Finding the Whole

Welcome back to Percent Foundations for Everyday Life! We have arrived at Lesson 8 of 8 — the final lesson in this course. Before we dive in, take a moment to appreciate how far we have come. We started by understanding what a percent even means, moved through conversions and mental math shortcuts, and then learned to calculate a part from a percent and to figure out what percent one number is of another.

Today we close the loop by tackling the third and final type of percent problem: finding the whole when we know only the part and the percent it represents. Imagine someone tells us, "I saved $45, and that was 15% of the original price." We know the piece and the rate, but the original price is missing. By the end of this lesson, we will have a clear method for recovering that missing whole every time.

Every percent problem involves three quantities: the part, the percent (rate), and the whole (base). In each problem, two of these are given and one is missing. Across the last three lessons of this course, we have met all three setups:

• Lesson 6 — Part missing: What is 20% of 150? → Multiply.
• Lesson 7 — Percent missing: 18 is what percent of 25? → Divide part by whole, then convert.
• This lesson — Whole missing: 45 is 15% of what number? → Divide part by decimal percent.

Recognizing which piece is missing is the single most important step before we do any arithmetic. Once we know which setup we are dealing with, the correct operation follows naturally.

In Lesson 6, our core relationship was:

When the whole is the unknown, we isolate it by dividing both sides by the decimal form of the percent. If we let $d$ stand for the percent written as a decimal (for example, $15\% \rightarrow d = 0.15$), the rearranged formula becomes:

The key operation is division, not multiplication. This is the exact reverse of what we did in Lesson 6, and keeping that contrast in mind will help us avoid mixing up the two setups.

Every "find the whole" problem follows the same three steps:

1. Identify the part (the piece we have) and the percent it represents.
2. Convert the percent to a decimal by dividing by $100$ (or shifting the decimal point two places left).
3. Divide the part by that decimal to get the whole.

Let's try it with our opening example: $45 is 15% of what amount?

The part is $45$ and the percent is $15\%$. Converting gives us $d = 0.15$. Now we divide:

The original amount was $300. We can verify by going in the forward direction: $0.15 \times 300 = 45$. It checks out.