Profit And Loss

1. First, calculate the marked price (M.P.) of the item. The item is marked 50% higher than its cost price (C.P.):

1. Let the cost price be and the original selling price be . The original profit is .
2. When the selling price is doubled, it becomes . The new profit is thus .
3. According to the problem, the new profit is triple the original profit. Hence, we have the equation: .
4. Expanding and simplifying the equation, we have:
   
   
   Add and subtract from both sides:
   
   
5. Now, the original selling price .
6. Profit is .
7. The profit percentage is calculated as:
   .

Let's solve the problem using the given conditions. We have:

• Total quantity of wheat = 900 kg.
• Let the quantity sold at a profit of 8% be x kg.
• Then the quantity sold at a 10% loss will be (900 - x) kg.

We are given that the total profit percentage is 6%. Let's set up the equation to find x:

First, we calculate the overall gain:

• Cost price of the total wheat = 900 kg.
• Selling price of the quantity sold at 8% profit = (x) × (1 + 8/100) = 1.08x.
• Selling price of the quantity sold at 10% loss = (900 - x) × (1 - 10/100) = 0.9(900 - x).

The total selling price considering a 6% profit would be the cost price + 6% profit:

• Total selling price = 900 × (1 + 6/100) = 954.

Let's equate the sum of individual selling prices to the total selling price:

1.08x + 0.9(900 - x) = 954

Expanding and solving the equation:

1.08x + 810 - 0.9x = 954

0.18x = 144

x = 800

Therefore, the quantity sold at a profit of 8% is 800 kg.

1. Calculate the number of days for each complete year:
   • 1997 (from Jan 11 to Dec 31) is a common year, leaving 354 days.
   • 1998 is also a common year, contributing 365 days.
   • 1999 is another common year, contributing 365 days.
   • 2000 (from Jan 1 to Jan 7) contributes 7 days.
2. Total days calculation:
   354 (1997) + 365 (1998) + 365 (1999) + 7 (2000) = 1091 days
3. Convert total days to weeks and extra days:
   • 1091 days ÷ 7 days/week = 155 weeks with a remainder of 6 days.
4. Determine the day:
   Since 6 extra days after a Sunday (Jan 11, 1997) leads to Saturday, January 7, 2000, was a Saturday.

1. Suppose the trader has n articles.
2. He sold half of his articles, which is n/2, at a 60% profit. Profit from these articles is
   Profit = Cost Price × 60%
   So, the Selling Price (SP) for these articles is
   SP = Cost Price × (1 + 60/100) = 1.6 × Cost Price.
3. The remaining articles are n/2. Out of these, half are sold at a 20% loss and half at cost price.
4. Thus, n/4 articles are sold at a 20% loss. So, the Selling Price for these articles is
   SP = Cost Price × (1 - 20/100) = 0.8 × Cost Price.
5. The rest, n/4 articles, are sold at the cost price (no profit, no loss), so
   SP = Cost Price.

• Total CP = n × Cost Price.
• Total SP = (n/2 × 1.6 × Cost Price) + (n/4 × 0.8 × Cost Price) + (n/4 × Cost Price).

• Total SP = (0.8n × Cost Price) + (0.2n × Cost Price) + (0.25n × Cost Price).

• Percentage Gain = ((1.25n × Cost Price - n × Cost Price) / (n × Cost Price)) × 100%.

• Percentage Gain = (0.25n × Cost Price / n × Cost Price) × 100% = 0.25 × 100% = 25%.

To determine the marginal cost when 70 geometry boxes are produced, we need to analyze the cost components involved as mentioned:

• Raw Material Cost: For 'x' units, each unit's cost is ₹(x² + 2). Total raw material cost for 'x' units is .
• Transportation Cost: Given as half the number of boxes produced, it is .
• Storage Cost: Fixed cost of ₹150 per day.

The total cost function for producing 'x' units is .

To find the marginal cost, compute the derivative of the cost function , denoted as . The expression for simplifies as follows:

• Combining terms:

Differentiate with respect to 'x':

• Simplify:
• Note the transportation term correct factor: checks the fixed computation.

Evaluate at :

Thus, the marginal cost when 70 geometry boxes are produced is ₹14,702.50.

To solve the given problem, we need to analyze the mathematical formulation and verify each statement step-by-step:

(A) The problem is formulated correctly.

- Profit function:

- Constraints:

• Space constraint:
• Budget constraint:
• Non-negativity constraint:

This statement is TRUE.

(B) To find the corner points of the feasible region, solve the following:

- Intersecting lines:

• and

Solving simultaneously:

From , solve for .

Substitute into the second equation:

Thus, .

Thus, the corner points are .

This statement is TRUE.

(C) Calculate the profit for purchasing 60 chairs only ( ).

Profit:

This statement is FALSE. The stated profit is incorrect.

(D) Calculate the profit for purchasing 50 tables only ( ).

Profit:

This statement is TRUE. Max profit is indeed Rs. 20,000.

Thus, the correct option is: (A), (B), and (D) only.