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Rakesh completes a math project in 10 days, while Ravi takes 15 days. Rakesh starts a project and works for 4 days alone, after 4 days Ravi joins and they work together for 2 days. Then a third person Anas joins and they three work together for 16/13 days and complete the project. In how many days can Anas complete a project alone?

Rakesh completes a math project in 10 days, while Ravi takes 15 days. Rakesh starts a project and works for 4 days alone, after 4 days Ravi joins and they work together for 2 days. Then a third person Anas joins and they three work together for 16/13 days and complete the project. In how many days can Anas complete a project alone? Correct Answer 20

Given,

Number of days by Rakesh to complete the project = 10 days

Number of days taken by Ravi to complete a project = 15 days

Concept used,

If someone completes work in X days

∴ Work done in 1 day = 1/X

Calculation

∵ Rakesh takes 10 days to complete a project

∴ Work done by Rakesh in a day = 1/10

∵ Ravi takes 15 days to complete a project

∴ Work done by Ravi in a day = 1/15

Let Anas takes X days to complete the project

Work done by Rakesh in 4days + work done by Rakesh and Ravi in 2 days + work done by all three persons in 16/13 days = 1

⇒ 4/10 + 2 × (1/10 + 1/15) + 16/13 × (1/10 + 1/15 + 1/X) = 1

⇒ 2/5 + 2 × (5/30) + 16/13 × (5/30 + 1/X) = 1

⇒ 1/3 + 16/13 × (1/6 + 1/X) = 1 – 2/5

⇒ 16/13 × (1/6 + 1/X) = 3/5

⇒ 16/13 × (1/6 + 1/X) = 3/5 – 1/3

⇒ 1/6 + 1/X = 13/16 × 4/15

⇒ 1/X = 13/60 – 1/6

⇒ 1/X = 1/20

⇒ X = 20

∴ Number of days taken by Anas to complete the project = 20 days

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Let the total work be 30 units (LCM of 10 and 15)

Efficiency of Rakesh = 30/10 = 3 units

Efficiency of Ravi = 30/15 = 2 units

Let the efficiency of Anas be x units

Total units = 30 = 3 × 4 + 5 × 2 + (3 + 2 + x) × 16/13

⇒ 8 = (5 + x) × 16/13

⇒ 13/2 - 5 = x

⇒ x = 3/2

Anas can complete the work in 30/(3/2) = 20 days

∴ Number of days taken by Anas to complete the project = 20 days

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