A Step-by-Step Guide: Converting Fractions to Decimals

Converting a fraction to a decimal in your head is one of the highest-return arithmetic skills for a trading assessment, because fractions turn up constantly and reaching for long division costs you the question. The method below is mechanical: the same three steps repeated until you stop, and it works for any fraction.

1
Divide, and keep the remainder. Integer-divide the numerator by the denominator. That gives the digit before the decimal point. Hold on to the remainder.
2
Multiply the remainder by 10, divide again. The result is your next decimal digit. Keep the new remainder.
3
Repeat until you stop. You stop on a remainder of 0, on a remainder you have seen before, or once you have enough digits.

The method

Integer division

The method rests on integer division, so it is worth being precise about it. Normally 10 divided by 4 is 2.5. In integer division you only ask how many whole times 4 fits into 10, which is 2, and what is left over, which is 2. That pair, the quotient and the remainder, is what drives every step below. In programming the remainder is the modulo.

One step of the method

$d_{i} = \left\lfloor \frac{10 \cdot r_{i-1}}{b} \right\rfloor, \qquad r_{i} = 10 \cdot r_{i-1} \bmod b$

Each pass produces one decimal digit and the remainder that feeds the next pass.

When to stop

Three things can end the process, and recognising which one you have hit is the whole trick:

The remainder is 0. There are no digits left and the decimal terminates.
A remainder repeats. Because you divide by the same denominator every time, a repeated remainder guarantees the digits from that point repeat forever. The repeating block is called the reptend.
You have enough digits. If you need 5 decimal places, work out the sixth and use it to round the fifth.

Worked examples

5/8, a terminating decimal

Step	Division	Digit	Remainder
1	5 ÷ 8	0	5
2	50 ÷ 8	6	2
3	20 ÷ 8	2	4
4	40 ÷ 8	5	0

The remainder hits 0, so the process ends: 5/8 = 0.625. This is a terminating, or non-repeating, decimal.

7/12, a repeating decimal

Step	Division	Digit	Remainder
1	7 ÷ 12	0	7
2	70 ÷ 12	5	10
3	100 ÷ 12	8	4
4	40 ÷ 12	3	4
5	40 ÷ 12	3	4

The remainder settles at 4 and stays there, so the 3 repeats: 7/12 = 0.58333…

1/7, a longer cycle

Step	Division	Digit	Remainder
1	1 ÷ 7	0	1
2	10 ÷ 7	1	3
3	30 ÷ 7	4	2
4	20 ÷ 7	2	6
5	60 ÷ 7	8	4
6	40 ÷ 7	5	5
7	50 ÷ 7	7	1

Step 7 returns the remainder to 1, which is where step 1 started, so the block 142857 repeats indefinitely: 1/7 = 0.142857142857…

13/17, the long case

Denominators like 17 produce long cycles, and this one runs 16 digits before it closes. It is worth working through once so the pattern-spotting becomes familiar rather than alarming.

Step	Division	Digit	Remainder
1	13 ÷ 17	0	13
2	130 ÷ 17	7	11
3	110 ÷ 17	6	8
4	80 ÷ 17	4	12
5	120 ÷ 17	7	1
6	10 ÷ 17	0	10
7	100 ÷ 17	5	15
8	150 ÷ 17	8	14
9	140 ÷ 17	8	4
10	40 ÷ 17	2	6
11	60 ÷ 17	3	9
12	90 ÷ 17	5	5
13	50 ÷ 17	2	16
14	160 ÷ 17	9	7
15	70 ÷ 17	4	2
16	20 ÷ 17	1	3
17	30 ÷ 17	1	13

Step 17 returns the remainder to 13, the value it started from, so the block repeats from there: 13/17 = 0.7647058823529411…

In an assessment you would almost never need all 16 digits. The useful skill is recognising after three or four steps that you are in a long cycle, taking the accuracy you need, and moving on.

Closing remarks

The method never changes, so the only variable is how quickly you execute it, and that comes from repetition. Practise on our fractions to decimals trainer, which drills exactly these conversions against a clock.

It pays off beyond fraction questions themselves. Firms ask for percentage and fraction conversions directly in mental maths tests, so the fluency feeds straight into the broader arithmetic work covered in our mental arithmetic guide and the Math Trainer.

A step-by-step guide: converting Fractions to Decimals