Question

[Urgent]
An engineer undertakes a project to build a road 15 km long in 300 days and employs 45 men for the purpose. After 100 days, he finds only 2.5 km of the road has been completed. Find the (approximate) number of extra men he must employ to finish the work in time.

Answer 1

$\begin{gathered}\large {\boxed{\sf{\mid{\overline {\underline {\star ANSWER ::}}}\mid}}}\end{gathered}$

CONCEPT :-

Work Done depends on :

1. No. of men
2. Time Taken / No. of days

$\large \boxed{ \bold{No. \: of \: men \: ∝ \: \frac{1}{time \: taken} ∝ \: distance}}$

GIVEN :-

• 15 Km of road ⟼ 300 days
• 2.5 Km of road ⟼ 100 days ⟼ 45 men

TO FIND :-

• Remaining (15 Km - 2.5 Km) 12.5 Km of road ⟼ (300 days - 100 days) 200 days ⟼ 'x men' .

SOLUTION :-

Let 'x' be the no. of men required to complete the remaining 12.5 Km of road in 200 days.

From the problem :-

• 45 men ⟼ 100 days ⟼ 2.5 Km
• 45 men ⟼ 40 days ⟼ 1 Km

[ By dividing the data by 2.5 to obtain the days required to complete 1 Km of the road ]

1 Km ⟼ 40 * 45 men days

12.5 Km ⟼ 40 * 45 * 12.5 men days

$\large \boxed{ \bold{No \: of \: men = 12.5 \times \frac{40 \times 45}{no .\: of \: days} }} \\ \\ = 12.5 \times \frac{40 \times 45}{200} \\ \\ = 12.5 \times 9 \\ \\ = 112.5$

No. of additional men needed = No. of men - 45

No. of additional men needed = 112.5 - 45

No. of additional men needed = 67.5

$\underline{ \large \boxed{ \bold{ =68 \: men (approx)}}}$

@SweetestBitter

Answer 2

Answer:

68 men

Step-by-step explanation:

Given,

1) An engineer undertakes a project to build a road 15 km long in 300 days and employs 45 men for the purpose.

2) After 100 days, he finds only 2.5 km of the road has been completed.

To Find :-

Number of extra men required to complete the work in time.

Solution :-

Here , we can say that 45 men completed 2.5km of road in 100 days.

→ Number of men (m₁) = 45

Work done(w₁) = 2.5km

Total time taken(Number of days)(t₁) = 100

Remaining work to be done (w₂)= 15km - 2.5km

= 12.5km

Remaining days left(t₂) = 300days - 100days

= 200days

Let, the men required be 'm₂'.

The required box equation is :-

$\begin{array}{|c|c|c|}\cline{1-3} \bf {Number\:of\:men}& \bf{Work\:done}& \bf{time(days)}\\ \cline{1-3} \sf 45& \sf 2.5km& \sf 100days\\ \cline{1-3} \sf ?(m_2)& \sf 12.5km& \sf 20days\\ \cline{1-3} \end{array}$

We can say that :-

Number of men is directly proportional to work done

Reason :- If men's increases , the work done by them also increases.

→ 45 : m₂ = 2.5 : 12.5

45 : m₂ = 25 : 125

45 : m₂ = 1 : 5

[Let it be equation - 1]

Number of men is inversly(in directly) proportional to time taken

Reason :- If men's Increases , the number of working days decreases as the work completes faster.

→ 45 : m₂ = Inverse of 100 : 200

45 : m₂ = 200 : 100

45 : m₂ = 2 : 1

∴ 45 : m₂ = Compound ratio of '1 : 5' and '2 : 1'

45 : m₂ = (1 : 5) × (2 : 1)

45 : m₂ =  (1 × 2) : (5 × 1)

45 : m₂ = 2 : 5

m₂ =  45 × 5/2

= 225/2

= 112.5

Let it be 113 (approx)

As we already kept 45 men , then the remaining men are :-

= 113 - 45

= 68

∴ Number of extra men required to finish the work in time is 68 men