### Question 11

SSC-CGL 2018 June 4 Shift 1

After giving two successive discounts each of x% on the marked price of an item, total discount is Rs. 259.20. If the face value of the object is Rs. 720, what will be the value of x?

- 25
- 24
- 18
- 20

### Solution in Brief

Starting price = Rs. 720. There are two successive disounts compounded at $\displaystyle x\%$. Final price by compound formula is $\displaystyle 720(1 - x)^2$

Hence the total discount is $\displaystyle 720(1 - x)^2 - 720$ = $\displaystyle 720x(x - 2)$, which can be equated to the given figure of 259.20 to get: 720x(2 - x) = 259.29

Next, we can cycle through the options to obtain x = 0.2 or 20% as the answer!

### Solution in Detail

Starting Marked Price = $\displaystyle 720$ Rs.

First $\displaystyle \%$ discount is $\displaystyle x$

$\displaystyle \therefore $ first discount = Rs. $\displaystyle 720 \times x \text{ . . . (1)}$

$\displaystyle \therefore $ Price = $\displaystyle 720 - 720x = 720(1 - x)$

Second discount is again of $\displaystyle x\%$

$\displaystyle \therefore = [720(1 - x)] x\text{. . . (2)}$

Total discount = $\displaystyle (1) + (2)$

$\displaystyle = 720 x + 720(1 - x) x$

$\displaystyle = 720 x [1 + (1 - x)]$

$\displaystyle = 720 x \times (2 - x)$

It has been given as 259.20

$\displaystyle \implies 720 x \times (2 - x) = 259.20$

Cycling through the options, x = 0.2 or 20% is the answer!

### Video Solution

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### Solution by Alternate Method

This is a standard shortcut formula that can be memorizedEffective rate after two successive and same discounts is $\displaystyle x (2 - x)$

Hence $\displaystyle 720 \times x(2 - x) = 259.20$

$\displaystyle \implies x (2 - x) = \frac{259.20}{720} = 0.36$

By cycling the options, or even, by observation, $\displaystyle x = 20 \%$ Ans

### Solution by Another Alternate Method

Discount is $\displaystyle \text{MP} \times [1 - (1 - x)^2]$

EXPLANATION: sale price after first discount is $\text{MP} - (\text{MP} \times x)$ = $\text{MP}(1 - x)$. This becomes marked price for the second discount, giving the final sale price of $[\text{MP}(1 - x)] \times (1 - x) = \text{MP}(1 - x)^2$, whence the above expression.Hence, $\displaystyle 720 \times \bigg[1 - (1 - x)^2\bigg] = 259.20$

$\displaystyle \implies 1 - (1 - x)^2 = \frac{259.20}{720} = 0.36$

$\displaystyle \implies 1 - x = \sqrt{1 - 0.36} = \sqrt {0.64} = 0.8$

$\displaystyle \implies x = 0.2$ or 20% Ans.

### Solution by Estimation

The overall discount is $\displaystyle \frac{259.20}{720} = 0.36$ or 36%

Each of the discounts approximates to 18%. But the case is of two successive discounts, hence, therefore, the first discount has to be, in any case, greater than 18%. Reject 18%, and try the next nearer option, i.e., 20%.

First discount of 20% gives discount = 20% of 720 = Re. 144

The second discount has to be further 20% less than the first discount, i.e, = 144 - 20% of 144 = 144 - 28.8 = 115.2.

The total overall discount is 144 + 115.2 = 259.20 Rs, which confirms 20% as the answer!

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