the work in =42/3 = 14 days Ans.
Ex.5 A can finish work in 24 days and B can finish work in 32 days. If they both started the work but 4 days before the completion of work A left then finds the total number of days for the whole work to complete.
Sol.LCM of 24 and 32 is 96.
One day work
A = 96/24 = 4
B = 96/32 = 3
A+B = 7
let us suppose they work together for D days. Work done by them is 7 * D
The remaining 4 days' work has been completed by B.
7*D + 3*4 = 96
D = 12 days
Total work done is completed in 12+4 = 16 days
Ex.6 A takes as much time as B and C to finish the job. and B together take 10 days and C takes 15 days then finds in how many days B can finish the job?
Sol. LCM of 10 and 15 is 30
One day work of
A+B = 30/10 = 3
C = 30/15 = 2
A =B+C
A+B = 3
B+C+B = 3
2B+C = 3
B = 1/2
B will finish 30 work in 30/1/2 = 60 days
Ex.7 A and B can complete work in 12 days and B and C can complete work in 16 days If A work for 5 days and B work for 7 days and C complete the remaining work in 13 days then find in how many days C would finish the job?
Sol.LCM of 12 and 16 is 48
One day work
A+B = 48/12 = 4
B+C = 48/16 = 3
5A+7B+13C = 48
5(A+B)+2(B+C)+11C = 48
11C = 22
C = 2
C will complete the work in 48/2 = 24 Days Ans.
Ex.8 A alone would take 27 hours more to complete the work than A and B work together. B takes 3 hours more to complete the work than A and B work together. then how many days A and B both can complete the work.
Sol. Let work done by
A+B = x
A = x+27
B = x+3
LCM of x, x+27, x+3 is x(x+27)(x+3)
(x+27)(x+3) = x(x+3) + x(x+27)
x2+27x+3x+81 = x2+3x+x2+27x
x2 = 81
x=9
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