How do you work out the old price of an item having been given the new price after a specified percentage change?

These problems can be worked out using the equation:
new = old x multiplier

new: is the new price of an item
old: is the old price of an item (before it was changed)
multiplier: is the percentage of the old price the new price represents

To calculate the multiplier, you must first know if there is an increase in price or a decrease in price:

Increase in price:
If the old price is increased to the new price

A t-shirt cost £10, it is sold for a 12% profit. What is it sold for?
Old = £10
New = ?
Multiplier = as there has been an increase, the new price is 100% + 12% = 112% of the old price. The multiplier is therefore (112%)/100 = 1.12
new = old x multiplier = 10 x 1.12 = £11.20

The question may also be asked as:
A t-shirt is sold for £11.20, which is a profit of 12% on the original price. What is the original price?
old = new/multiplier = 11.20/1.12 = £10
Or, a t-shirt is bought for £10 and sold for £11.20. What is the percentage profit?
multiplier = new/old = 11.2/10 = 1.12
percentage = multiplier x 100 = 1.12 x 100 = 112% <-- so the new is 112% of the old, which is a profit of 112 - 100 = 12% (as the old value is always 100% of the price)

Decrease in price:
If the old price is decreased to the new price

A t-shirt is on sale. The original price is £15, there is 20% off in the sale. How much does the t-shirt cost?
Old = £15
New = ?
Multiplier = the new is going to be 100% - 20% = 80% of the old. So the multiplier is 80/100 = 0.8
new = old x multiplier = 15 x 0.8 = £12

The question may also be asked as:
A t-shirt costs £12 in a 20% off sale. What is the original cost of the t-shirt?
old = new/multiplier = 12/0.8 = £15
Or, a t-shirt in a sale costs £12, the original price is £15. What percentage has been taken from the original price?
multiplier = new/old = 12/15 = 0.8
percentage = 0.8 x 100 = 80% <-- so the new price is 80% of the old price, which gives a percentage change of 80 - 100 = -20%, or a decrease of 20%.
So, there is 20% off in the sale.

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