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.
- 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$
- 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.
- 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?
Best Answer
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.