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Converting Between Fractions, Decimals and Percents

One number, three notations. The anchors worth memorising, how to scale a denominator to 100, and the rule that says which fractions terminate as decimals.

By Published Updated 5 min read

Reign Creative LLC makes IQ Test & Brain Training Games, which this article links to, so we have a commercial interest in it. How we write and check articles.

Key takeaways

  • Fractions, decimals and percents are three notations for one number, and the conversions between them are a division, a shift of two decimal places, and a reduction over 100.
  • Around seventeen conversions cover most everyday work; memorising them turns conversion into recall and removes the division entirely.
  • Scaling the denominator to 100 is faster than dividing whenever the denominator is a factor of 100, which covers halves, quarters, fifths, tenths, twentieths and twenty-fifths.
  • A reduced fraction terminates as a decimal exactly when its denominator has no prime factors other than 2 and 5, because a terminating decimal is a fraction over a power of ten.
  • When a fraction does repeat, the length of the repeating block is fixed by the denominator, which is why one seventh repeats in blocks of six and one eleventh in blocks of two.

A fraction, a decimal and a percent are three ways of writing the same number, and each conversion is one move. Fraction to decimal is a division. Decimal to percent is a shift of two places. Percent to fraction is the number over 100, reduced. The skill worth having is not the procedure but knowing which conversions to do by memory instead.

Three worked in a line: 3/8 divides to 0.375, which is 37.5%. Going the other way, 0.625 is 625/1000, which reduces by 125 to 5/8, and is 62.5%. And 45% is 45/100, which reduces by 5 to 9/20, and is 0.45.

The anchors worth knowing outright

The table below lists the conversions that come up often enough to be worth holding in memory rather than computing. Each row is the same number in three notations, with thirds and sixths rounded at the percent column because they repeat forever.

Fraction Decimal Percent
1/2 0.5 50%
1/3 0.333… 33.3%
2/3 0.666… 66.7%
1/4 0.25 25%
3/4 0.75 75%
1/5 0.2 20%
2/5 0.4 40%
3/5 0.6 60%
4/5 0.8 80%
1/6 0.1666… 16.7%
5/6 0.8333… 83.3%
1/8 0.125 12.5%
3/8 0.375 37.5%
5/8 0.625 62.5%
7/8 0.875 87.5%
1/10 0.1 10%
1/20 0.05 5%

Seventeen rows. Once they are recall rather than calculation, most conversion questions never reach the division stage at all, and the rest can be built from them.

Converting without dividing

Scale the denominator to 100. This works whenever the denominator is a factor of 100. For 17/25, multiply top and bottom by 4 to get 68/100, so 68%. For 9/20, multiply by 5 to get 45/100, so 45%. Halves, quarters, fifths, tenths, twentieths, twenty-fifths and fiftieths all yield to this.

Split into anchors when it will not scale. Sixteenths do not divide 100, but they decompose. 5/16 is 1/4 plus 1/16. A quarter is 25%, and a sixteenth is half of an eighth, so half of 12.5%, which is 6.25%. Adding gives 31.25%, and dividing 5 by 16 confirms 0.3125.

Halve a known anchor. Anything whose denominator doubles a familiar one is one halving away. Twelfths come from sixths, sixteenths from eighths, fortieths from twentieths.

Which fractions terminate, and why

This is the rule almost nobody is taught, and it settles in advance whether a division will ever end.

A fraction in lowest terms terminates as a decimal exactly when its denominator's only prime factors are 2 and 5. MathWorld states the condition in the same form: the numbers with finite decimal expansions are fractions whose denominator is a power of two times a power of five.

The reason is short. A terminating decimal is, by definition, a fraction over a power of ten. Every power of ten is a product of twos and fives and nothing else. So a fraction can be rewritten over a power of ten only when its denominator is built from twos and fives.

  • 7/40 terminates, because 40 is 8 × 5. It comes to 0.175.
  • 7/12 does not, because 12 carries a factor of 3. It comes to 0.58333…

Reduce before applying the rule. 6/15 appears to carry a 3, but it reduces to 2/5, which terminates at 0.4. The test applies to the fraction in lowest terms, which is one more reason the ordering method in simplifying a fraction quickly earns its place first.

How long the repeat runs

When a fraction does repeat, the block length is not arbitrary. MathWorld ties it to how powers of ten behave modulo the denominator, so the length is a property of the denominator rather than of the numerator.

  • 1/7 repeats in blocks of six: 0.142857…
  • 1/11 repeats in blocks of two: 0.0909…
  • 1/13 repeats in blocks of six: 0.076923…

A practical consequence is that every seventh has the same six digits in a rotated order, which is a pattern set out with the full cycle in mental math tricks that actually work. The related point about deciding before you divide whether a decimal will ever terminate is covered from the division side in how to divide in your head.

Percent to fraction, and why you would bother

Because a division is often easier than a multiplication.

37.5% of 240. As a decimal it is 0.375 × 240, which is three separate multiplications and a carry. As a fraction it is 3/8 of 240: divide 240 by 8 to get 30, multiply by 3 to get 90.

The techniques library in our app states the underlying principle bluntly — memorise the big anchors as fractions, because dividing beats multiplying. That is the entire argument for learning the table above in the fraction direction as well as the percent direction.

Forward percentage work, including tips, discounts and the difference between percent and percentage points, is covered separately in percentages in your head. The conversion triangle is the layer underneath it, and comparing two fractions once they are converted is treated in comparing two fractions without a common denominator. The full method set is in the mental math training guide, with the rest of this hub under mental math and the broader subject under education and brain training.

Why fluency across all three notations matters

The What Works Clearinghouse practice guide on fractions instruction places the idea that fractions are numbers — points on a line, comparable with any other number — at the centre of its recommendations, alongside using number lines as a central representation. Conversion fluency is what makes that concrete: a number you can write three ways is a number you can place, compare and estimate with.

There is a documented failure mode on the other side. In a study of sixth and eighth graders published in Developmental Psychology, Siegler and Pyke found that around a third of children estimated fraction magnitudes from the numerator alone or the denominator alone, a pattern associated with weaker fraction arithmetic. Conversion is a useful antidote precisely because a decimal has no separate top and bottom to misread.

Practising the conversions

IQ Test & Brain Training Games, which is our own app, built by Reign Creative, ships Decimal Converter — an accuracy-format mode of ten questions with Easy, Medium and Hard settings. An accuracy format rather than a timed sprint suits conversion work, because the thing being drilled is recall of a fixed set rather than speed on a generated stream.

For the memorisation half, the Flash Cards mode carries a Fraction ↔ Percent deck described in the app as common fraction conversions, holding seventeen fixed pairs from one half through to five sixths, with twelve cards to a session and a flip-then-grade format you mark yourself. There is also a Quick Percents deck covering 10, 20, 25 and 50 percent of generated bases.

The app is an educational game and a practice tool rather than a course. What its drills record is score, accuracy, streak, response time and personal bests on the modes you actually play.

Good to know

Frequently asked questions

How do you convert a fraction to a percent without dividing?
Scale the denominator to 100 if you can. For 17/25, multiplying top and bottom by 4 gives 68/100, which is 68 percent. This works for any denominator that divides 100 — halves, quarters, fifths, tenths, twentieths, twenty-fifths and fiftieths. When the denominator will not scale cleanly, split the fraction into anchors you already know instead.
Which fractions terminate as decimals?
A fraction in lowest terms terminates exactly when its denominator's only prime factors are 2 and 5. The reason is that a terminating decimal is a fraction over a power of ten, and every power of ten factorises into twos and fives. So 7/40 terminates because 40 is 8 times 5, while 7/12 repeats because 12 carries a factor of 3.
Why does one seventh repeat in blocks of six digits?
Because the repeating block length is determined by how the powers of ten behave when divided by the denominator. For 7 that cycle runs through six values before returning to the start, so the decimal repeats every six digits. For 11 the cycle is two values long, which is why one eleventh is 0.0909 repeating.
Why convert a percent back to a fraction?
Because dividing is often easier than multiplying. Working out 37.5 percent of 240 as a decimal multiplication is fiddly, but 37.5 percent is 3/8, and three eighths of 240 is 240 divided by 8, which is 30, times 3, which is 90. The fraction form turns an awkward multiplication into a clean division.

Sources

  1. Decimal Expansion (opens in a new tab)MathWorld — A Wolfram Resource — accessed
  2. Repeating Decimal (opens in a new tab)MathWorld — A Wolfram Resource — accessed
  3. Developing Effective Fractions Instruction for Kindergarten Through 8th Grade (WWC Practice Guide) (opens in a new tab)What Works Clearinghouse, Institute of Education Sciences, U.S. Department of Education — accessed
  4. Developmental and Individual Differences in Understanding of Fractions (opens in a new tab)Developmental Psychology (American Psychological Association), via PubMed Central — accessed

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