Math CentralQuandaries & Queries


Question from Margaret:

What is the formula to find the 75% off full price item that was already marked 30% off?

Hi Margaret,

I'm not sure what you need. Do you know the original price? Do you know the 30% off price?

Do you want to find the price if the 30% off price is reduced by 75% or do you want to know the what the price would be original price were reduced by 75%? Perhaps you want something else.


Margaret wrote back

Penny,Thank you for your response. I work retail. Our employee discount is 75%. If there is an item that's on sale we can't override the price so I have to figure out what that item would be to give the 75% off price.

An example,

An item is $\$100,$ it's marked down 30% or $\$70,$ the employee discount would be 75% of $\$100$ for a total of $\$25.$ Is there a formula I can use to come up the correct price? I know the end number but have to do this with percentages.


Thanks Margaret. I understand now.

Suppose the original price of the item is $\$p,$ in your example $\$p = \$100,$ and the sale price is $\$s,$ in your example $\$s = \$70.$ Suppose also that the employee price is $\$e,$ in your example $\$e = \$25.$ You know $\$s$ and you want to determine $\$p$ so that you can calculate $\$e.$

Since the sale is 30% off the relationship between $\$p$ and $\$s$ is that $\$s$ is 70% of $\$p$ or

\[s = 0.70 \times p.\]


\[p = \frac{s}{0.70}.\]

Since employees get a 75% discount, $\$e$ is 25% of $\$p,$ and hence

\[e = 0.25 \times p.\]

Putting the two expressions together gives

\[e = 0.25 \times \frac{s}{0.70} = \frac{0.25}{0.70} \times p = \frac{25}{70} \times p\]

which simplifies to

\[e = \frac{5}{14} \times p.\]

I hope this is what you are looking for.



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