Problem Setup

Let the cost price (CP) of each article be $Rs. 100$. Let the premium price be $P$. The sale price is $\frac{P}{2}$. The total duration is 1 year (12 months). Articles are sold at premium price for the first 8 months and at sale price for the next 4 months. Assume 'n' articles are sold each month. Total articles sold per year = $12n$. The net profit at the end of the year is $20\%$.

Calculate Total Cost Price (CP)

The total cost price for all articles sold in a year is:

Total CP = (Number of articles sold per year) $\times$ (CP per article)

Total CP = $(12n) \times (Rs. 100) = 1200n$

Calculate Total Selling Price (SP)

The net profit is $20\%$ of the total CP.

Profit = $20\% \times$ Total CP

Profit = $0.20 \times (1200n) = 240n$

The total selling price (SP) is the sum of the total CP and the profit.

Total SP = Total CP + Profit

Total SP = $1200n + 240n = 1440n$

Determine SP from Selling Periods

The total SP can also be calculated based on the selling prices during different periods:

• SP from premium price period (8 months): $(8n \times P)$
• SP from sale price period (4 months): $(4n \times \frac{P}{2}) = 2nP$
• Total SP = $(8nP) + (2nP) = 10nP$

Solve for Premium Price (P)

Equate the two expressions for Total SP:

$10nP = 1440n$

Divide both sides by $n$ (since $n$ is the number of articles, $n \neq 0$):

$10P = 1440$

Solve for $P$:

$P = \frac{1440}{10}$

$P = 144$

Therefore, the premium price of the article is Rs. 144.