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.
Completing the Percent Trio
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.
Rearranging the Core Formula
In Lesson 6, our core relationship was:
Part=100Percent×Whole
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%→d=0.15), the rearranged formula becomes:
Whole=dPart
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.
The Step-by-Step Method
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Every "find the whole" problem follows the same three steps:
Identify the part (the piece we have) and the percent it represents.
Convert the percent to a decimal by dividing by 100 (or shifting the decimal point two places left).
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:
45÷0.15=300
The original amount was $300. We can verify by going in the forward direction: 0.15×300=45. It checks out.
More Worked Examples
Checking Our Answers
Recognizing the "Find the Whole" Problem
The trickiest part of reverse percent problems is not the arithmetic — it is recognizing that we need to divide in the first place. Here are common phrases that signal the whole is the missing piece:
Phrasing
Why the Whole Is Missing
"27 is 15% of what number?"
Asks directly for the base
"15% of some amount is 27."
The base is unnamed
"She saved $60, which was 20%. What was the original price?"
Original quantity is unknown
Compare these with the other two problem types. If we see "what is 15% of 180?" the part is missing, so we multiply. If we see "27 is what percent of 180?" the percent is missing, so we divide the part by the whole. Getting comfortable spotting these clues will help us choose the right operation before we even pick up a calculator.
Recovering a Restaurant Bill
Common Mistakes to Avoid
Before we head into practice, let's flag two errors that come up often with this problem type.
Multiplying instead of dividing. When we see a percent, our first instinct may be to multiply (as we practiced in Lesson 6). But when the whole is missing, we must divide. A helpful gut check: if the part is smaller than we expect the whole to be, yet our answer comes out even smaller, we almost certainly multiplied when we should have divided.
Swapping the part and the whole. Just as in Lesson 7, look for the word "of" to locate the whole. In "9 is 18% of what?" the word "of" points to the unknown whole. Misidentifying these roles will send the entire calculation in the wrong direction.
Conclusion and Next Steps
In this lesson, we completed the trio of core percent problems by learning to find the whole when given a part and a percent. The method comes down to two moves: convert the percent to a decimal, then divide the part by that decimal. We also practiced verifying our answers with a quick forward check and discussed how to tell this problem type apart from the other two setups.
The practice exercises up next will walk us through identifying which type of percent problem we are facing, filling in the key conversion and division steps, computing answers independently across a range of percent values, and writing out a full solution for a tipping scenario. Let's finish this course strong!
Let's build fluency with a few more problems that span different percent values.
Example 1 — 12 is 40% of what number?
Convert: 40%=0.40. Divide: 12÷0.40=30. The whole is 30.
Example 2 — 27 is 15% of what number?
Convert: 15%=0.15. Divide: 27÷0.15=180. The whole is 180.
Example 3 — 6 is 2% of what number?
Convert: 2%=0.02. Divide: 6÷0.02=300. The whole is 300.
Notice a pattern: the smaller the percent, the larger the whole turns out to be relative to the part. That makes intuitive sense. If 6 represents only 2% of something, that something must be quite large.
A quick forward check is the easiest way to catch arithmetic mistakes. After finding the whole, multiply it by the decimal percent and see if the original part comes back:
Check: Whole×d=Part?
For Example 2 above: 180×0.15=27. ✓ For Example 3: 300×0.02=6. ✓
This takes only a few seconds and catches most errors before they cause trouble. Building this habit now will pay off in the practice exercises and well beyond this course.
Let's apply this skill to a situation many of us encounter regularly. Suppose we know that someone left a $9 tip and that it was an 18% tip. What was the bill before the tip?
The part is 9 (the tip amount) and the percent is 18%, so d=0.18. Dividing gives us:
9÷0.18=50
The original bill was $50. Forward check: 0.18×50=9. ✓
The same logic applies whenever we know a dollar amount and the rate it represents — a tax amount and tax rate, a discount amount and discount rate, or a commission and commission rate. In every case, we divide the known amount by the decimal percent to recover the base it came from.
