The proportion
part/whole = percent/100. Fill in any three pieces and cross-multiplying frees the fourth. It works in every direction with no new rules.
Percentages topic
Parts, Wholes, and Percentages
One proportion connects all three amounts: part over whole equals percent over 100. Pick any two and the explorer below solves the third — with the proportion, the equation, and every step shown in words you can say out loud.
- percent proportion
- part over whole
- percent over 100
- percent equation
Interactive explorer
Part–Whole–Percent Explorer
One proportion connects all three amounts. Choose the piece you want to find, fill in the other two, and watch part over whole line up with percent over 100.
Try an example
The percent must be at least 1 so the whole stays a fair share.
The percent proportion
3 is 50% of 6
3 out of 6 is the same share as 50 out of 100.
36part over whole
=50100percent over 100
The part inside its whole50% filled
The percent out of 10050% filled
- 13 is 50% of the whole — that is 50 out of every 100.
- 2One percent of the whole is 3 ÷ 50 = 0.06.
- 3The whole is 100 of those shares: 0.06 × 100 = 6.
For parents: Say: “We know the part and the percent. One percent of the whole is the part divided by the percent — then 100 of those shares make the whole.”
How the percent proportion works
Part over whole equals percent over 100
A percent already tells you the whole it compares against: 100. So every percent question is really two fractions that stay equal — the share of your whole on one side, the share of 100 on the other. Children who see the two bars fill to the same length stop asking which number goes where.
The percent equation
part = percent × whole, with the percent as hundredths. The same idea written as a one-step multiplication once the percent is a decimal.
Start the problem
Name what is given, then choose the missing piece
Every part–whole–percent problem hides one missing amount. Read the sentence once, label the two numbers you have, and the first move writes itself.
Two fractions stay equal: part over whole, percent over 100
Write the proportion, then decide which piece is missing.
3/4 = 75/100
The share of the whole and the percent out of 100 name the same amount. Fill in the two known pieces of part/whole = percent/100, and the missing one can be solved by reasoning or by cross-multiplying.
The part is missing
Multiply the whole by the percent as hundredths.
30% of 60 → 0.30 × 60 = 18
This is the percent equation: part = percent × whole. It is the proportion shortcut once the percent has been rewritten as a decimal or a fraction out of 100.
The whole is missing
Find one percent of the whole, then scale to 100.
18 is 90% of what? → 18 ÷ 90 = 0.2, then 0.2 × 100 = 20
Dividing the part by the percent measures a single 1% share of the whole. Multiplying that share by 100 rebuilds the whole. The proportion shows the same move: part/whole = percent/100, so whole = part × 100 ÷ percent.
Problem guides
Worked examples for the three directions, each with the wording that keeps the proportion attached to a picture.
The percent proportion
Solve a percent proportion
n/20 = 40/100 → n × 100 = 20 × 40 → n = 8
Name the part and the whole, write them as part over whole, and set that equal to the percent over 100. Cross-multiply, then divide by the remaining number to free the missing piece.
For parents: Say: “The top numbers count the same-sized share of two wholes — one is our whole, the other is always 100.”
Finding the whole
Find the whole when you know a part and its percent
9 is 30% of what? → 9 ÷ 30 = 0.3, then 0.3 × 100 = 30
The whole is always bigger than the part when the percent is below 100. Divide part by percent to weigh one percent, then multiply by 100 to hold all of them.
For parents: Say: “If 9 blocks is 30 shares out of 100, one share is tiny. Count 100 of those shares to rebuild the whole.”
The percent equation
Use the percent equation with an example
part = percent × whole → 0.25 × 40 = 10
Rewrite the percent as a decimal by moving the decimal point two places left, then multiply by the whole. The equation and the proportion always agree, so one can check the other.
For parents: Say: “25% is just 0.25 — a quarter. So this question is really a quarter of 40.”
Common mistakes
Keep the fractions pointing the right way
- Cross-multiplying along the wrong diagonal, such as pairing part with percent instead of part with 100.
- Dividing the percent by the part when finding the whole, which flips the proportion upside down.
- Forgetting that “of” means the whole in the percent equation, so 30% of 60 becomes 30 × 60.
- Treating the percent as a plain number instead of hundredths when rewriting part = percent × whole.
Check the result
The proportion is its own answer key
Because the method is an equation, every answer can prove itself. These checks take a few seconds and catch almost every slip.
Plug the answer back into the proportion
Replace the missing piece with your answer and check that part/whole really equals percent/100. For 18 is 90% of 20: 18/20 = 90/100, and both sides name the same share.
Compare sizes before you compute
When the percent is below 100, the part must be smaller than the whole. If a find-the-whole answer comes out smaller than the part, the division went the wrong way round.
Estimate with a friendly percent first
50% is half, 25% is a quarter, 10% is a tenth. If 9 is about a third-ish of the whole, the whole should be near 30 — a good bracket for the exact answer.
Practice order
Write the proportion first, say which piece is missing, and only then solve. Predict whether the answer will be bigger or smaller before you compute.
- 1.Proportion: Solve n/20 = 40/100
- 2.Find the whole: 9 is 30% of what number?
- 3.Percent equation: part = 0.25 × 40
- 4.Check by plugging back: Is 18/20 equal to 90/100?
Explain it out loud
“Part over whole equals percent over 100.”
One sentence covers every direction. These prompts help your child decide what is missing before any arithmetic starts.
- Which piece is missing — the part, the whole, or the percent?
- What does the bottom of each fraction mean: our whole, and always 100?
- Should the whole be bigger or smaller than the part here? How do you know?
- Can you plug the answer back into the proportion to prove it?
Keep exploring
The two percent questions on the 100-square grid
The proportion on this page is the written method behind the same two questions. See both directions shaded square by square in the visual explorer.
Percents back to fractions
The proportion calls the percent “over 100” — the percent-to-fraction calculator shows exactly what that fraction looks like in lowest terms.
The fraction foundation
Equivalent fractions are the reason the proportion stays balanced. Revisit that idea visually before relying on cross-multiplying.
Back to the percentages hub
More percent guides are being built page by page on the same foundation — percent difference, percent error, percentage increase, and the percent equation.
Common questions
Parts, Wholes, and Percentages FAQ
It is the statement part/whole = percent/100. The part compared to its whole stays equal to the percent compared to 100, which lets you solve for any missing piece.