The amount before the change

After a 20% discount, a pair of trousers costs 120 ₪.
How much did it cost before the discount?

• The price before the discount is unknown: call it $\displaystyle x$.
• A 20% discount is multiplying by 0.8, so the price after the discount is $\displaystyle 0.8x$.
• It is 120 ₪: $\displaystyle {0.8x}={120}$.
• Divide by 0.8: $\displaystyle {x}={150}$. The trousers cost 150 ₪.

A 20% discount is 20% of the price before the discount, not of 120.
So adding twenty percent of 120 to it does not bring back the original price:

• $\displaystyle {1.2\times120}={144}$ – not 150.
• 20% of 120 is 24 ₪, but the discount was 20% of 150: $\displaystyle {150-120}={30}$ ₪.

Solve.

After a 25% rise, a school has 500 students.
How many students did it have before the rise?

• A 25% rise is multiplying by 1.25: $\displaystyle {1.25x}={500}$.
• $\displaystyle {x}={500:1.25}={400}$ students.

On most goods in Israel you pay VAT (value added tax): the price to the buyer is 18% more than the price without VAT.

• A computer costs 2000 ₪ without VAT. With VAT: $\displaystyle {1.18\times2000}={2360}$ ₪.
• Repairing a washing machine cost 590 ₪ with VAT. Without VAT: $\displaystyle {1.18x}={590}$, so $\displaystyle {x}={500}$ ₪.

After a 30% discount, a bicycle costs 840 ₪.
How much did it cost before the discount?

A restaurant bill, with VAT, is 236 ₪.
How much is it without VAT?

After a price rise of 8%, a fridge costs 4320 ₪.
By how many shekels did the price rise?

After a discount, a pair of shoes costs 180 ₪.
Ron says: “The discount was 10%, so the price before the discount was $\displaystyle {180+18}={198}$ ₪.”
Is he right?
Explain.