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Answer :
Let's solve each problem one by one:
Finding the marked price of an article with successive discounts
The trader gives two successive discounts of 20% and 10% on the marked price.
Let's assume the marked price is [tex]x[/tex].
After the first discount of 20%, the price becomes [tex]0.8x[/tex].
After the second discount of 10%, the price becomes [tex]0.9 \times 0.8x = 0.72x[/tex].
The final amount the trader receives is Rs.108.
So, [tex]0.72x = 108[/tex].
Solving for [tex]x[/tex]:
[tex]x = \frac{108}{0.72} = 150[/tex]
Therefore, the marked price of the article is Rs.150.
Calculating the profit percentage for a dishonest merchant
The merchant uses 900 grams instead of 1 kilogram (1000 grams).
The true cost should be based on 1 kg, but he gains 100 grams on every kg sold.
The effective weight sold is 900 grams, where he makes a profit on the missing 100 grams out of 1000 grams.
Therefore, the profit percentage [tex]P[/tex] is calculated by:
[tex]P = \frac{100}{900} \times 100\% = 11.11\%[/tex]
Hence, the profit percentage is 11.11%.
Finding the actual profit percentage when profit is one-fifth of the selling price (SP)
Let the selling price be [tex]SP[/tex].
Profit is [tex]\frac{1}{5} \times SP[/tex].
Therefore, [tex]SP = CP + \frac{SP}{5}[/tex], where [tex]CP[/tex] is the cost price.
Solving, [tex]CP = \frac{4}{5}SP[/tex].
Profit percentage is:
[tex]P\% = \left( \frac{SP - CP}{CP} \right) \times 100\%[/tex]
[tex]P\% = \left( \frac{SP - \frac{4}{5}SP}{\frac{4}{5}SP} \right) \times 100\% = 25\%[/tex]
Therefore, the actual profit percentage is 25%.
These explanations step through the processes needed to solve each problem, employing basic principles of percentage calculations and algebra. By understanding these solutions, students can apply similar methods to a wide variety of related problems.
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