Question

# A discount series of 10%, 20% and 40% is equal to a single discount ofA. 50%B. 56.80%C. 60%D. 70.28%

Hint: We will assume that the total cost of items on which discount is levied is Rs. 100. Then considering 10% discount on the total cost, we get $\dfrac{10}{100}\times 100=10$, hence we get the cost of the item after 10% discount as Rs. 100 – Rs. 10 = Rs. 90. Now we can take a 20% discount on Rs. 90 just like we did in the case of 10%. Similarly we will find the cost of the item after a 40% discount and then add all three values in order to find the value of the single discount.

Complete step-by-step solution -
In the question we are given a discount series of 10%, 20% and 40%. We have to find the single discount that is equivalent to all the three discounts. Let us assume that the initial cost of a material is Rs. 100. We are considering Rs. 100 as it is easy for us to relate and solve the question.
Now, we have a first discount of 10% on Rs. 100. So we get, $\dfrac{10}{100}\times 100=Rs.10$. So in the first step we get a discount of Rs. 10. Therefore, we get the cost of the item after a discount of Rs. 10 as, Rs. 100 – Rs. 10 = Rs. 90.
Now we have to find the second discount of 20% on Rs. 90. So, we get, $\dfrac{20}{100}\times 90=Rs.18$. So in the second step we get a discount of Rs. 18. Therefore, we get the cost of the item after a discount of Rs. 18 as, Rs. 90 – Rs. 18 = Rs. 72.
Now we have to find the third discount of 40% on Rs. 72. So we get, $\dfrac{40}{100}\times 72=Rs.\text{ }28.80$. So in the third step we get a discount of Rs. 28.80.
Now we have to find the total discount which is the sum of all the three discount values. So, total discount = Rs. 10 + Rs. 18 + Rs. 28.80 = Rs. 56.80.
Therefore, with the initial price assumed as Rs. 100, the single discount is equivalent to the total discount, Rs. 56.80. We can write Rs. 56.80 in percentage as 56.80% as we had assumed the initial price as Rs. 100. So we have found out the total discount offered as 56.80%. Hence the single equivalent discount is 56.80%. So option (B) is the correct answer.

Note: This question can also be solved by using an alternative method. As we have assumed the cost price of the item as Rs. 100, after applying the discount of 10%, 20% and 40%, we get the cost of the item as (100 - 40) % of (100 - 20) % of (100 - 10) % of Rs. 100.
$=60%\times 80%\times 90%\times 100$
$=\dfrac{60}{100}\times \dfrac{80}{100}\times \dfrac{90}{100}\times 100$
$=\dfrac{6\times 8\times 9}{10}$
= 43.20%
So as we had assumed initially, that the cost price of the item is Rs. 100, after all the three discounts, the cost price of the item will get reduced to Rs. 43.20.
Hence, we can calculate the discount offered as Rs. 100 – Rs. 43.20 = Rs. 56.80.
Thus the single discount of 56.80% is equivalent to all three discounts.