If the successive discount is 10% and 15%, then what is the single equivalent rate of discount?
Understand the Problem
The question is asking how to calculate the single equivalent rate of discount when two successive discounts of 10% and 15% are applied. This involves using the formula for successive discounts to find the overall discount percentage.
Answer
The equivalent single rate of discount is $23.5\%$.
Answer for screen readers
The equivalent single rate of discount for successive discounts of 10% and 15% is $23.5%$.
Steps to Solve
- Define the Discounts
Let the original price of an item be denoted as $P$. We will apply the two discounts of 10% and 15% successively.
- Calculate the first discount
The first discount is 10% of $P$. Thus, after the first discount, the price becomes:
$$ P_1 = P - 0.1P = 0.9P $$
- Calculate the second discount
The second discount of 15% is applied to the new price $P_1$. Therefore, after applying this 15% discount, the price becomes:
$$ P_2 = P_1 - 0.15P_1 = 0.85P_1 = 0.85 \times 0.9P = 0.765P $$
- Determine the final equivalent discount
To find the equivalent single discount, we can compare the final price $P_2$ with the original price $P$. The equivalent discount is given by:
$$ \text{Equivalent Discount} = P - P_2 = P - 0.765P = 0.235P $$
Now, to express this as a percentage, we divide by the original price:
$$ \text{Equivalent Discount %} = \frac{0.235P}{P} \times 100% = 23.5% $$
The equivalent single rate of discount for successive discounts of 10% and 15% is $23.5%$.
More Information
When applying successive discounts, the overall discount is not simply the sum of the individual discounts. Instead, you multiply the remaining amounts to get the final result. In this case, obtaining a total discount of 23.5% is a common example in finance and retail.
Tips
- Assuming Direct Addition: A common mistake is to simply add the two discount percentages (10% + 15%) to get 25%. Remember, the discounts are applied successively, not directly added.
- Confusion Over New Prices: Students sometimes overlook that each subsequent discount is based on the new price, not the original price.
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