Understanding the Item Discount Problem

The question asks us to find the value of a specific discount percentage, denoted by 'x', applied during a festival sale. We are given the original price of an item, the effect of two consecutive discounts (the x% sale discount and a 10% coupon discount), and the final price after both discounts are applied.

Let's break down the information provided:

• Original price of the item: Rs. 400
• First discount: x% off the regular price (Rs. 400)
• Second discount: 10% off the price after the first discount
• Final price after both discounts: Rs. 216

We need to determine the value of x.

Step-by-Step Solution to Find the Discount Value x

To solve this, we will apply the discounts sequentially as described in the problem and set up an equation with the final price.

Step 1: Calculate the price after the x% sale discount

A discount of x% on the original price of Rs. 400 means the price is reduced by $\frac{x}{100} \times 400$.

The price after the x% discount will be:

Original Price - Sale Discount Amount

$\text{Price after } x\% \text{ discount} = 400 - \left(\frac{x}{100} \times 400\right)$

This can also be written as:

$\text{Price after } x\% \text{ discount} = 400 \times \left(1 - \frac{x}{100}\right)$

Step 2: Apply the 10% coupon discount

The 10% coupon discount is applied to the price obtained after the first discount (from Step 1). Let's call the price after the x% discount as $P_1$. So, $P_1 = 400 \times \left(1 - \frac{x}{100}\right)$.

A 10% discount on $P_1$ means the price is reduced by $\frac{10}{100} \times P_1$.

The price after the 10% coupon will be:

$P_1$ - Coupon Discount Amount

$\text{Final Price} = P_1 - \left(\frac{10}{100} \times P_1\right)$

This can also be written as:

$\text{Final Price} = P_1 \times \left(1 - \frac{10}{100}\right)$

Substitute the expression for $P_1$ from Step 1:

$\text{Final Price} = \left[400 \times \left(1 - \frac{x}{100}\right)\right] \times \left(1 - \frac{10}{100}\right)$

Step 3: Set up the equation using the given final price

We are given that the final price after both discounts is Rs. 216. So, we can set the expression from Step 2 equal to 216:

$\left[400 \times \left(1 - \frac{x}{100}\right)\right] \times \left(1 - \frac{10}{100}\right) = 216$

Detailed Calculation for the Discount Percentage x

Now, let's solve the equation for x:

First, simplify the term $\left(1 - \frac{10}{100}\right)$:

$1 - \frac{10}{100} = 1 - 0.1 = 0.9$

Substitute this back into the equation:

$\left[400 \times \left(1 - \frac{x}{100}\right)\right] \times 0.9 = 216$

Now, divide both sides by 0.9:

$400 \times \left(1 - \frac{x}{100}\right) = \frac{216}{0.9}$

$400 \times \left(1 - \frac{x}{100}\right) = 240$

Next, divide both sides by 400:

$1 - \frac{x}{100} = \frac{240}{400}$

Simplify the fraction $\frac{240}{400}$:

$\frac{240}{400} = \frac{24}{40} = \frac{3}{5} = 0.6$

So, the equation becomes:

$1 - \frac{x}{100} = 0.6$

Subtract 0.6 from 1 to isolate $\frac{x}{100}$:

$\frac{x}{100} = 1 - 0.6$

$\frac{x}{100} = 0.4$

Finally, multiply by 100 to find x:

$x = 0.4 \times 100$

$x = 40$

Thus, the value of x is 40.

Revision Table: Key Concepts

Additional Information on Sequential Discounts

It is important to understand that when multiple discounts are applied sequentially, they are not simply added together. A 10% discount followed by a 20% discount on an item is not equivalent to a single 30% discount. Each subsequent discount is calculated on the already reduced price.

For example, on an item of Rs. 100:

• A single 30% discount makes the price Rs. $100 \times (1 - 0.30) = 100 \times 0.70 = 70$.
• A 10% discount followed by a 20% discount:
  • Price after 10% discount: Rs. $100 \times (1 - 0.10) = 100 \times 0.90 = 90$.
  • Price after subsequent 20% discount on Rs. 90: Rs. $90 \times (1 - 0.20) = 90 \times 0.80 = 72$.

As you can see, Rs. 70 is different from Rs. 72. Our problem involves sequential discounts, which is why we applied the 10% coupon discount to the price remaining after the x% sale discount.