a) A road 300 meter long was constructed by 45 men in 27 days working 8 hours a day. How many men will be required to construct one kilometer of the same road in 20 days if they work 9 hours a day?…

a) Solution to the Work Problem

Step 1: Understand the Problem

We are given:

• Initial scenario:
  • Length of road = 300 meters
  • Number of men = 45
  • Number of days = 27
  • Hours worked per day = 8
• New scenario:
  • Length of road = 1 kilometer (1000 meters)
  • Number of days = 20
  • Hours worked per day = 9
  • Number of men required = ?

We need to find the number of men required to construct 1 km of the same road in 20 days, working 9 hours/day.

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Step 2: Calculate Total Work in Man-Hours

The total work done can be measured in man-hours, which is the product of the number of men, the number of days, and the number of hours worked per day.

For the initial scenario: $$\text{Total work}=\text{Number of men}\times \text{Number of days}\times \text{Hours per day}$$ $$=45\times 27\times 8=9720\text{ man-hours}$$

This amount of work constructs 300 meters of the road.

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Step 3: Find Work Required for 1 km (1000 meters)

Since the road is of the same type, the work required is directly proportional to the length.

$$\text{Work for 1000 meters}=\frac{1000}{300}\times 9720=\frac{10}{3}\times 9720=32400\text{ man-hours}$$

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Step 4: Calculate Required Number of Men

Now, we need to find the number of men ($M$) required to complete 32400 man-hours of work in 20 days, working 9 hours/day.

$$M\times 20\times 9=32400$$ $$M\times 180=32400$$ $$M=\frac{32400}{180}=180$$

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Step 5: Final Answer

180 men are required to construct 1 km of the same road in 20 days, working 9 hours/day.

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Figure: Work Proportion (Man-Hours vs. Road Length)

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b) Solution to the Profit and Discount Problem

Step 1: Understand the Problem

We are given:

• Marked Price (MP) = Rs. 6200
• Discount = 15%
• Profit = 20% (on Cost Price, CP)

We need to find the Cost Price (CP) of the article.

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Step 2: Calculate Selling Price (SP) After Discount

The discount is applied to the marked price.

$$\text{Discount Amount}=15\%\text{ of }6200=0.15\times 6200=930$$

$$\text{Selling Price (SP)}=\text{MP}-\text{Discount}=6200-930=5270$$

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Step 3: Relate Selling Price to Cost Price

The dealer makes a 20% profit on the cost price. This means:

$$\text{SP}=\text{CP}+20\%\text{ of CP}=1.2\times \text{CP}$$

We know SP = Rs. 5270, so:

$$1.2\times \text{CP}=5270$$

$$\text{CP}=\frac{5270}{1.2}=4391.67$$

Since prices are typically rounded to two decimal places (paise in Nepali currency), we can write:

$$\text{CP}=\text{Rs. }4391.67$$

However, in many contexts (especially in exams), we keep it as a fraction for precision:

$$\text{CP}=\frac{5270}{1.2}=\frac{52700}{12}=\frac{13175}{3}\approx 4391.67$$

But let's verify the calculation more carefully:

$$1.2\times \text{CP}=5270⟹\text{CP}=\frac{5270}{1.2}=\frac{52700}{12}=\frac{13175}{3}=4391.\overline{6}$$

So, the exact cost price is Rs. 4391.67 (rounded to two decimal places).

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Step 4: Final Answer

The cost price of the article is Rs. 4391.67.

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Figure: Profit and Discount Relationship