Calculating the Printed Price with Multiple Discounts

Let's break down this problem involving discounts on a product to find its original printed price. We are given the final selling price after two consecutive discounts: first on the printed price to get the label price, and then another on the label price to get the final selling price.

Understanding the Pricing Terms

• Printed Price: This is the initial price marked on the product before any discounts. This is what we need to find.
• Label Price: This is the price after the first discount is applied to the printed price.
• Selling Price: This is the final price the customer pays after all discounts are applied to the label price.

Step-by-Step Calculation

Let \(P\) be the Printed Price (in Rs.).

Step 1: Calculate the Label Price

The label price is at a 20% discount on the printed price. This means the label price is \(100\% - 20\% = 80\%\) of the printed price.

Label Price \(L = P - 20\%\) of \(P\)

In terms of calculation:

\(L = P - 0.20 \times P\)

\(L = (1 - 0.20) \times P\)

\(L = 0.80 \times P\)

Step 2: Calculate the Selling Price from the Label Price

The shopkeeper gave an additional 10% discount on the label price at the time of sale. This means the selling price is \(100\% - 10\% = 90\%\) of the label price.

Selling Price \(S = L - 10\%\) of \(L\)

In terms of calculation:

\(S = L - 0.10 \times L\)

\(S = (1 - 0.10) \times L\)

\(S = 0.90 \times L\)

Step 3: Use the Given Selling Price to Find the Label Price

We are given that the customer bought the product at Rs. 468. So, \(S = 468\).

Substitute this into the equation from Step 2:

\(468 = 0.90 \times L\)

Now, solve for \(L\):

\(L = \frac{468}{0.90}\)

\(L = \frac{468}{\frac{90}{100}}\)

\(L = \frac{468 \times 100}{90}\)

\(L = \frac{46800}{90}\)

\(L = 520\)

So, the Label Price was Rs. 520.

Step 4: Use the Label Price to Find the Printed Price

Now we use the equation from Step 1:

\(L = 0.80 \times P\)

Substitute the calculated value of \(L = 520\):

\(520 = 0.80 \times P\)

Now, solve for \(P\):

\(P = \frac{520}{0.80}\)

\(P = \frac{520}{\frac{80}{100}}\)

\(P = \frac{520 \times 100}{80}\)

\(P = \frac{52000}{80}\)

\(P = 650\)

So, the Printed Price of the product is Rs. 650.

Summary of Prices

The calculated selling price (Rs. 468) matches the given selling price, confirming our steps are correct.

Therefore, the printed price of the product is Rs. 650.

Revision Table: Key Concepts

Additional Information: Successive Discounts

It is important to note that two successive discounts of 20% and 10% are not equivalent to a single discount of \(20\% + 10\% = 30\%\). This is because the second discount is applied to a smaller base (the label price), not the original printed price.

Let's verify this with an example:

If the printed price was Rs. 100:

• Single 30% discount: Selling price = \(100 - 0.30 \times 100 = 100 - 30 = 70\)
• Successive 20% and 10% discounts:
  • Price after 20% discount (Label Price) = \(100 - 0.20 \times 100 = 80\)
  • Price after additional 10% discount (Selling Price) = \(80 - 0.10 \times 80 = 80 - 8 = 72\)

As you can see, Rs. 70 is different from Rs. 72. The effective single discount equivalent to successive discounts of \(d_1\%\) and \(d_2\%\) is given by the formula:

\(Effective Discount = (d_1 + d_2 - \frac{d_1 \times d_2}{100})\%\)

For 20% and 10%:

\(Effective Discount = (20 + 10 - \frac{20 \times 10}{100})\%\)

\(Effective Discount = (30 - \frac{200}{100})\%\)

\(Effective Discount = (30 - 2)\%\)

\(Effective Discount = 28\%\)

This means the customer effectively received a 28% discount on the original printed price. Let's check this: 28% of Rs. 650 is \(0.28 \times 650 = 182\). Selling Price = \(650 - 182 = 468\), which matches the given selling price.