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Find a single discount equivalent to following successive discounts of 50%, 10% and 20% in percent?(a) 54%(b) 64%(c) 74%(d) 84%

Last updated date: 20th Jun 2024
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Hint: We start solving the problem by assuming a variable for the cost of the material. We then apply a 50% discount to this price by using the fact that a% of b as $\dfrac{a}{100}\times b$ to get a new price. We then apply a 10% discount to this new price using the similar fact to get updated price. We then apply 20% to this updated price to get the final price of the product. We then find the total equivalent discount by using $\dfrac{\text{initial price }-\text{ final price }}{\text{initial price}}\times 100$.

According to the problem, we are asked to find the single discount equivalent to successive discounts of 50%, 10% and 20%.
Let us assume the price for which we apply the successive discounts be ‘x’.
Let us apply 50% to this price. So, the new price will be $\left( 100-50 \right)\%=50\%$ of ‘x’.
We know that a% of b is defined as $\dfrac{a}{100}\times b$.
So, the new price will be $\dfrac{50}{100}\times x=0.5x$.
Now, let us apply a 10% discount to this new price. So, the updated price will be $\left( 100-10 \right)\%=90\%$ of $0.5x$.
So, the updated price will be $\dfrac{90}{100}\times 0.5x=0.45x$.
Now, let us apply a 20% discount to this updated price. So, the final price will be $\left( 100-20 \right)\%=80\%$ of $0.45x$.
So, the final price will be $\dfrac{80}{100}\times 0.45x=0.36x$.
We know that the percent change in the price will be the final equivalent discount provided.
So, we have final equivalent discount = $\dfrac{\text{initial price }-\text{ final price }}{\text{initial price}}\times 100$.
$\Rightarrow$ Final equivalent discount = $\dfrac{x-0.36x}{x}\times 100$.
$\Rightarrow$ Final equivalent discount = $\dfrac{0.64x}{x}\times 100$.
$\Rightarrow$ Final equivalent discount = $64\%$.
So, we have found the single equivalent discount of the successive discounts 50%, 10% and 20% as 64%.

So, the correct answer is “Option b”.

Note: We should know that if we apply a% discount to any price, then a% of it will be reduced and $\left( 100-a \right)\%$ will be retained back. We should not make calculation mistakes while solving this problem. We should keep in mind that the amount reduced from the original price will be the total discount applied on it. Similarly, we can expect problems to find the equivalent discount if a 10% discount is applied on this final price.