Three ways of writing the same amount.
| Fraction | Decimal | Percent |
|---|---|---|
| 38 | 0.375 | 37.5% |
| 925 | 0.36 | 36% |
| 37100 | 0.37 | 37% |
Percent means “per hundred”, so a percent is a fraction with 100 already underneath. That single fact drives every conversion on this page.
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Fractions, Decimals and Percents
One number, three costumes — and comparing them means putting them all in the same one.
Arrange from smallest
Arrange from smallest: 38, 0.36, 37%
Put everything in one form: 38 = 0.375, 0.36 stays, 37% = 0.37. Then 0.36 < 0.37 < 0.375, so the order is 0.36, 37%, 38. C is the trap: it turns 37% into 0.037 instead of 0.37 (moving the point three places, not two), and 0.037 < 0.36 < 0.375 puts the percent first. Converting all three to decimals first is the only reliable way to compare mixed forms.
Fraction to decimal: just divide.
38 = 3 ÷ 8
top by bottom, always that way
= 0.375
it terminates
0.375 × 100 = 37.5%
shift the point two places right
Which fractions terminate
Only those whose denominator, in lowest terms, is built from 2s and 5s — halves, quarters, fifths, eighths, twentieths. Anything with a 3 or a 7 in there repeats forever. Reduce first: 1435 has a 7 underneath, but it is 25 = 0.4 and terminates.
To compare, put everything in one form.
Decimals are the easiest to line up, so send everything there.
38
3 ÷ 8 = 0.375
largest
37%
37 ÷ 100 = 0.370
middle
0.36
0.360
smallest
Pad the decimals to the same length
0.375, 0.37 and 0.36 are hard to rank at a glance; 0.375, 0.370 and 0.360 are not. Writing the extra zeros costs nothing and removes the guessing.
The ones to know without working.
| Fraction | Decimal | Percent |
|---|---|---|
| 12 | 0.5 | 50% |
| 13 | 0.333… | 33⅓% |
| 14 | 0.25 | 25% |
| 15 | 0.2 | 20% |
| 16 | 0.1666… | 16⅔% |
| 18 | 0.125 | 12.5% |
| 23 | 0.666… | 66⅔% |
| 34 | 0.75 | 75% |
| 38 | 0.375 | 37.5% |
| 58 | 0.625 | 62.5% |
The eighths go 0.125, 0.250, 0.375, 0.500, 0.625, 0.750, 0.875 — each one 0.125 more than the last. Learn that row and half the conversions on any paper are instant.
Decimal to fraction: count the places, or use nines.
0.36 = 36100
two places → hundredths
= 925
divide both by 4
A repeating block goes over the same number of nines.
0.454545… — the block is 45
two digits repeating
= 4599
two digits → two nines
= 511
divide both by 9
Where the nines come from
Let x = 0.4545…. Then 100x = 45.4545…, and subtracting leaves 99x = 45. The rule is that subtraction, already done: 0.3̄ = 39 = 13, and 0.123123… = 123999.
The percents that catch people.
| Written | Really is | Not |
|---|---|---|
| 0.5% | 0.005 | 0.5 |
| 125% | 1.25 — percents may pass 100 | 12.5 or 0.125 |
| 33⅓% | 13, exactly | 0.33 |
| 2.5% | 0.025 = 140 | 0.25 |
A percent of a percent
“50% of 20%” is 10%, not 70% and not 10. Percents multiply as decimals: 0.5 × 0.2 = 0.1. Adding them only works when both are parts of the same whole.
Variations: a late repeat, a tiny percent, and numbers above 1.
The repeat starts late: write 0.2333… as a fraction.
x = 0.2333…, 10x = 2.333…, 100x = 23.333…
line up two copies with the same tail
100x − 10x: 90x = 21
the 3s cancel
x = 2190 = 730
check: 7 ÷ 30 = 0.2333… ✓
What is different: the 2 does not repeat, so 39 alone is wrong. Shift until the repeating tails line up, then subtract.
A percent below 1%: write 0.75% as a fraction.
0.75% = 0.75 ÷ 100 = 0.0075
still divide by 100
= 7510000 = 3400
÷ 25
What is different: the percent sign has a decimal in front. Reading 0.75% as 0.75 gives 34 — a hundred times too big.
Numbers bigger than 1: 1.35 and 2⅜ in all three forms.
1.35 = 135100 = 2720 = 1 720
two places → hundredths, ÷ 5
1.35 × 100 = 135%
more than the whole
2⅜ = 2 + 0.375 = 2.375 = 237.5%
convert the fraction part, keep the whole
What is different: the whole number rides along unchanged, and the percent passes 100. Writing 2⅜ as 23.75% or 2.375% loses the whole.
Percents of fractions, and mixed-form equations.
“What percent of A is B” is always part ÷ whole — even when A and B are fractions. What percent of 45 is 310?
whole = 45 (after “of”), part = 310
label before dividing
310 ÷ 45 = 310 × 54 = 1540 = 38
flip the second
38 = 37.5%
a known eighth
Three-quarters of a number is 6 more than 45% of it. Find the number.
0.75n − 0.45n = 6
put both in decimals
0.30n = 6, n = 20
check: 15 is 6 more than 9 ✓
Dividing the wrong way round
(45) ÷ (310) = 83 ≈ 266.7% answers “what percent of 310 is 45”. Multiplying, 45 × 310 = 24%, answers nothing at all. The number after of always goes underneath.
On the exam
Expect repeating decimals to convert, a list in three forms to order, “what percent of” with fractions, and equations that mix a fraction with a percent. Send everything to one form — decimals for ordering, fractions for exact answers — and reduce a fraction before judging whether it terminates.
Check: convert back the other way.
511
5 ÷ 11 = 0.4545…
the decimal returns ✓
925
9 ÷ 25 = 0.36
✓
order
0.360, 0.370, 0.375
rising ✓
Now you try
Practice problem
Arrange from largest
Arrange from largest: 58, 0.6, 62%, 23
As decimals: 23 ≈ 0.667, 58 = 0.625, 62% = 0.62, 0.6. Largest first gives 23, 58, 62%, 0.6. D is the same list ascending — check which end the question asked for before you pick.