Let me try to make some things clear-
The guy, the retailer, gets the product from a dealer for say $80$. He wants a profit of say $20$ making it $100$ . Now, He has to pay tax. He doesn't want to pay tax himself, but will try to get the customer to pay the tax. He adds say $20 \%$ tax and finally, a customer buys it for $120$.
Cost Price - $80$
Selling price - $100$
Selling price(inclusive of all taxes) - $120$
There are two selling prices - One in terms of the retailer and one in terms of the customer. The retailer will give the $20$ to the tax and finally, he thinks the selling price is just $100$ . Customer though got it for $120$. This is his selling price.
Basically, it depends on the problem, as to - from whose side is it asking. I hope I am clear.
The most important similarity between the first question and this question is that the total Selling Price can be equated. There, we equated what you gave to the shopkeeper and what he received from you. Here, we will equate the seller's $SP$ when he used a normal yard stick, and the $SP$ when he used the modified yard stick, since he never changed his selling price. You have understood correctly that the amount is written on the pricetag. I will use the same terminology. Here, $CP$ is the cost price of one inch for the seller.
When he uses a normal yard stick:
Total Cost of 36 inches (1 yard)? Well, that's just $36CP$.
He gets $50\%$ profit, so profit = $\frac 12 \times 36CP$.
Add this to the total cost price to get the total selling price of $36$ inches.
$$36CP + \frac 12 36CP = 36CP\left(1+\frac 12\right) = \color{blue}{\left(\frac 32\right)36CP}$$
Now, the shopkeeper changes the yard stick, but keeps the pricetag same. So, he sells $(36+x)$ inches for the same price obtained above. Now, his profit is $20\%$ of $(36+x)CP$ (see the last part of this answer if you're confused why we took $(36+x$). That is $(20/100) \times (36+x)CP$. Add this to the total cost price to get the total selling price:
$$(36+x)CP+(20/100) \times (36+x)CP = (36+x)CP\left(1+\frac 15\right) = \color{blue}{(6/5) \times (36+x)CP}$$
The two things (in blue) are the amounts written on the pricetag. So they must be equal. Notice that when you equate them, the $CP$ just cancels out, leaving you with a simple linear equation in $x$.
Just in case you want to know why we took $36+x$ in the second part:
He measures cloth a $36+x$ inches long stick and calls it $36$ inches. However, he has brought it from a dealer at $(36+x)CP$ (because the dealer is using a proper yardstick). But now he tells you it's $36$ inches, and sells at the same amount mentioned on the price tag for $36$ inches. Due to this stupidity, his profits reduce.
Best Answer
Alright, let's call the original cost price $x$ as you've done, and perform all the indicated transformations.
The dealer sells his juice in such a way as to make $30\%$ profit after paying tax of $13\%$ of cost of his returns. Thus, his selling price is $(100+30+13)\% x=1.43x.$
The shopkeeper increases the price by $15\%,$ so they now have $115\%×1.43x.$ They dilute the juice by $25\%,$ so it follows that per litre, they are actually selling $3/4$ of juice for $1.15×1.43x.$ Thus, for a complete litre of juice, the true selling price is $4/3×1.15×1.43x.$ After giving a discount of $4\%,$ the selling price is now $(100-4)\%×4/3×1.15×1.43x=0.96×4/3×1.15×1.43x,$ and this is equal to $\text{Rs}98.67.$ Thus, we must solve $$0.96×4/3×1.15×1.43x=98.67$$ for the original cost price $x.$