# Calculation of successive discount percentage

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?

• 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 $\sqrt{0.36} = 0.6$. Commented Sep 20, 2021 at 23:55

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.