Choosing the Correct Base
Introduction
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.
Why the Base Matters More Than Ever
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.
Spotting the Correct Base in Multi-Number Problems
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:
| Step | Percent | Base (the number it applies to) | Calculation |
|---|---|---|---|
| Discount | 25% | Original price = $80 | |
| Tax | 8% | Discounted price = $60 |
![[Flow diagram showing that the 25% discount uses $80 as the base and the 8% tax uses $60 as the base]](https://k3-production-bucket.s3.us-east-1.amazonaws.com/uploads/qGiPy3T7vtcbZiedm_discount_graph.png)
Notice that the tax is not calculated on the original $80. The problem tells us tax is on the discounted price, so the base for the second percent is the intermediate result of $60. If we mistakenly used $80 as the base for both percents, we would overcharge by $1.60 — a small slip with a real consequence.
The key habit to develop is to pause before each percent step and ask: "Which amount does this percent describe a portion of?" The answer to that question is your base for that step.
