Question

Three successive discounts of \[6\% ,{\text{ }}10\% ,{\text{ }}15\% \] is equal to a single discount of ?

Answer 1

Hint: In order to solve this question, firstly we will assume any price on which we will find the three successive discounts i.e., \[6\% ,{\text{ }}10\% ,{\text{ }}15\% \] .So, we will assume the price to be \[100\] and we will calculate all the three discount values using the formula i.e., \[\left( {a - discount\% a} \right)\] where \[a\] is the number we assumed i.e., \[100\] we will find for \[6\% \] .After that the value we get after calculating for \[6\% \] will be the new value of \[a\] which is to be used to calculate for \[10\% \] and similarly the new value we get after calculating for \[10\% \] will be the value used to calculate for \[15\% \] .At last we will use the same formula where we have to calculate \[discount\% \] Thus, after simplification we get the required result.

Complete step by step answer:
Here, first we will assume any price on which we will find three successive discounts. So, let us assume the price to be \[100\].Now, first we calculate for \[6\% \] using the formula i.e., \[\left( {a - discount\% a} \right)\].
Here, \[a = 100\]
\[{\text{ }}6\% \] discount on \[100\] will be given as
\[\Rightarrow 100 - \left( {6\% \times 100{\text{ }}} \right) - - - \left( 1 \right)\]

Now we know that \[x\% = \dfrac{x}{{100}}\]
So, we can write \[6\% \] as \[\dfrac{6}{{100}}\]
Equation \[\left( 1 \right)\] becomes,
\[100 - \left( {\dfrac{6}{{100}} \times 100} \right){\text{ }} = 100 - 6 = 94\]
Now, we will calculate for \[10\% \] .
So, the new value of \[a\] will be \[94\]
\[{\text{ 10}}\% \] discount on \[94\] will be given as:
\[94 - \left( {10\% \times 94} \right){\text{ }} - - - \left( 2 \right)\]
Here we can write \[10\% \] as \[\dfrac{{10}}{{100}}\]

Equation \[\left( 2 \right)\] becomes,
\[94 - \left( {\dfrac{{10}}{{100}} \times 94} \right){\text{ }} = \left( {94 - 9.4} \right) = 84.6\]
At last, we will calculate for \[15\% \]
So, the new value of \[a\] will be \[84.6\]
\[{\text{ 15}}\% \] discount on \[84.6\] will be given as:
\[84.6 - \left( {15\% \times 84.6} \right){\text{ }} - - - \left( 3 \right)\]
Here we can write \[15\% \] as \[\dfrac{{15}}{{100}}\]
Equation \[\left( 3 \right)\] becomes,
\[84.6 - \left( {\dfrac{{15}}{{100}} \times 84.6} \right){\text{ }} = \left( {84.6 - 12.69} \right) = 71.91\]

Now, we will calculate how much percent of \[100\] is equal to \[71.91\] and use the same formula i.e., \[\left( {a - discount\% a} \right)\] to calculate the \[discount\% \]
So, after substituting the values, we get
\[100 - \left( {discount\% {\text{ }} \times 100} \right){\text{ = 71}}{\text{.91}}\]
Here we can write \[discount\% \] as \[\dfrac{{discount}}{{100}}\]
\[{\text{ }}100 - \left( {\dfrac{{discount}}{{100}}{\text{ }} \times 100} \right){\text{ = 71}}{\text{.91}}\]
On simplification we get,
\[100 - 71.91 = {\text{ }}discount\]
\[ \therefore discount = 28.09\] which is the required answer.

Hence, three successive discounts of \[6\% ,{\text{ }}10\% ,{\text{ }}15\% \] is equal to a single discount of \[28.09\% \].

Note: This problem can also be solved by assuming the initial price as some unknown variable. Also, in order to solve such problems never try to add the discounts to get the single discount.
This problem can be solved by another method also:
i.e., first assume any price on which we will find three successive discounts.
So, let us assume the price to be \[100\]
Now, we will find the sales price after three successive discounts using formula,
i.e., \[{\text{Price}} \times \left( {1 - \dfrac{x}{{100}}} \right)\left( {1 - \dfrac{y}{{100}}} \right)\left( {1 - \dfrac{z}{{100}}} \right)\]
where \[x,{\text{ }}y,{\text{ }}z\] are the three successive discounts respectively.
So, here \[x = 6\% ,{\text{ }}y = 10\% ,{\text{ }}z = 15\% \]
\[\therefore \] sales price \[ = 100\left( {1 - \dfrac{6}{{100}}} \right)\left( {1 - \dfrac{{10}}{{100}}} \right)\left( {1 - \dfrac{{15}}{{100}}} \right)\]
On simplification we get
sales price \[ = 71.91\]
Now to get the required discount, we will subtract \[71.91\] from \[100\]
So, \[discount = 100 - 71.91 = 28.09\% \] which is the required answer.