Why a percent is just a ratio with the same partner every time — a number out of one hundred.
In Lesson 4.3 you compared two quantities with a ratio, and you saw that a ratio can name any comparison — 3 to 5, 7 to 12, 9 to 200. Percentages make one quiet but powerful decision: always compare against one hundred. Fix the second number at 100 forever, and every comparison in the world can be read off the same scale. That is exactly why news, prices, batteries, test scores, and tips all speak in percents — once everything is "out of 100," you can line any two of them up at a glance.
By the end of this lesson you will be able to do five things: explain what a percent means as a number out of 100; convert freely among percents, fractions, and decimals; find what percent one number is of another; find a given percent of a number; and work backward from a part and a percent to the whole. We keep one steady habit of color throughout: the part — the first quantity is amber, the whole, the "out of 100" is blue, and a third quantity, when one appears, is purple. An increase is green; a decrease is red.
The word percent comes straight from Latin: per centum, "per hundred." So % is nothing more than a tidy stamp meaning "out of 100." When a battery reads 80%, picture its charge cut into 100 equal slivers; 80 of them are full. When a store sign says 25% off, think 25 dollars saved on every 100 dollars of price.
That last sentence hides the key insight. A percent is really a ratio — the same kind of comparison from Lesson 4.3 — but a ratio whose second term is always locked at 100. "80%" is the ratio 80 to 100. And a ratio with second term 100 is also, word for word, a fraction over 100:
The hundred-grid is the picture to keep in your head. It is a square cut into 100 small cells — a perfect "out of 100" board. Shade N cells and you are literally looking at N percent.
A grid has 37 of its 100 cells shaded. What percent is shaded, and what fraction, and what decimal?
Out of 100, 37 are shaded, so that is 37% = 37100 = 0.37. All three are the same amount, just dressed differently.
A percent is a number out of 100 — at once a ratio with second term 100 and a fraction over 100. So N% = N100. Hold onto that one sentence and everything else in this lesson is just careful arithmetic with it.
Slide to shade any number of the 100 cells. The grid is the percent: watch the part and the same amount written as a percent, a fraction over 100, and a decimal.
Because a percent is just "out of 100," moving between the three forms is mechanical once you see the moves. Start from a percent like 75% and you can reach the other two in one step each.
75% = 75100 = 34 = 0.75
Percent → fraction. Drop the % and write the number over 100; then reduce. 75% = 75100, and dividing top and bottom by 25 gives 34.
Percent → decimal. A percent is already "hundredths," so dividing by 100 just moves the point two places LEFT. 75% → 75. → 0.75. Going the other way, decimal → percent multiplies by 100, which moves the point two places RIGHT: 0.75 → 75 → 75%.
Fraction → percent. If the denominator already divides 100, just rescale it to 100. 35: multiply top and bottom by 20 to get 60100 = 60%. If it does not divide 100 neatly, just divide and then ×100: 13 = 1 ÷ 3 = 0.333… → 33.3% (a repeating decimal, so we round).
A handful of conversions come up so often they are worth knowing cold:
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/3 | 0.333… | 33.3% |
| 1/10 | 0.1 | 10% |
| 1/1 | 1.0 | 100% |
The shift is always two places, even when it forces a zero. 5% is 0.05, not 0.5 — you must pass two places, and there is only one digit, so a 0 fills the tenths place. Likewise 120% = 1.2, a number bigger than 1.
Step the value and pick which form you are starting from. See it as a percent, a fraction over 100 (reduced), and a decimal all at once — and watch the two-place shift between percent and decimal.
Often you have two real numbers and want the comparison as a percent: 45 correct out of 50 questions, 18 girls in a class of 24, 3 rainy days out of 12. The question is always the same shape: "What percent of B is A?" And the recipe is always the same:
percent = partwhole × 100%
Divide the part by the whole to get a decimal, then multiply by 100 to turn that decimal into a percent. For 45 out of 50: 45 ÷ 50 = 0.9, and 0.9 × 100 = 90%. You scored 90%.
What percent of 24 is 18?
18 ÷ 24 = 0.75, and 0.75 × 100 = 75%. (Shortcut: 1824 reduces to 34, which we know is 75%.)
And what percent of 8 is 3?
3 ÷ 8 = 0.375, so 37.5%. Not every comparison lands on a whole percent — that is fine.
The whole is the number that comes right after the word "of", and it goes on the bottom. "What percent of 50 is 45?" → 50 is the whole. Flipping them gives a different (wrong) answer: 45 ÷ 50 = 90%, but 50 ÷ 45 ≈ 111% — that would answer a different question, "45 is what percent of itself plus more." When in doubt: the part is usually the smaller, the whole is the total you are measuring against.
Set the part A and the whole B. The widget computes A ÷ B × 100 and rounds to one decimal place. The bar fills to show "A out of B."
The reverse situation is just as common: you know the percent and want the actual amount. "15% of 200 students," "a 20% tip on a $40 meal," "8% sales tax." The move is to turn the percent into a decimal and multiply:
part = whole × percent ⟶ 15% of 200 = 200 × 0.15 = 30
Why multiply? Because "15% of 200" means 15 hundredths of 200, and "of" with fractions and decimals always means multiply. Writing 15% as 0.15 and multiplying does exactly that.
You rarely need a calculator for the friendly percents. Build the one you want from these:
• 10% — move the point one place left (10% of 80 = 8).
• 1% — move the point two places left (1% of 80 = 0.8).
• 50% — just halve it (50% of 80 = 40).
• 5% — half of 10% (5% of 80 = 4).
• 20% — double 10% (20% of 80 = 16); 15% = 10% + 5%.
Leave a 20% tip on a $45 meal.
Find 10% of 45 first — move the point one place left: $4.50. Double it for 20%: $9.00. (Check by multiplying: 45 × 0.20 = 9. ✓)
Set a percent and a whole. The widget shades that percent of the bar and shows the arithmetic whole × percent.
The trickiest of the three — and the most satisfying — runs backward. You are told a part and what percent of the whole it is, and you must find the whole. "12% of my goal is 6 dollars saved — what is the goal?" "30% of the votes is 21 people — how many voted?"
Set it up with the very same equation as Section 4.4.4, but now the unknown is the whole:
part = percent × whole ⟶ whole = part ÷ percent
Since multiplying by the percent made the part, dividing by the percent undoes it. If 12% is 6, then the whole = 6 ÷ 0.12 = 50. Check it forward: 12% of 50 = 50 × 0.12 = 6. ✓
30% of a number is 21. Find the number.
whole = 21 ÷ 0.30 = 70. Check: 30% of 70 = 70 × 0.30 = 21. ✓
A handy mental version: if 30% is 21, then 10% is 21 ÷ 3 = 7, so 100% is 7 × 10 = 70. Same answer, no decimals.
It is tempting to multiply 6 × 0.12 here, but that shrinks the part instead of growing it back to the whole. Ask yourself: the whole must be bigger than the part (when the percent is under 100%), and dividing by a number less than 1 makes things bigger — so dividing is right. 6 ÷ 0.12 = 50, comfortably larger than 6. ✓
Set the known part and its percent. The widget computes whole = part ÷ percent and draws the part as a slice of the recovered whole.
A percent is a number out of 100 — a ratio with second term 100 and a fraction over 100 — so N% = N100. To convert, drop the % over 100 (and reduce) for a fraction, or shift the point two places — left for percent→decimal, right for decimal→percent. To find what percent A is of B, compute part ÷ whole × 100. To find a percent of a number, multiply whole × percent. And to recover the whole, divide part ÷ percent. Three questions, one little equation — part = percent × whole — read three different ways.
Now that you know what percentages mean, 4.5 Percentages in Action puts them to work: percent increase and decrease, discounts and markups, sales tax, simple interest, and the famous trap of "50% off then 50% off." You will use every move from this lesson on real money.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson introduces percentages as a rate per 100 and is aligned to the U.S. Common Core standards 6.RP.A.3c (find a percent of a quantity as a rate per 100; solve problems involving finding the whole given a part and the percent) and 7.RP.A.3 (use proportional relationships to solve multistep percent problems). Throughout, a percent is presented first as a special ratio with its second term fixed at 100 — connecting directly to the ratio work of Lessons 4.1–4.3 — and equally as a fraction over 100, so that converting among percents, fractions, and decimals is one idea rather than three rules. The hundred-grid and bar models are used deliberately so students see that 80% is 80 of 100 and that "the whole" is whatever quantity follows the word "of." The three signature problem types (what-percent, percent-of, find-the-whole) are all derived from the single relationship part = percent × whole, read three ways. The most common misconceptions are addressed head-on: confusing which number is the whole (Section 4.4.3), losing a place in the two-step point shift (Section 4.4.2), and the belief that two successive 50% discounts give the coat away free (Exercises and the quiz) — each percent acts on a different whole, a misconception worth surfacing before the percent-change work of Lesson 4.5.