Overview
A percent is a ratio out of 100. Identify the reference quantity before multiplying: a discount uses the original price, while a percent increase compares the change with the original amount. Proportions keep a rate constant by relating quantities with their units.
Consider this example: A $72 jacket is discounted by 25%. What is the sale price, in dollars? A 25% discount removes 18 dollars. The buyer pays 75% of 72, or 54 dollars. The choice 18 is the discount amount rather than the sale price.
The method also handles a different case: A club’s membership grows from 80 to 92. What is the percent increase? The increase is 12 members. Relative to the original 80, 12/80 = 0.15 = 15%. The choice 12 confuses a count with a percent.
For a further application, consider: A quantity increases by 20% and then decreases by 20% of its new value. What percent of the original quantity remains? The successive multipliers are 1.20 and 0.80. Their product is 0.96, so 96% remains. Adding +20% and −20% overlooks the changed base.
An increase followed by an equal percent decrease does not restore the original amount because the second percent uses a different base. In a new model, label the original quantity and the rate. Check a result by recomputing the part-to-whole ratio or by reversing the stated change.
When the final amount is given, reverse the multiplier rather than subtracting the stated percent from the final amount. An item costs $59 after an 18% tax. If p is its price before tax, the tax is 0.18p and the total is 1.18p = 59. Dividing by 1.18 gives p = 50 dollars. Check: 18% of $50 is $9, making the total $59. Subtracting 18% of $59 would use a different tax base. A proportion is another way to preserve a fixed relationship. On a map, 3 centimeters represents 12 kilometers, so the rate is 12/3 = 4 kilometers per centimeter. For 7.5 centimeters, multiply 7.5 centimeters by 4 kilometers per centimeter to get 30 kilometers. The centimeter units cancel; using the reciprocal rate would fail that unit check.