A and B together, B and C together, and A and C together can do a certain piece of work in x, x+6 and 63/2 days , respectively. A,B and C together can complete the work in 24 days. A,B and C start the work together and after 12 day A left the job and 24 days before the complition on the work C also left. In how many day the work is completed?
Answers
Given : A and B together, B and C together, and A and C together can do a certain piece of work in x, x+6 and 63/2 days , respectively
To find : In how many day the work is completed
Solution:
Let say A , B , & C complete work in A , B & C Days respectively
Hence their 1 day work are 1/A , 1/B & 1/C
1/A + 1/B = 1/x
1/B + 1/C = 1/(x + 6 )
1/A + 1/C = 2/63
Adding all 2 ( 1/A + 1/B + 1/C) = 1/x + 1/(x + 6) + 2/63
1/A + 1/B + 1/C = 1/24
=> 2(1/24) = 1/x + 1/(x + 6) + 2/63
=> 1/12 = 1/x + 1/(x + 6) + 2/63
=> 1/x + 1/(x + 6) = 1/12 - 2/63
=> (2x + 6)/x(x + 6) = (21 - 8)/252
=> 504x + 1512 = 13x² + 78x
=> 13x² -426x - 1512 = 0
=> 13x² - 468x + 42x - 1512 = 0
=> 13x(x - 36) + 42(x - 36) = 0
=> x = 36
1/A + 1/B = 1/36 1/B + 1/C = 1/42 1/A + 1/C = 2/63
1/A + 1/B + 1/C = 1/24
=> 1/C = 1/24 - 1/36 = (3 - 2)/72 = 1/72
=> C = 72
1/A = 1/24 - 1/42 = (7 - 4)/168 = 3/168 = 1/56
=> A = 56
1/A + 1/B = 1/36 => 1/B = 1/36 - 1/56 = 5/504
=> B = 100.8
Work Done in 12 Days = 12 (1/24) = 1/2
B Worked alone for 24 Days after C left
Work done by B = 24 (5/504) = 5/21
Work done by B & C = 1 - 1/2 - 5/21 = 1/2 - 5/21
= (21 - 10)/42
= 11/42
=> B & C worked for 11 Days
Total Days Worked = 12 + 11 + 24
= 47
in 47 Days Work completed
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The work was completed in 47 days.
Step-by-step explanation:
From question, we have that,
1/A + 1/B = 1/x → (equation 1)
1/B + 1/C = 1/(x + 6) → (equation 2)
1/A + 1/C = 2/63 → (equation 3)
On adding equation 1, 2 and 3, we get,
2(1/A + 1/B + 1/C) = 1/x + 1/(x + 6) + 2/63
From question, A, B and C together can complete the work in 24 days.
2(1/24) = 1/x + 1/(x + 6) + 2/63
1/12 = 1/x + 1/(x + 6) + 2/63
1/x + 1/(x + 6) = 1/12 - 2/63
(2x + 6)/(x(x + 6)) = 13/252
252(2x + 6) = 13(x(x + 6))
504x + 1512 = 13x² + 78x
13x² - 426x - 1512 = 0
13x² - 468x + 42x - 1512 = 0
(13x + 42)(x - 36) = 0
The negative value is not considered.
∴ x = 36
Now, the value of C is:
1/36 + 1/C = 1/24 ⇒ 1/C = 1/24 - /36
∴ C = 72
Now, the value of B is:
1/B + 2/63 = 1/24 ⇒ 1/B = 1/24 - 2/63
∴ B = 100.8
Now, the value of A is:
1/A + 1/42 = 1/36 ⇒ 1/A = 1/36 - 1/42
∴ A = 56
From question, the work done by A in 12 days.
⇒ 24 (5/504) = 5/21
From question, the number of days worked by B and C.
⇒ (21 - 10)/42 = 11/42
The number of days worked is:
∴ D = 11 + 12 + 24 = 47 days