The correct answer is option (B):
\(22\frac{11}{12} \%\)
Let's break down how to solve this problem step-by-step. The key here is to relate the selling price and profit percentage to find the cost price for each article.
1. Represent Selling Prices: Let the selling prices of the four articles be 3x, 5x, 6x, and 8x respectively.
2. Calculate Cost Prices: We'll use the profit percentages to find the cost price (CP) for each article. Remember that Profit % = ((Selling Price - Cost Price) / Cost Price) * 100
* Article 1: 20% profit means 1.20 * CP1 = 3x, so CP1 = 3x / 1.20 = 2.5x
* Article 2: 25% profit means 1.25 * CP2 = 5x, so CP2 = 5x / 1.25 = 4x
* Article 3: 20% profit means 1.20 * CP3 = 6x, so CP3 = 6x / 1.20 = 5x
* Article 4: 25% profit means 1.25 * CP4 = 8x, so CP4 = 8x / 1.25 = 6.4x
3. Calculate Total Cost Price: The total cost price (TCP) of all four articles is CP1 + CP2 + CP3 + CP4 = 2.5x + 4x + 5x + 6.4x = 17.9x
4. Calculate Total Selling Price: The total selling price (TSP) of all four articles is 3x + 5x + 6x + 8x = 22x
5. Calculate Overall Profit: Overall Profit = TSP - TCP = 22x - 17.9x = 4.1x
6. Calculate Overall Profit Percentage: Overall Profit Percentage = (Overall Profit / Total Cost Price) * 100 = (4.1x / 17.9x) * 100 = (4.1/17.9) * 100 approximately equal to 22.905%.
7. Convert the Result to a Mixed Fraction: (4.1/17.9)*100 = 410/17.9, this is actually 410/179. Then 410 divided by 179 results in the quotient is 2 with the remainder of 52. Therefore 410/179 = 2 + 52/179. The correct value is approximately 22.8%, thus the value 22 and 11/12 is close to the approximate result. Now let's do the exact calculation to see which option is the closest.
8. Calculate (4.1/17.9)*100.
(41/10) / (179/10) * 100
(41/179) * 100
4100/179 = 22 and 112/179, after doing the division. Then, we need to compare the given answers and see which one is the closest.
The calculation is actually incorrect.
Overall Profit Percentage = (Total Selling Price - Total Cost Price) / Total Cost Price * 100
= (22x - 17.9x) / 17.9x * 100
= (4.1x / 17.9x) * 100
= (41/179) * 100
= 4100/179
= 22 112/179.
Now let's check the options
22 11/12 = 22 + (11/12) = 22 + 0.916666 = 22.916666
The closest option is 22 11/12.
Therefore, the correct answer is 22 11/12%.