Welcome to Multi-Step Percent Problems, the final course in this learning path! By reaching this point, you have built a solid toolkit of percent skills, from converting between fractions, decimals, and percents all the way through discounts, interest, and percent change in the news. That is a lot of ground covered, and you should feel proud of the progress.

In this first lesson, we will tackle the skill that underpins every topic in this course: choosing the correct base. It might sound simple, but picking the wrong base is the single most common source of errors in multi-step percent problems. Our goal here is to build the habit of figuring out which number a given percent should be applied to, especially when a problem hands us more than one number to consider.

Every percent calculation has three ingredients: the percent, the base (the whole that the percent refers to), and the part (the result). In a formula, their relationship looks like this:

When a problem contains only one dollar amount, choosing the base is straightforward — "find 20% of $50" leaves no room for confusion. But real life rarely stays that simple. A single purchase might involve an original price, a discounted price, and a tax rate. A paycheck scenario might mention a base salary, a raise, and a bonus. Each percent in the problem may attach to a different number, and picking the wrong one changes the answer entirely.

Consider a quick example. A jacket has an original price of $80. The store applies a 25% discount, and then the state charges 8% sales tax on the discounted price. Two percents appear here, and each one has its own base:

Everyday percent problems almost always contain language that points directly to the correct base. Here are a few common phrases and what they signal:

• "…of the original price" → Use the starting amount before any changes.
• "…of the sale price" or "…on the reduced price" → Use the amount after the discount has been subtracted.
• "…of his new salary" → Use the salary after a raise has been applied.
• "…of the total before tip" → Use the subtotal, not a number that already includes a tip.

Let's see this in action with a wage example. Suppose Priya earns $4,000 per month. She receives a 5% raise, and then her employer calculates a 10% bonus on her new monthly pay. The raise applies to the original salary:

The most frequent mistake in multi-step percent problems is using the original amount as the base for every percent, even after the amount has changed. This happens because the original number feels like the "main" number in the problem, and it is tempting to keep coming back to it.

A quick labeling strategy can guard against this error. Before doing any arithmetic, identify the quantities in the problem:

1. Original amount — the starting value.
2. Intermediate amount(s) — any value produced by applying a percent change to the original.
3. Final amount — the end result after all adjustments.

Then, for each percent in the problem, write down which labeled amount it belongs to. This small step takes only a few seconds and dramatically reduces mistakes. Think of it as choosing the right ingredient before you start cooking — the recipe will not work if you grab the wrong one.

In this lesson, we established the most important habit for multi-step percent problems: always identify the base before you calculate. We saw that problems with multiple numbers require us to match each percent to the specific quantity it describes — whether that is the original amount, an intermediate result, or some other value entirely. Language clues in the problem, such as "of the sale price" or "on her new salary," are your best friends here, so reading carefully is just as important as computing correctly.

Up next, you will put this skill into practice with exercises that challenge you to pick the right base in realistic scenarios. Get ready to slow down, read closely, and build the kind of careful thinking that makes multi-step percent problems feel manageable.