(i) Let us assume the printed price of the article be Rs. x

The retailer marks up the price by 10% on the printed price

So, the marked price by the retailer

\[=\text{ }Rs.\text{ }x\text{ }+\text{ }10%\text{ }of\text{ }Rs.\text{ }x\]

\[=\text{ }Rs.\text{ }x\text{ }+\text{ }Rs.\text{ }0.1x\]

or,

\[=\text{ }Rs.\text{ }1.1x\]

Due to competition the retailer allows discount of 5% on the marked price, then

The selling price of the article

\[~=\text{ }Rs.\text{ }1.1x\text{ }\text{ }discount\]

Discount

\[=\text{ }5%\text{ }of\text{ }Rs.\text{ }1.1x\]

\[=\text{ }5%\text{ }of\text{ }Rs.\text{ }1.1x\]

or,

\[=\text{ }Rs.~\left( 5/100 \right)\text{ }x\text{ }1.1x\]

\[=\text{ }Rs.\text{ }0.055x\]

The rate of GST

\[=\text{ }12%\]

The tax (under GST) for the purchase

= 12% of the selling price set by the retailer

\[=\text{ }12%\text{ }of\text{ }Rs.~\left( 1.1x\text{ }\text{ }0.055x \right)\]

\[=\text{ }Rs.~\left( 12/100 \right)\text{ }x\text{ }\left( 1.045x \right)\]

Thus, the price of the article inclusive of GST

\[=\text{ }Rs.\text{ }1.045x\text{ }+\text{ }Rs.~\left( 12/100 \right)\text{ }x\text{ }\left( 1.045x \right)\]

Given, buyer pays Rs. 468.16 for the article inclusive of tax (under GST)

So,

\[1.045x\text{ }+\text{ }\left( 12/100 \right)\text{ }x\text{ }\left( 1.045x \right)\text{ }=\text{ }468.16\]

or,

\[1.045x\text{ }+\text{ }0.1254x\text{ }=\text{ }468.16\]

\[1.1704x\text{ }=\text{ }468.16\]

or,

\[x\text{ }=\text{ }468.16/1.1704\]

or,

\[x\text{ }=\text{ }Rs.\text{ }400\]

Therefore, the printed price of the article is Rs. 400

(ii) The retailer buys at 15% discount of the printed price,

and

sells at 5% discount for the marked price of 10% on the printed price

So,

Bought at

\[=\text{ }400\text{ }\text{ }15%\text{ }of\text{ }Rs.\text{ }400\]

\[=\text{ }Rs.\text{ }400\text{ }\text{ }Rs.\text{ }60\text{ }=\text{ }Rs.\text{ }340\]

Sold at

\[\begin{array}{*{35}{l}}

=\text{ }\left( Rs.\text{ }400\text{ }+\text{ }10%\text{ }of\text{ }Rs.\text{ }400 \right)\text{ }\text{ }5%  \\

of\text{ }\left( Rs.\text{ }400\text{ }+\text{ }10%\text{ }of\text{ }Rs.\text{ }400 \right)  \\

\end{array}\]

\[=\text{ }Rs.\text{ }\left( 400\text{ }+\text{ }40 \right)\text{ }\text{ }\left[ \left( 5/100 \right)\text{ }x\text{ }Rs.\text{ }400\text{ }+\text{ }40 \right)]\]

\[=\text{ }Rs.\text{ }440\text{ }\text{ }Rs.~\left( 0.05\text{ }x\text{ }440 \right)\]

or,

\[=\text{ }Rs.\text{ }440\text{ }\text{ }Rs.\text{ }22\]

\[=\text{ }Rs.\text{ }418\]

So, profit = Selling price – cost price

\[=\text{ }Rs.\text{ }418\text{ }\text{ }Rs.\text{ }340\text{ }=\text{ }Rs.\text{ }78\]

Hence, the profit percentage

\[=\text{ }\left( 78/340 \right)\text{ }x\text{ }100\text{ }=\text{ }22.94%\]