Answer

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Hint: We know that there is a formula which relates discount percentage, discount and marked price and that formula is given by \[Discount\text{ }percentage=\dfrac{Discount}{marked\text{ }price}\times 100\] where, discount can be given by this formula $discount=marked\text{ }price-purchased\text{ }price$.

Complete step-by-step solution:

Arun bought a pair of skates at a sale with a discount of 20% and the amount paid by Arun was Rs.1,600.

We use this formula to find the discount \[Discount\text{ }percentage=\dfrac{Discount}{marked\text{ }price}\times 100\].

So, \[Discount=\dfrac{Discount\text{ }percentage\times marked\text{ }price}{100}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)\]

We know that discount percentage=20 and let the marked price be x and let it be equation 2 as from the above question.

Now we will substitute discount percentage and marked price in equation 1.

\[Discount=\dfrac{20\times x}{100}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 3 \right)\]

Now, simplifying the equation 3 i.e. dividing 20 by 100 we will get $\dfrac{1}{5}$ which is a discount price.

\[Discount=\dfrac{x}{5}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 4 \right)\]

So, \[discount=\dfrac{x}{5}\]

We use this formula $discount=marked\text{ }price-purchased\text{ }price\cdot \cdot \cdot \cdot \cdot \cdot \left( 5 \right)$ to find marked price.

Purchased price which is already given in the question and from equation 4, discount=$\dfrac{x}{5}$ and the purchased price is 1600.

From equation 5 $marked\text{ }price=discount+purchased\text{ }price\cdot \cdot \cdot \cdot \cdot (6)$.

We know that marked price=x and discount=$\dfrac{x}{5}$.

Now, we will substitute the values in equation 6 we will get,

$\Rightarrow x=\dfrac{x}{5}+1600$

Now simplifying it becomes $x-\dfrac{x}{5}=1600$

Further simplification $\Rightarrow \dfrac{4x}{5}=1600$

Now finding x,

$\Rightarrow x=\dfrac{5}{4}\times 1600$

Now cancelling 1600 with 4 as 1600 is multiple of 4.

$\therefore marked\text{ }price=2000$ .

The marked price on a pair of skates is 2000.

Note: There is an alternative way to solve the above problem.

Arun paid 80% of the marked price because he got a 20% discount on the marked price. So, finally 80% of marked price is equal to paid price.

Let the marked price be x and the paid price is 1600 which is in the question.

80% of x=paid price

So,$\Rightarrow \dfrac{80}{100}\times x=1600$

Now simplifying the above (i.e. dividing 80 with 100) we will get

$\Rightarrow x=1600\times \dfrac{5}{4}$

Now multiplying with $\dfrac{5}{4}$ on both sides.

We will get x=2000.

i.e. marked price on price of skates=Rs.2000.

Complete step-by-step solution:

Arun bought a pair of skates at a sale with a discount of 20% and the amount paid by Arun was Rs.1,600.

We use this formula to find the discount \[Discount\text{ }percentage=\dfrac{Discount}{marked\text{ }price}\times 100\].

So, \[Discount=\dfrac{Discount\text{ }percentage\times marked\text{ }price}{100}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)\]

We know that discount percentage=20 and let the marked price be x and let it be equation 2 as from the above question.

Now we will substitute discount percentage and marked price in equation 1.

\[Discount=\dfrac{20\times x}{100}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 3 \right)\]

Now, simplifying the equation 3 i.e. dividing 20 by 100 we will get $\dfrac{1}{5}$ which is a discount price.

\[Discount=\dfrac{x}{5}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 4 \right)\]

So, \[discount=\dfrac{x}{5}\]

We use this formula $discount=marked\text{ }price-purchased\text{ }price\cdot \cdot \cdot \cdot \cdot \cdot \left( 5 \right)$ to find marked price.

Purchased price which is already given in the question and from equation 4, discount=$\dfrac{x}{5}$ and the purchased price is 1600.

From equation 5 $marked\text{ }price=discount+purchased\text{ }price\cdot \cdot \cdot \cdot \cdot (6)$.

We know that marked price=x and discount=$\dfrac{x}{5}$.

Now, we will substitute the values in equation 6 we will get,

$\Rightarrow x=\dfrac{x}{5}+1600$

Now simplifying it becomes $x-\dfrac{x}{5}=1600$

Further simplification $\Rightarrow \dfrac{4x}{5}=1600$

Now finding x,

$\Rightarrow x=\dfrac{5}{4}\times 1600$

Now cancelling 1600 with 4 as 1600 is multiple of 4.

$\therefore marked\text{ }price=2000$ .

The marked price on a pair of skates is 2000.

Note: There is an alternative way to solve the above problem.

Arun paid 80% of the marked price because he got a 20% discount on the marked price. So, finally 80% of marked price is equal to paid price.

Let the marked price be x and the paid price is 1600 which is in the question.

80% of x=paid price

So,$\Rightarrow \dfrac{80}{100}\times x=1600$

Now simplifying the above (i.e. dividing 80 with 100) we will get

$\Rightarrow x=1600\times \dfrac{5}{4}$

Now multiplying with $\dfrac{5}{4}$ on both sides.

We will get x=2000.

i.e. marked price on price of skates=Rs.2000.

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