The marked price on a book is ₹1,000. In a book fair, it is available for sale with a discount scheme offering two successive discounts of 12% and 8%. What is the final selling price (in ₹) of the book for a customer (rounded off to the nearest integer)?
810
Understanding Successive Discounts on Book Price
This problem involves calculating the final selling price of a book after applying two successive discounts. Successive discounts mean that the second discount is applied not on the original marked price, but on the price after the first discount has been applied.
Let's break down the steps to find the final selling price of the book.
Calculating the Price After the First Discount
The marked price of the book is ​1,000.
The first discount offered is 12%.
First discount amount = 12% of ​1,000
We can calculate this as:
\(\text{Discount}_1 = \text{Marked Price} \times \frac{\text{Discount Rate}_1}{100}\)
\(\text{Discount}_1 = 1000 \times \frac{12}{100} = 1000 \times 0.12 = 120\)
So, the first discount amount is ​120.
The price of the book after the first discount is the marked price minus the first discount amount:
\(\text{Price after Discount}_1 = \text{Marked Price} - \text{Discount}_1\)
\(\text{Price after Discount}_1 = 1000 - 120 = 880\)
The price after the first discount is ​880.
Calculating the Price After the Second Successive Discount
The second discount offered is 8%. This discount is applied to the price after the first discount, which is ​880.
Second discount amount = 8% of ​880
We calculate this as:
\(\text{Discount}_2 = \text{Price after Discount}_1 \times \frac{\text{Discount Rate}_2}{100}\)
\(\text{Discount}_2 = 880 \times \frac{8}{100} = 880 \times 0.08\)
To calculate \(880 \times 0.08\):
\(880 \times 0.08 = (800 + 80) \times 0.08 = (800 \times 0.08) + (80 \times 0.08)\)
\(= 64 + 6.4 = 70.4\)
So, the second discount amount is ​70.4.
The final selling price is the price after the first discount minus the second discount amount:
\(\text{Final Selling Price} = \text{Price after Discount}_1 - \text{Discount}_2\)
\(\text{Final Selling Price} = 880 - 70.4 = 809.6\)
The final selling price before rounding is ​809.6.
Rounding Off the Final Selling Price
The question asks us to round off the final selling price to the nearest integer.
The calculated price is ​809.6.
- The digit after the decimal point is 6.
- Since 6 is 5 or greater, we round up the integer part.
​809.6 rounded to the nearest integer is ​810.
Therefore, the final selling price of the book for the customer is ​810.
Alternative Method Using Effective Discount Rate
Successive discounts can also be calculated using a single effective discount rate. If two successive discounts are \(d_1\%\) and \(d_2\%\), the effective discount rate \(D\%\) is given by:
\(D = d_1 + d_2 - \frac{d_1 \times d_2}{100}\)
In this case, \(d_1 = 12\%\) and \(d_2 = 8\%\).
\(D = 12 + 8 - \frac{12 \times 8}{100}\)
\(D = 20 - \frac{96}{100}\)
\(D = 20 - 0.96\)
\(D = 19.04\%\)
The effective discount rate is 19.04%.
Now, calculate the total discount amount based on the original marked price:
\(\text{Total Discount} = \text{Marked Price} \times \frac{D}{100}\)
\(\text{Total Discount} = 1000 \times \frac{19.04}{100} = 1000 \times 0.1904 = 190.4\)
The total discount amount is ​190.4.
The final selling price is the marked price minus the total discount:
\(\text{Final Selling Price} = \text{Marked Price} - \text{Total Discount}\)
\(\text{Final Selling Price} = 1000 - 190.4 = 809.6\)
Rounding ​809.6 to the nearest integer gives ​810.
Both methods yield the same final selling price, ​810.
Summary of Successive Discount Calculation
| Item | Value |
|---|---|
| Marked Price | ​1000 |
| First Discount Rate | 12% |
| Amount after 12% Discount (\(1000 \times (1 - 0.12)\)) | ​880 |
| Second Discount Rate | 8% |
| Final Selling Price (\(880 \times (1 - 0.08)\)) | ​809.6 |
| Rounded Final Selling Price (Nearest Integer) | ​810 |
Revision Table: Key Concepts for Successive Discounts
| Concept | Explanation |
|---|---|
| Marked Price (MP) | The original price of an item before any discounts. |
| Selling Price (SP) | The price at which an item is sold after discounts. |
| Discount | A reduction in the marked price. Usually expressed as a percentage. |
| Successive Discounts | Applying one discount after another, where each subsequent discount is calculated on the reduced price, not the original marked price. |
| Calculation Method 1 | Apply discounts one by one on the current price. |
| Calculation Method 2 | Calculate the single effective discount rate for successive discounts and apply it to the marked price. Effective discount is less than the sum of individual discounts. |
Additional Information on Discounts and Pricing
Understanding discounts is crucial in commercial mathematics and real-life shopping scenarios. Successive discounts are common in sales, especially during festive seasons or clearance events. It's important for customers to understand how these discounts are applied to calculate the final price correctly.
- A single discount of the sum of successive discounts (e.g., 12% + 8% = 20%) would result in a lower selling price than successive discounts. In this case, a 20% discount on ​1000 would be ​200, leading to a final price of ​800. This is different from the ​810 calculated with successive discounts, highlighting that successive discounts are generally less beneficial than a single discount equal to their sum.
- The order of applying successive discounts does not affect the final selling price. Applying 8% first and then 12% would yield the same result: \(1000 \times (1 - 0.08) \times (1 - 0.12) = 1000 \times 0.92 \times 0.88 = 920 \times 0.88 = 809.6\).
Calculating final prices accurately ensures you know the actual cost of an item on sale.
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