a) A contractor has to the complete the work of a road in 16 days. He employed 30 persons for 12 days and completes 57 of the work. How many more persons should be employed now in order to complete…

a) Road Construction Problem (Work and Men)

Step 1: Understand the Problem

• Total work duration: 16 days
• Workers initially employed: 30 persons
• Work done in 12 days: $\frac{5}{7}$ of the total work
• Goal: Find how many additional workers are needed to complete the remaining work in the remaining time (4 days).

Step 2: Calculate Total Work in Man-Days

Work is measured in man-days (number of workers × number of days).

• Work done in 12 days by 30 workers: $$30\text{ workers}\times 12\text{ days}=360\text{ man-days}$$ This completes $\frac{5}{7}$ of the work.

• Total work (W) in man-days: $$\frac{5}{7}W=360⟹W=360\times \frac{7}{5}=504\text{ man-days}$$

Step 3: Calculate Remaining Work

• Remaining work: $$504-360=144\text{ man-days}$$

• Remaining time: $$16-12=4\text{ days}$$

Step 4: Find Required Workers for Remaining Work

Let $x$ be the additional workers needed. Total workers now = $30+x$.

• Work equation for remaining 4 days: $$(30+x)\times 4=144$$

• Solve for $x$: $$30+x=\frac{144}{4}=36$$ $$x=36-30=6$$

Final Answer

6 more persons should be employed to complete the work on time.

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b) Orange Selling Problem (Profit and Loss)

Step 1: Understand the Problem

• Total oranges bought: 100 kg
• Cost per kg: Rs 72
• Rotten oranges: 10 kg (unsellable)
• Goal: Find the selling price per kg of the remaining oranges to gain 15% profit on the total cost.

Step 2: Calculate Total Cost Price (CP)

$$\text{CP}=100\text{ kg}\times \text{Rs }72/\text{kg}=\text{Rs }7,200$$

Step 3: Calculate Desired Selling Price (SP) for 15% Profit

$$\text{Profit}=15\%\text{ of CP}=0.15\times 7,200=\text{Rs }1,080$$ $$\text{Total SP}=\text{CP}+\text{Profit}=7,200+1,080=\text{Rs }8,280$$

Step 4: Calculate Sellable Quantity

$$\text{Sellable oranges}=100\text{ kg}-10\text{ kg}=90\text{ kg}$$

Step 5: Find Selling Price per kg

$$\text{SP per kg}=\frac{\text{Total SP}}{\text{Sellable quantity}}=\frac{8,280}{90}=\text{Rs }92$$

Final Answer

The remaining 90 kg of oranges must be sold at Rs 92 per kg to achieve a 15% profit on the total outlay.