Question:
Catherine can finish a work in 100 days. Heather can finish the same work in 105 days while Diane can finish the work in 110 days. How long will it take to finish it if they work together?
Correct Answer
34 626/661 days or 34.947 days
Solution And Explanation
Solution
Given,
The number of days required to finish a piece of work by Catherine = 100 days
And, the number of days required to finish the same work by Heather = 105 days
And, the number of days required to finish the same work by Diane = 110 days
Thus, the number of days required to finish the work by all of them if they work together = ?
Here,
∵ In 100 days the work is done by Catherine = 1
∴ The work done by Catherine in 1 day = 1/100
Similarly,
∵ In 105 days the work is done by Heather = 1
∴ The work done by Heather in 1 day = 1/105
Similarly,
∵ In 110 days the work is done by Diane = 1
∴ The work done by Diane in 1 day = 1/110 part
Now, the work done by Catherine, Heather, and Diane together in 1 day
= Catherine's 1 day work + Heather's 1 day work + Diane's 1 day work
= 1/100 + 1/105 + 1/110
= 231 + 220 + 210/23100
= 661/23100 part of work
This means in 1 day, 661/23100 part of work is done by Catherine, Heather and Diane working together.
Now, the number of days required to finish 661/23100 part of work by Catherine, Heather, and Diane working together = 1
∴ the number of days required to finish the whole work (1 work) by Catherine + Heather + Diane together
= 1/661/23100
= 1 × 23100/661 = 23100/661 days
= 34 626/661 days = or 34.947 days
Thus, Catherine, Heather, and Diane working together will finish the total work (1 work) in 34 626/661 days = or 34.947 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 Catherine = 100 days
And the number of days required to finish the same work by Heather = 105 days
And the number of days required to finish the work by Diane = 110 days
Thus, the number of days required to finish the work by Catherine, Heather, and Diane together = ?
Here a = 100 days
And, b = 105 days
And, c = 110 days
Thus, using formula a × b × c/ab + ac + bc days
The number of days required to finish the work when Catherine, Heather, and Diane working together
= 100 × 105 × 110/100 × 105 + 100 × 110 + 105 × 110 days
= 1155000/10500 + 11000 + 11550 days
= 1155000/33050 days
= 1155000/33050 days
= 1155000 ÷ 50/33050 ÷ 50 = 23100/661 days
= 34 626/661 days or 34.947
Thus, Catherine, Heather, and Diane together will finish the work in 34 626/661 days or 34.947 days Answer
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