15 men can mow the grass of a lawn in 6 days and 12 boys can mow the grass of the same lawn in 15 days. How many days will 5 men and 8 boys take to mow the grass of the same lawn?

15 men can mow the grass of a lawn in 6 days and 12 boys can mow the grass of the same lawn in 15 days. How many days will 5 men and 8 boys take to mow the grass of the same lawn? Correct Answer 10 days

Given:

15 men take 6 days to mow the lawn’s grass

& 12 boys take 15 days to mow the lawn’s grass.

Concept:

For a given amount of work, the number of persons is inversely proportional to the number of working days, i.e. when the number of persons increases then working days decrease and vice - versa. 

Working efficiency is the work done by an individual in one day and this efficiency is inversely proportional to the number of days to complete a work.

It means that a person who takes fewer days to complete a work is said to be more efficient than a person who takes more days to complete the same work.

Formula used:

(M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂

Where, M₁, M₂ = Number of men, D₁, D₂ = Number of days, H₁, H₂ = Number of hours, W₁, W₂ = Work Done

Calculation:

Here, 15 men take 6 days to mow the lawn’s grass.

or 90 men take 1 day to mow the lawn’s grass.

Similarly, 12 boys take 15 days to mow the lawn’s grass.

or 180 boys take 1 day to mow the lawn’s grass.

So, we can make the inference that, 90 men = 180 boys

⇒ 1 man = 2 boys

Now, 5 men + 8 boys = 5 men + (4 × 2) boys

⇒ 5 men + 4 men

⇒ 9 men

Now, using (M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂

We will simplify the formula to the following,

M₁ × D₁ = M₂ × D₂                          (As no change has been specified in the amount of work and number of hours)

⇒ 15 × 6 = 9 × D₂

⇒ D₂ = 90/9

⇒ D₂ = 10 days.

∴ The time required by 5 men and 8 boys to mow the grass of the lawn is 10 days.

Additional Information

We can avoid remembering the formula, (M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂ by understanding the logic of Time and Work.

For example:

Suppose we have, 15 men take 6 days to mow the lawn’s grass and we want to find the number of days (d) taken by 9 men to mow the same lawn’s grass.

So, d = 6 × (15/9)

⇒ d = 10 days

Here, we multiply 6 by a fraction (made up of the number of men in the given conditions) such that the fraction is greater than 1 as the number of men is decreasing which will thereby increase the number of days required to do the work.

On the other hand, if the number of men would have been increasing then we would have multiplied 6 by a fraction (made up of the number of men in the given conditions) such that the fraction is smaller than 1 as the number of men is increasing which will thereby decrease the number of days required to do the work.

Important Points

In these type of particular questions, we have to find the number of men or boys required to do the work in one day, so as to find a relation between the working capacity of the men and the boys. After that, we can use the conditions given in the question to find what is required.