8. Reverse Percent: Finding the Original

Reverse percent problems give the final amount and ask for the original. These feel harder because the original is hidden.

Use this idea:
\[ \text{final}=\text{multiplier}\times\text{original}. \]

If a price after 20% off is 64 dollars, then the customer paid 80% of the original:
\[ 64=0.80x. \]
Solve:
\[ x=\frac{64}{0.80}=80. \]
The original price was 80 dollars.

Tax reverse example: A total after 5% tax is 84 dollars. The total is 105% of the original:
\[ 84=1.05x,\qquad x=80. \]

Case study - paycheck raise: After a 10% raise, a worker earns 22 dollars per hour. The new wage is 110% of the old wage:
\[ 22=1.10x,\qquad x=20. \]
The old wage was 20 dollars per hour.

Common mistake: subtracting 20% from 64 to undo a 20% discount. That gives 51.20, but 64 is already the discounted value. You must divide by the paid percent.