How do you convert a fraction to a decimal?

The fraction bar is a division sign that got a haircut. That is the whole method, and it is the part most explanations bury under three paragraphs of preamble. 3/4 means 3 divided by 4. Everything below is about doing that division quickly, doing it without a calculator, and knowing in advance whether it will ever finish.

The shortcut worth trying first

Before dividing anything, check whether the denominator scales up to 10, 100 or 1000. If it does, multiply the top and the bottom by the same number and read the answer straight off the place value:

3/4    × 25/25   =   75/100    =  0.75
7/20   ×  5/5    =   35/100    =  0.35
9/25   ×  4/4    =   36/100    =  0.36
3/8    × 125/125 =  375/1000   =  0.375

denominators 2, 4, 5, 8, 20, 25, 40, 50, 125 all scale cleanly

This is exactly converting a decimal to a fraction run backwards. That post takes 0.75 apart into 75/100 and reduces it to 3/4. Here you scale 3/4 back up to 75/100 and stop. Same two numbers, opposite direction.

Long division, for everything else

When the denominator will not scale, divide by hand. Put a decimal point and some zeros after the numerator, then work left to right. Take 5/8:

       0 . 6 2 5
     ---------------
  8  ) 5 . 0 0 0

  50 ÷ 8  =  6  remainder 2      write 6
  20 ÷ 8  =  2  remainder 4      write 2
  40 ÷ 8  =  5  remainder 0      write 5, and stop

  5/8  =  0.625

  check:  0.625 × 8 = 5

The remainder of zero is the finish line. Multiply your answer back by the denominator to check it; if you do not land exactly on the numerator, you dropped a digit somewhere.

Now the same procedure on 4/11, which does not finish:

  40 ÷ 11  =  3  remainder 7      write 3
  70 ÷ 11  =  6  remainder 4      write 6
  40 ÷ 11  =  3  remainder 7      we have been here before

  4/11  =  0.363636...

The remainder 4 came back, so the digits from that point are doomed to repeat. This is the practical test: watch the remainders, not the digits. A repeat in the remainders means a repeat in the answer, and you can stop dividing and write the block instead.

It also puts a ceiling on the chaos. Dividing by 11 can only ever produce remainders 1 through 10, so the repeating block can never be longer than 10 digits. Dividing by 7 gives at most 6, and 1/7 uses all of them: 0.142857142857...

Which fractions actually terminate

You can tell before you start. Reduce the fraction to lowest terms, then factor the denominator. Decimals are built out of tenths, and 10 is 2 × 5, so a denominator made only of 2s and 5s terminates. Anything else repeats.

5/8     8  = 2 × 2 × 2          terminates   0.625
7/20   20  = 2 × 2 × 5          terminates   0.35
3/40   40  = 2 × 2 × 2 × 5      terminates   0.075
5/16   16  = 2 × 2 × 2 × 2      terminates   0.3125

1/3     3  = 3                  repeats       0.333...
1/6     6  = 2 × 3              repeats       0.1666...
3/14   14  = 2 × 7              repeats       0.2142857142857...

Reducing first matters. 6/15 looks doomed because 15 contains a 3, but it reduces to 2/5, and 2/5 is 0.4 exactly. The 3 was never really in the denominator.

Mixed numbers keep their whole part

Convert the fraction, leave the whole number where it is, and add them at the end:

2 3/8   ->  keep 2,  convert 3/8

3/8   =  0.375
2 3/8 =  2 + 0.375  =  2.375

or convert the improper fraction:  2 × 8 + 3 = 19,  19 ÷ 8 = 2.375

Both routes land on the same number, which is a useful check when the whole part is large enough to be worth losing track of.

The conversions worth memorising

The percent column is free once you have the decimal: shift the point two places right. 17/20 is 0.85, so a score of 17 out of 20 is 85%. That is the real reason this conversion earns its keep. Fractions are how quantities arrive, and decimals and percentages are how anyone compares them.

What the decimal throws away

Every terminating fraction survives the trip intact. The repeating ones do not, because your screen has a finite number of digits and the decimal does not:

1/3   =  0.333333333333...      the true value
      ->  0.3333333333          what a 10 digit display shows

1/3 + 1/3 + 1/3        =  1          exactly
0.3333333333 × 3       =  0.9999999999    short by 1e-10

One third of a pizza, three times, is a pizza. The decimal version is not, and no number of digits fixes it. Convert early and the error rides along through every step after it, which is the same failure that makes adding fractions as decimals unreliable and the same one behind 0.1 + 0.2 not landing on 0.3.

The fix is not more decimal places. It is not converting until the end. A calculator doing exact rational arithmetic holds 1/3 as 1/3 through the whole calculation and only renders a decimal when you ask to see one, so 1/3 + 1/3 + 1/3 comes back as 1 and 0.1 + 0.2 comes back as exactly 0.3.

So which form do you want

Decimals win when you are comparing, sorting or dealing with money, because 0.375 and 0.4 sort at a glance and 3/8 and 2/5 do not. Fractions win when the denominator has a 3 or a 7 in it, when the quantity is genuinely a ratio, and any time the value has more arithmetic ahead of it. Convert last, not first.

Type 5/8, tap once, see 0.625

Prism's fraction mode flips between fraction and decimal on the same result and computes exactly underneath, so nothing rounds until you look at it. Fractions are free forever, no subscription. See the app.