Question : Helen can finish a work in 84 days. Katherine can finish the same work in 89 days while Christine can finish the work in 94 days. How long will it take to finish it if they work together?
Correct Answer 29 7171/11869 days or 29.604 days
Solution & Explanation
Solution
Given,
The number of days required to finish a piece of work by Helen = 84 days
And, the number of days required to finish the same work by Katherine = 89 days
And, the number of days required to finish the same work by Christine = 94 days
Thus, the number of days required to finish the work by all of them if they work together = ?
Here,
∵ In 84 days the work is done by Helen = 1
∴ The work done by Helen in 1 day = 1/84
Similarly,
∵ In 89 days the work is done by Katherine = 1
∴ The work done by Katherine in 1 day = 1/89
Similarly,
∵ In 94 days the work is done by Christine = 1
∴ The work done by Christine in 1 day = 1/94 part
Now, the work done by Helen, Katherine, and Christine together in 1 day
= Helen's 1 day work + Katherine's 1 day work + Christine's 1 day work
= 1/84 + 1/89 + 1/94
= 4183 + 3948 + 3738/351372
= 11869/351372 part of work
This means in 1 day, 11869/351372 part of work is done by Helen, Katherine and Christine working together.
Now, the number of days required to finish 11869/351372 part of work by Helen, Katherine, and Christine working together = 1
∴ the number of days required to finish the whole work (1 work) by Helen + Katherine + Christine together
= 1/11869/351372
= 1 × 351372/11869 = 351372/11869 days
= 29 7171/11869 days = or 29.604 days
Thus, Helen, Katherine, and Christine working together will finish the total work (1 work) in 29 7171/11869 days = or 29.604 days Answer
Shortcut Trick to solve the problems based on time and work using formula
Case: If A can finish a work in a days, B can finish the same work in b days, and C can finish the work in c days
then working together the number of days required to finish the work
= a × b × c/ab + ac + bc days.
Here given,
The number of days required to finish the work by Helen = 84 days
And the number of days required to finish the same work by Katherine = 89 days
And the number of days required to finish the work by Christine = 94 days
Thus, the number of days required to finish the work by Helen, Katherine, and Christine together = ?
Here a = 84 days
And, b = 89 days
And, c = 94 days
Thus, using formula a × b × c/ab + ac + bc days
The number of days required to finish the work when Helen, Katherine, and Christine working together
= 84 × 89 × 94/84 × 89 + 84 × 94 + 89 × 94 days
= 702744/7476 + 7896 + 8366 days
= 702744/23738 days
= 702744/23738 days
= 702744 ÷ 2/23738 ÷ 2 = 351372/11869 days
= 29 7171/11869 days or 29.604
Thus, Helen, Katherine, and Christine together will finish the work in 29 7171/11869 days or 29.604 days Answer
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