Remainders as Decimals: How to Continue Division After a Remainder

To turn a remainder into a decimal, put a decimal point after the quotient, add a zero to the remainder, and keep dividing. Dividing 43 by 8 gives 5 remainder 3. Continuing past that point turns the 3 into 30, 60 and then 40, producing the digits 3, 7 and 5, so the answer is 5.375.

Nothing new is needed. The remainder is not discarded, it is carried into a new column and divided again, exactly as it was in the whole-number part of the division.

This guide works through the continuation step by step, explains why some divisions stop and others repeat forever, shows how to predict which you will get before starting, and covers rounding, with ten practice questions.

The Method in 4 Steps

1. Divide as usual until you run out of digits and are left with a remainder.
2. Write a decimal point in the answer, directly above the decimal point in the dividend.
3. Bring down a zero. The dividend has an unlimited supply of them: 43 is the same number as 43.000.
4. Repeat the divide, multiply, subtract, bring down cycle until the remainder reaches 0 or a pattern repeats.

The step that makes this work is number 3. Attaching a zero to the remainder multiplies it by 10, which moves it into the next place value down. That is the same shift that happens every time you bring down a digit in ordinary long division with remainders.

Worked Example 1: A Division That Stops

Problem. Write 43 ÷ 8 as a decimal.

The whole-number part. 8 fits into 43 five times, using 40, leaving 3.

43 ÷ 8 = 5 remainder 3

Now continue. Place the decimal point after the 5 and bring down a zero.

The remainder reached 0, so the division is finished.

Answer. 43 ÷ 8 = 5.375

Verification. 5.375 × 8 = 43. ✓

Worked Example 2: A Division That Repeats

Problem. Write 25 ÷ 6 as a decimal.

25 ÷ 6 = 4 remainder 1

Continue:

The remainder is stuck at 4. Every further step produces another 6, forever.

Answer. 25 ÷ 6 = 4.1666… = 4.16̄, where the bar marks the digit that repeats.

The moment a remainder reappears, the digits from that point repeat exactly. That is the signal to stop dividing and write the bar.

Worked Example 3: A Longer Repeating Block

Problem. Write 100 ÷ 7 as a decimal.

100 ÷ 7 = 14 remainder 2

Continuing gives the remainders 2, 6, 4, 5, 1, 3 and then 2 again:

Answer. 100 ÷ 7 = 14.285714285714… = 14.285714, with all six digits repeating.

The repeating block can never be longer than divisor − 1 digits, because only that many different non-zero remainders exist. Dividing by 7 gives at most 6 digits, and here it gives exactly 6.

Worked Example 4: When the Quotient Is Zero

Problem. Write 7 ÷ 40 as a decimal.

The divisor is larger than the dividend, so the whole-number quotient is 0 and the entire dividend is the remainder.

7 ÷ 40 = 0 remainder 7

Write “0.” and continue:

Answer. 7 ÷ 40 = 0.175

Terminating or Repeating: How to Tell in Advance

A decimal stops only when a remainder of 0 is reached. Whether that ever happens is decided entirely by the divisor.

Write the fraction in its simplest form. If the denominator’s only prime factors are 2 and 5, the decimal terminates. Any other prime factor makes it repeat.

The reason is that our number system is base 10, and 10 = 2 × 5. A denominator built only from 2s and 5s divides some power of 10 exactly, so the division can finish.

Simplify first. 6 ÷ 15 looks as though it will repeat, since 15 contains a 3. But 6/15 reduces to 2/5, and that terminates at 0.4. The test only works on the reduced form.

How to Write a Repeating Decimal

Three notations are in common use:

4.1666…          dots to show it continues
4.16̄             a bar over the repeating digit
4.16(6)          the repeating part in brackets

The bar goes over only the digits that repeat. For 100 ÷ 7 the whole six digit block repeats, so the bar covers 285714. For 7 ÷ 12 = 0.58333… only the 3 repeats, so the bar covers just the 3 and the 58 stands outside it.

Writing 4.166 without any marking is a different, smaller number, so the notation is not decoration.

Rounding a Decimal Answer

A repeating decimal has no end, so a practical answer needs rounding.

Rule. Look at the first digit you are dropping. If it is 5 or more, round the last kept digit up. Otherwise leave it alone.

Round only at the very end. Rounding partway through and then continuing to calculate compounds the error, sometimes badly.

For money, round to 2 decimal places. For most schoolwork, 2 or 3 places is expected unless the question says otherwise.

Decimal or Fraction?

The same remainder can be written either way, and both are exact in the fraction form.

43 ÷ 8  =  5 3/8  =  5.375        both exact
25 ÷ 6  =  4 1/6  =  4.1666…      only the fraction is exact

That second line is the key difference. A fraction is always exact. A decimal is exact only when it terminates, and otherwise it has to be rounded or marked with a bar.

Use a decimal for money, measurement, and anything being compared or added to other decimals. Use a fraction when the value must be exact, especially with thirds, sixths, sevenths and ninths. Building the fraction form is covered in remainders as fractions.

Common Mistakes

• Writing the remainder after the decimal point. 43 ÷ 8 is 5 remainder 3, but that is not 5.3. The remainder must be divided, not just relocated.
• Forgetting the decimal point in the answer. It goes directly above its position in the dividend, before the first brought-down zero.
• Stopping too early. If the remainder is not 0 and no pattern has appeared, the division is not finished.
• Missing a repeat. Watch the remainders, not the digits. A remainder appearing twice means the block between them repeats.
• Putting the bar over the wrong digits. In 0.58333… only the 3 repeats.
• Testing an unsimplified fraction for termination. Reduce first, then look at the denominator’s prime factors.
• Rounding partway through. Carry the full value and round once at the end.

Practice Questions

Write each as a decimal. Mark any repeating block, and say whether the decimal terminates or repeats.

1. 17 ÷ 4
2. 246 ÷ 8
3. 1 ÷ 3
4. 89 ÷ 16
5. 5 ÷ 11
6. 47 ÷ 9
7. 31 ÷ 25
8. 7 ÷ 12
9. 5 ÷ 7
10. Round the answer to question 9 to 3 decimal places.

Answers

1. 17 ÷ 4 = 4 r 1, and 10 ÷ 4 = 2 r 2, then 20 ÷ 4 = 5 r 0. 4.25, terminating (4 = 2²).
2. 246 ÷ 8 = 30 r 6, then 60 ÷ 8 = 7 r 4, then 40 ÷ 8 = 5 r 0. 30.75, terminating.
3. 1 ÷ 3 = 0 r 1, and 10 ÷ 3 = 3 r 1 forever. 0.3̄, repeating (3 is not 2 or 5).
4. 89 ÷ 16 = 5 r 9, and continuing gives 5, 6, 2, 5. 5.5625, terminating (16 = 2⁴).
5. 5 ÷ 11 = 0 r 5, and the remainders alternate 5, 6, 5, 6. 0.45, with both digits repeating.
6. 47 ÷ 9 = 5 r 2, and the remainder stays 2 forever. 5.2̄, repeating.
7. 31 ÷ 25 = 1 r 6, then 60 ÷ 25 = 2 r 10, then 100 ÷ 25 = 4 r 0. 1.24, terminating (25 = 5²).
8. 7 ÷ 12 = 0 r 7. Continuing gives 5, then 8, then 3 repeating. 0.583̄, repeating, because 12 = 2² × 3.
9. 5 ÷ 7 = 0 r 5, and the six digit block repeats. 0.714285, repeating.
10. 0.714285… to 3 decimal places. The fourth digit is 2, which is below 5, so nothing rounds up: 0.714.

The Remainder Calculator reports the quotient and remainder for any division, and the Long Division Calculator carries the tableau on into the decimal places.

Remainder to Decimal FAQ

How do you turn a remainder into a decimal?

Place a decimal point after the quotient, attach a zero to the remainder, and divide again. Repeat until the remainder is 0 or a remainder repeats. For 43 ÷ 8 remainder 3, the continuation gives 30, 60 and 40, producing 5.375.

Why do you add zeros to the remainder?

Adding a zero multiplies the remainder by 10, which moves it into the next place value down. It is the same shift as bringing down a digit, and it is valid because 43 and 43.000 are the same number.

Why do some divisions never end?

Because the remainder never reaches 0. There are only finitely many possible remainders below the divisor, so one must eventually reappear, and from that point the digits repeat. Dividing by 7 gives a block of at most 6 digits.

How do I know if a decimal will terminate before I start?

Reduce the fraction, then factor the denominator. If it contains only 2s and 5s, the decimal terminates. Any other prime factor guarantees a repeating decimal. So 8, 16, 20 and 25 terminate, while 3, 6, 7, 9 and 11 repeat.

What does the bar over a decimal mean?

It marks the digits that repeat without end. 0.3̄ means 0.3333… and 0.45 with a bar over both digits means 0.454545…. Only the repeating digits go under the bar.

Is the decimal more accurate than the remainder?

No. A quotient with a remainder is exact, and so is a fraction. A terminating decimal is exact too, but a rounded repeating decimal is an approximation. 1/3 is exact where 0.333 is not.

What is the difference between the remainder and the digits after the decimal point?

They carry the same information in different units. In 43 ÷ 8, the remainder 3 counts leftover whole units, and the decimal .375 expresses that same leftover as a fraction of the divisor. Dividing 3 by 8 gives 0.375, which is why the two agree.

Can a decimal answer be turned back into a remainder?

Yes. Multiply the fractional part by the divisor. For 5.375 with divisor 8, that gives 0.375 × 8 = 3, the original remainder. That reversal is the basis of finding the remainder on a calculator.