Mixing Percents and Amounts

Welcome back to Multi-Step Percent Problems! You have already worked through three of the five lessons in this course, so you are well past the halfway mark. Great progress!

So far, we have practiced choosing the correct base for a percent calculation, recovering an original amount after a percent change, and applying consecutive percent changes one after another. In each of those lessons, every adjustment was a percentage. In real life, though, not every adjustment is a percentage. Sometimes a transaction mixes a percent-based step with a flat dollar amount — think of a store coupon for 20% off followed by a $5 delivery fee, or a $10 loyalty credit followed by 8% sales tax. This lesson focuses on exactly that kind of mix: one percent adjustment and one fixed-dollar adjustment, applied in a stated order.

Before we dive into calculations, let's build some intuition. When two adjustments are both fixed dollar amounts, order does not matter. Subtracting $5 and then subtracting $10 gives the same result as subtracting $10 first. Similarly, as you saw in the previous lesson on consecutive percent changes, the order of two percent multiplications does not change the final product.

However, the moment we combine a percent step with a fixed-dollar step, order does matter. A percentage is calculated on whatever amount exists at that moment, so the dollar value it produces depends on when it is applied. A fixed-dollar adjustment, on the other hand, always adds or subtracts the same number regardless of timing. This single difference is the key idea for the entire lesson.

Let's start with a common pattern: apply the percent change first, then apply the fixed-dollar change.

Example 1. A jacket costs $80. A store offers 25% off, and there is a flat $6 shipping fee added afterward.

1. Percent step first. The discount is 25% of $80:

The price after the discount is $80 − $20 = $60.

2. Fixed-dollar step second. Add the $6 shipping fee:

The final amount is $66.

Notice that the percent was calculated on the original price, because the problem told us the discount comes first. The shipping fee is then simply tacked on to whatever remains.