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The cash difference between the selling prices of an article at a profit of $4%$ and $6%$ is $Rs\text{ 3}$. Find the ratio of the two selling prices.

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Hint: Here we apply the concept of percentages and profit and loss. We need to find out the ratio of the two selling prices with the given information.
Selling price: It is nothing but the price at which the product or service is sold to the buyer.

Complete step by step answer:
Given: The cash difference between the selling prices of an article at a profit of $4%$ and $6%$ is $Rs\text{ 3}$.
Required Formula:
Selling Price=$\dfrac{100+(profit\,\%)}{100}\times \cos t\text{ }price$

Let the cost price of an article be $x$.
Selling Price of an article sold at a profit of $4%$$={{S}_{1}}$
Selling Price of an article sold at a profit of $6%$$={{S}_{2}}$
According to the question,
$\begin{align}
  & {{S}_{1}}=\dfrac{104}{100}x \\
 & {{S}_{2}}=\dfrac{106}{100}x \\
\end{align}$
Since the cash difference between the selling prices of an article at a profit of $4%$ and $6%$ is $Rs\text{ 3}$.
$\begin{align}
  & \Rightarrow {{S}_{2}}-{{S}_{1}}=3 \\
 & \Rightarrow \dfrac{106}{100}x-\dfrac{104}{100}x=3 \\
 & \Rightarrow \dfrac{2x}{100}=3 \\
 & \Rightarrow x=150 \\
\end{align}$
Ratio of ${{S}_{1}}\text{ }to\text{ }{{S}_{2}}$ becomes
$\begin{align}
  & \Rightarrow \dfrac{{{S}_{1}}}{{{S}_{2}}}=\dfrac{\dfrac{104x}{100}}{\dfrac{106x}{100}} \\
 & \Rightarrow \dfrac{{{S}_{1}}}{{{S}_{2}}}=\dfrac{104}{106} \\
 & \Rightarrow \dfrac{{{S}_{1}}}{{{S}_{2}}}=\dfrac{52}{53} \\
\end{align}$

Note: In such types of questions which involves percentages and profit and loss, the knowledge about profit and loss is needed. Accordingly, apply the appropriate formula to be applied with the known values and solve them to get the unknown value by satisfying the condition involved in the question. Here as many calculations are involved, we will need to be vigilant with the arithmetic calculations involved.
Profit/ gain = Selling price - Cost price
Loss = Cost price - Selling price
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