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Two furniture stores are running double discount sales. ABC offers a discount of 60%, followed by another discount of 10% on that discounted price.

XYZ offers a discount of 50%, followed by another discount of 25% on that discounted price.

  1. Explain why a sofa with an original price of £800 will cost £288 at ABC.

Final cost $ = (800*\frac{40}{100})*\frac{90}{100} = £288$

  1. How much will the same sofa cost at XYZ?

Final cost $ = (800*\frac{50}{100})*\frac{75}{100} = £300$

XYZ decides to change its discount structure. It wants to offer exactly the same overall percentage savings as ABC. It proposes to offer a discount of x%, followed by another discount of x% on that discounted price.

  1. Write down an equation for x, and solve it to find the value of x.

ABC overall percentage discount $= \frac{800-288}{800} * 100 = 64$%

XYZ final cost 'C' with new discounting: $C =(800*\frac{100-x}{100}) * \frac{100-x}{100} = 288$

Therefore: $8x^2 - 1600x +51200=0$, roots of x 160, 40, gives x = 40%

Seems a bit overly convoluted to come to that result, if it's correct, any thoughts on a simpler/cleaner approach?

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    $\begingroup$ For the 3rd problem, there is a shortcut. Suppose that instead of focusing on the discount, you focus on the percentage of the price after the discount. For example, a discount of $d$% results in a price of $e$% of the original, where $d + e = 100$%. Then, you can solve for $e$ immediately: $e^2 = 0.36 \implies e = 0.6 \implies d = 0.4$. So, the price goes from $100$% to $36$% after two consecutive discounts of $40$% precisely because $\sqrt{0.36} = 0.6$. $\endgroup$ Commented Sep 20, 2021 at 23:55

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The first two parts are as direct as it can get apart from using decimal numbers ($0.4\cdot 0.9=0.36$ might be easier than $\frac{40}{100}\cdot\frac{90}{100}$).

For part 3, ABC overall percentage discount doesn't seem to be used? So I am not sure why it is calculated.

Apart from that, notice that $288=800\cdot 0.36$ (from part (1)). We can use this along with some squared terms grouping to simplify greatly here:

$$C=800\cdot\frac{100-x}{100}\cdot\frac{100-x}{100}=800\cdot 0.36$$

$$(\frac{100-x}{100})^2=0.36$$

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

$$x=40$$

To avoid expanding the whole equation.

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