First, the question says that a product was sold from January to April with sales of 120, 135, 150, and 165 units respectively. The cost price is Rs. 240 per unit, and there's a fixed marked price throughout. Discounts were given in January, February, and March: 20%, 10%, and 5%, respectively. April had no discount. The total profit from these four months is Rs. 138,825. We need to find the marked price per unit.
Alright, so profit is calculated as (Selling Price - Cost Price) multiplied by the number of units sold. Since the cost price is fixed, and the selling price depends on the discount given each month, I can model this for each month and sum them up.
Let the marked price as M. Then, the selling price each month would be:
January: M - 20% of M = 0.8M
February: M - 10% of M = 0.9M
March: M - 5% of M = 0.95M
So, for each month, the profit per unit is (Selling Price - Cost Price). Then, multiply by the number of units sold to get the total profit for that month. Adding all four months' profits should give the total profit of Rs. 138,825.
Let me write this out as equations.
For January: Profit per unit = 0.8M - 240 Total profit = (0.8M - 240) * 120
February: Profit per unit = 0.9M - 240 Total profit = (0.9M - 240) * 135
March: Profit per unit = 0.95M - 240 Total profit = (0.95M - 240) * 150
April: Profit per unit = M - 240 Total profit = (M - 240) * 165
Adding all these up:
(0.8M - 240)120 + (0.9M - 240)135 + (0.95M - 240)150 + (M - 240)165 = 138,825
Now, I need to expand each term and then combine like terms.
Compute each term separately.
First term: (0.8M - 240)120 = 0.8M120 - 240*120 = 96M - 28,800
Second term: (0.9M - 240)135 = 0.9M135 - 240*135 = 121.5M - 32,400
Third term: (0.95M - 240)150 = 0.95M150 - 240*150 = 142.5M - 36,000
Fourth term: (M - 240)165 = M165 - 240*165 = 165M - 39,600
Now, adding all these together:
(96M - 28,800) + (121.5M - 32,400) + (142.5M - 36,000) + (165M - 39,600)
Let me combine the M terms:
96M + 121.5M + 142.5M + 165M = (96 + 121.5 + 142.5 + 165) M
Calculating the coefficients:
96 + 121.5 = 217.5
217.5 + 142.5 = 360
360 + 165 = 525
So, total M terms: 525M
Now, the constants:
-28,800 -32,400 -36,000 -39,600
Adding them up:
-28,800 -32,400 = -61,200
-61,200 -36,000 = -97,200
-97,200 -39,600 = -136,800
So, the equation becomes:
525M - 136,800 = 138,825
Now, solving for M:
525M = 138,825 + 136,800
138,825 + 136,800 = 275,625
So, 525M = 275,625
Divide both sides by 525:
M = 275,625 / 525
Compute that.
First, simplify numerator and denominator by dividing numerator and denominator by 75.
525 ÷ 75 = 7
275,625 ÷ 75: Computing 275,625 ÷ 75.
75 3,675 = 275,625 (since 753,000=225,000; 75*675=50,625; 225,000+50,625=275,625)
So, 275,625 ÷ 75 = 3,675
Thus, M = 3,675 / 7 = 525
So, M is 525.
Looking at the options, D is 525. So, the correct answer is D.
Calculating 525 525: 525500=262,500; 525*25=13,125; total 275,62
