Find a single discount percentage equivalent to successive discounts of 10%, 20% and 25%.
46%
Calculating Equivalent Single Discount
When multiple discounts are applied one after another on an item's price, these are called successive discounts. To find the single discount percentage that would result in the same final price, we need to calculate the overall percentage reduction from the original price.
Let's find the equivalent single discount for successive discounts of 10%, 20%, and 25%.
Method 1: Assuming an Initial Price
We can assume an initial price for the item, for example, ₹100, to make the calculations easy.
- First Discount (10%):
The price after a 10% discount on ₹100 is:
\(\text{Price after 1st discount} = ₹100 - 10\% \text{ of } ₹100\)
\( = ₹100 - ₹\frac{10}{100} \times 100\)
\( = ₹100 - ₹10\)
\( = ₹90\)
- Second Discount (20%):
The second discount of 20% is applied to the new price, which is ₹90.
\(\text{Discount amount} = 20\% \text{ of } ₹90\)
\( = ₹\frac{20}{100} \times 90\)
\( = ₹\frac{1}{5} \times 90\)
\( = ₹18\)
The price after the second discount is:
\(\text{Price after 2nd discount} = ₹90 - ₹18\)
\( = ₹72\)
- Third Discount (25%):
The third discount of 25% is applied to the latest price, which is ₹72.
\(\text{Discount amount} = 25\% \text{ of } ₹72\)
\( = ₹\frac{25}{100} \times 72\)
\( = ₹\frac{1}{4} \times 72\)
\( = ₹18\)
The final price after the third discount is:
\(\text{Final Price} = ₹72 - ₹18\)
\( = ₹54\)
The original price was ₹100, and the final price after successive discounts is ₹54.
Total discount amount = Original Price - Final Price
\( = ₹100 - ₹54\)
\( = ₹46\)
The equivalent single discount percentage is calculated on the original price:
\(\text{Equivalent Single Discount} = \frac{\text{Total Discount}}{\text{Original Price}} \times 100\%\)
\( = \frac{₹46}{₹100} \times 100\%\)
\( = 46\%\)
Method 2: Using the Formula for Successive Discounts
The formula for the equivalent single discount for two successive discounts, \(d_1\)% and \(d_2\)%, is given by:
\(\text{Equivalent Discount} = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\%\)
We have three successive discounts: 10%, 20%, and 25%.
First, let's find the equivalent discount for the first two discounts, 10% and 20%.
Let \(d_{12}\) be the equivalent discount for 10% and 20%.
\(d_{12} = \left(10 + 20 - \frac{10 \times 20}{100}\right)\%\)
\( = \left(30 - \frac{200}{100}\right)\%\)
\( = (30 - 2)\%\)
\( = 28\%\)
So, successive discounts of 10% and 20% are equivalent to a single discount of 28%.
Now, we find the equivalent discount for this 28% and the third discount of 25%.
Let \(D\) be the equivalent single discount for 28% and 25%.
\(D = \left(28 + 25 - \frac{28 \times 25}{100}\right)\%\)
\( = \left(53 - \frac{700}{100}\right)\%\)
\( = (53 - 7)\%\)
\( = 46\%\)
Both methods give the same result.
Summary of Successive Discount Calculation
Applying discounts of 10%, 20%, and 25% successively results in a final price that is 46% less than the original price. Therefore, the single equivalent discount is 46%.
| Discount | Calculation (starting with ₹100) | Price After Discount |
|---|---|---|
| 10% | \(100 \times (1 - 0.10) = 100 \times 0.90\) | ₹90 |
| 20% (on ₹90) | \(90 \times (1 - 0.20) = 90 \times 0.80\) | ₹72 |
| 25% (on ₹72) | \(72 \times (1 - 0.25) = 72 \times 0.75\) | ₹54 |
The total reduction is \(₹100 - ₹54 = ₹46\), which is 46% of the original ₹100.
Revision Table: Key Discount Concepts
| Concept | Description | Calculation Example |
|---|---|---|
| Single Discount | A direct percentage reduction on the original price. | 10% off ₹200 = \(200 \times \frac{10}{100} = ₹20\) discount. Final price \(₹180\). |
| Successive Discounts | Multiple discounts applied one after another on the reduced price each time. | 10% off ₹100 then 20% off the new price. 10% off ₹100 is ₹10 (\(₹90\)). 20% off ₹90 is ₹18 (\(₹72\)). Final price \(₹72\). |
| Equivalent Single Discount | The one discount percentage that gives the same final price as applying successive discounts. | As calculated above, 10%, 20%, and 25% successive discounts are equivalent to a single 46% discount. |
Additional Information on Discounts
Discounts are common in retail and sales scenarios. Understanding how successive discounts work is important because they are not simply additive. For example, a 10% discount followed by a 20% discount is NOT equivalent to a 30% discount.
The order of successive discounts does not affect the final price. Applying a 10% discount then a 20% discount yields the same result as applying a 20% discount then a 10% discount.
For example, using ₹100:
- 10% off, then 20% off: \(100 \xrightarrow{-10\%} 90 \xrightarrow{-20\%} 72\)
- 20% off, then 10% off: \(100 \xrightarrow{-20\%} 80 \xrightarrow{-10\%} 72\)
In general, for successive discounts \(d_1\), \(d_2\), \(d_3\), ..., the final price as a fraction of the original price is \((1 - \frac{d_1}{100})(1 - \frac{d_2}{100})(1 - \frac{d_3}{100})...\). The equivalent single discount \(D\) is then \(1 - (1 - \frac{d_1}{100})(1 - \frac{d_2}{100})(1 - \frac{d_3}{100})...\) expressed as a percentage.
Using this formula for 10%, 20%, 25%:
Final price fraction \( = (1 - \frac{10}{100})(1 - \frac{20}{100})(1 - \frac{25}{100})\)
\( = (1 - 0.10)(1 - 0.20)(1 - 0.25)\)
\( = (0.90)(0.80)(0.75)\)
\( = 0.72 \times 0.75\)
\( = 0.54\)
This means the final price is 0.54 times the original price, or 54% of the original price.
The total discount is \(1 - 0.54 = 0.46\), which is 46% of the original price.
Understanding successive discounts helps consumers and businesses calculate actual price reductions accurately.
Important Questions from Discount and MP
The marked price of an article is Rs. 1500. A shopkeeper sells it by giving 20% discount on its marked price. If the cost price of the article is Rs. 991, then his profit (in Rs.) is:
Surbhi sold an article for Rs. 176 after giving 12% discount on its marked price. Had she not given any discount; she would have earned a profit of 25%. What is the cost price (in Rs.) of the article?
The marked price of an article is Rs. 240. A shopkeeper sells it by allowing 18% discount on its marked price and still gains 23%. What is the cost price (in Rs.) of the article?
Three shopkeepers A, B and C marked on an identical article at Rs. 4820. A, B and C sold their article on successive discounts of 20% and 20%; 25% and 15%; 30% and 10% respectively. Which shopkeeper gives the maximum discount and how much (in Rs.)?
A shopkeeper marks the price of the article in such a way that after allowing 28% discount, he wants a gain of 12%. If the marked price is Rs. 224, then the cost price of the article is:
