Percentage change and interest

Convert the percentage discount into money: price × p ÷ 100. Compare it with the fixed discount. The larger monetary amount saves more.

Discount amount = original price × p ÷ 100. Subtract this amount from the original price. New price = original price × (1 − p/100).

Discount amount = original price × p ÷ 100. Subtract this amount from the original price. New price = original price × (1 − p/100).

Saved amount = target × p ÷ 100. Remaining amount = target − saved amount, or target × (100 − p) ÷ 100.

Simple interest is calculated from the original deposit each year. Interest = deposit × p ÷ 100 × years. Final balance = original deposit + total interest.

A discount percentage is calculated from the original price. The price paid after the discount is only the remaining fraction of that price.

1. Let p be the discount percentage and K the price after the discount.
2. Find the multiplier for the remaining fraction: q = (100 − p) / 100 = 1 − p / 100.
3. If X is the original price, then K = X · q. Recover the original price by dividing: X = K / q = K · 100 / (100 − p).
4. Check by multiplying the original price you found by q: this must reproduce the discounted price given in the question.

Do not add the same percentage to the reduced price: that percentage would be calculated from a different, smaller amount and would not restore the original price. For a discount greater than 0% but less than 100%, the original price must exceed the discounted price. If needed, round money to cents only at the end.

The price increases twice in succession, so the percentage is not calculated from the original price both times. The second increase is calculated from the price obtained after the first year.

1. Convert the percentage to a multiplier: 1 + p/100.
2. Multiply the original price by this multiplier to find the price after the first year.
3. Multiply that result by the same multiplier once more.

Formula: final price = original price × (1 + p/100)².

A change in percentage points is the difference between two percentage rates: new rate − old rate. For example, increasing from 20% to 25% is an increase of 5 percentage points.

Multiply the original price by (1 + p/100), then multiply the increased price by (1 − p/100). The percentages use different bases, so they do not cancel. For example, increasing €100 by 10% and then decreasing it by 10% gives €99.

A change in percentage points is the difference between two percentage rates: new rate − old rate. For example, increasing from 20% to 25% is an increase of 5 percentage points.