Recurring decimals to fractions: worked examples and method

One digit

A single repeating digit sits over 9. 0.777... is 7/9.

Two digits

A two-digit block sits over 99, then you cancel.

Cancel

45/99 cancels by 9 to 5/11. The decimal and the cancelled fraction are the same value.

Check a claim

If someone writes both 0.777... and 7/9, those two are equal.

From a fraction

Divide the numerator by the denominator until the remainder repeats, then write the block.

Exact form

Leave the cancelled fraction. Do not round the decimal.

Read the dots before you multiply

The dots tell you the length of the repeating block, and that length tells you whether to multiply by 10, 100 or 1000. One repeating digit, as in 0.dot7, needs × 10. Two repeating digits, as in 0.dot1dot6, need × 100. Three would need × 1000. You want the repeating block to sit just in front of the decimal point on the second line, in the same rhythm as the first line.

0.dot1dot6 is not 0.1666ldots The two dots mean both the 1 and the 6 are inside the block, so the number is 0.161616ldots Writing 0.1dot6 puts the dot only on the 6, so the 1 happens once and then sixes continue. Those are different numbers: 16/99 and 1/6. Copy the dots onto your x line before you choose the multiplier.

A terminating decimal such as 0.16 is 16/100 = 4/25. It does not need the subtraction method. If the question prints dots, or writes 'recurring', use the algebra. If it prints a decimal that stops, use place value. Mixing the two writes a terminating decimal over 99 and gets a number that is slightly different.

Method: 1. The dots mark the digits that repeat. 2. One repeating digit means multiply by 10. 3. Two repeating digits means multiply by 100.

One digit on repeat

Let x = 0.dot7 = 0.7777ldots Then 10x = 7.7777ldots Subtract the first line from the second. The endless sevens cancel: 10x - x = 7, so 9x = 7. Therefore x = 7/9. The denominator is 9 because you multiplied by 10 and then took one x away, leaving 9 lots of x.

The same pattern gives every single-digit recurrence. 0.dot4 = 4/9 and 0.dot2 = 2/9. The numerator is the repeating digit and the denominator is 9. Cancel if you can. 0.dot6 = 6/9 = 2/3. Always check whether the fraction simplifies. 7 and 9 share no common factor, so 7/9 is already in lowest terms.

A quick decimal check: 7 ÷ 9 starts 0.777ldots If your fraction gives a different first digit, the subtraction went wrong, usually because 10x was written as 7.777ldots and then x was subtracted as if it were 0.7 exactly. Use the full recurring tail on both lines so the tails really do cancel.

Method: 1. Let x be the recurring decimal. 2. Multiply by 10 and subtract the original line so the repeating tail cancels. 3. Solve for x and simplify the fraction.

A two-digit block

Let x = 0.dot1dot6 = 0.161616ldots Two digits repeat, so multiply by 100: 100x = 16.161616ldots Subtract: 100x - x = 16, because the tails 0.161616ldots cancel. So 99x = 16 and x = 16/99. The denominator is 99 whenever a two-digit block starts immediately after the decimal point.

16 and 99 have no common factor greater than 1. 16 = 2^4 and 99 = 9 × 11. The fraction 16/99 is in lowest terms. If the numerator and denominator share a factor, cancel it. 0.dot3dot6 = 36/99 = 4/11. The method always produces a correct fraction. Lowest terms is a separate last step.

Do not multiply by 10 for a two-digit block and hope. 10x = 1.61616ldots does not line up with x = 0.161616ldots Subtracting those leaves a recurring decimal still on the right-hand side, so you have not reached a whole number. The multiplier must shift by the length of the block.

Method: 1. Let x be the decimal and multiply by 100 when two digits repeat. 2. Subtract x so the repeating block cancels. 3. The denominator is 99 before you cancel.

A digit that does not repeat

For 0.1dot6 = 0.1666ldots the 1 is not in the loop. Let x = 0.1666ldots First multiply by 10 so the non-repeating digit clears the decimal point: 10x = 1.6666ldots Now the sixes recur. Multiply by 10 again: 100x = 16.6666ldots Subtract the 10x line from the 100x line, not the original x line. 100x - 10x = 15, so 90x = 15.

Then x = 15/90. Cancel by 15: 15 ÷ 15 = 1 and 90 ÷ 15 = 6, so x = 1/6. The check is the division you already know: 1 ÷ 6 = 0.1666ldots Subtracting 100x - x instead of 100x - 10x would leave the non-repeating digit inside the tail and the right-hand side would not be a whole number.

Which fractions recur? Cancel the fraction first. If the denominator is then only twos and fives, the decimal terminates, because those are the primes in 10. 1/2, 1/4, 1/5, 1/8, 3/20 all terminate. If any other prime remains, such as 3, 7 or 11, the decimal recurs. 1/3, 1/6, 1/9 and 1/11 all recur. 1/6 has a 3 left after the factor of 2 is accounted for, so it recurs.

Method: 1. Multiply first so the non-repeating digits move past the decimal point. 2. Multiply again by 10 for each repeating digit, then subtract. 3. Simplify the fraction you get.

Blocks of three or more repeating digits

The rule for converting a recurring decimal is to multiply by 10 to the power of the number of digits in the repeating block. One repeating digit means multiply by 10, two digits means 100 and three digits means 1000. For 0.dot18dot5, which means 0.185185185ldots, the block is 185, so use 1000x. Then 1000x = 185.185185ldots and x = 0.185185ldots, so subtracting gives 999x = 185 and x = 185/999.

The dots tell you the block. A dot goes over the first digit and the last digit of the repeating block, and every digit between them repeats too. Always write out the decimal to at least six places so you can see the block before you multiply. Finally cancel the fraction. When the denominator is 999, remember that 999 = 27 × 37, so a numerator that is a multiple of 37 cancels. Here 185 = 5 × 37, so the fraction is 5/27.

Method: 1. Write out the decimal to at least six places and mark the repeating block. 2. Count the digits in the block and multiply x by 10 to that power. 3. Subtract x so the recurring tails cancel, leaving a whole number. 4. Divide to find x and cancel the fraction fully.

Tip: The denominator before cancelling is always made of nines, one nine for each digit in the block.

From a fraction to a recurring decimal

A fraction in its simplest form gives a recurring decimal when its denominator has a prime factor other than 2 or 5. So 5/6 recurs because 6 = 2 × 3, while 3/8 terminates because 8 = 2^3. The fraction 7/12 recurs because 12 = 2^2 × 3, and so does 4/11. To convert, divide the numerator by the denominator with short or long division. Add zeros after the point and keep dividing until a remainder repeats. The digits between the repeated remainders are the block.

Use the dot notation correctly. The fraction 5/6 = 0.8333ldots is written 0.8dot3, with one dot over the 3 because only the 3 repeats. The fraction 4/11 = 0.363636ldots is written 0.dot3dot6, with dots over the first and last digit of the block. A mark scheme gives the method mark for showing the division and the accuracy mark for the correct dots. Rounding to 0.83 or 0.8333 earns no accuracy mark.

Method: 1. Divide the numerator by the denominator, adding zeros after the decimal point. 2. Stop when a remainder repeats. 3. Identify which digits repeat. 4. Write the answer with dots over the first and last digit of the repeating block.

Proofs: show that and prove that

When a question says "Show that 0.6dot3 = 19/30" or "Prove that", you must write full algebra, not just the final fraction. Begin with "let x = 0.6333ldots", multiply twice so the repeating parts line up, subtract, solve for x, simplify and state the conclusion. Each line earns credit: the first for correct multiples, the second for the subtraction, and the last for the simplified fraction that matches the target.

Because the target fraction is given, you can check backwards on scrap paper by dividing 19 by 30. The written proof must still run forwards. For a decimal like 0.6dot3, where one digit does not repeat, multiply by 10 and by 100 so that the recurring tails match. Their difference, 100x - 10x = 90x, is a whole number. Finish with a sentence such as "so 0.6dot3 = 19/30" so the conclusion is explicit.

Method: 1. Write "let x =" and the decimal to six places. 2. Multiply by two powers of 10 so that the recurring tails are identical. 3. Subtract the smaller multiple from the larger and solve for x. 4. Simplify, compare with the target and write a short conclusion.

Using recurring decimals inside other problems

Some questions place recurring decimals inside a calculation, such as adding 0.dot7 and 0.dot2dot7, writing 2.dot4 as a fraction, or ordering 0.45, 0.4dot5 and 5/11. Convert each recurring decimal to a fraction first, then use a common denominator. For a number bigger than 1, such as 2.dot4, keep the whole number part and convert only the recurring part: 2.dot4 = 2 + 4/9 = 22/9.

The word hence means use the result you already proved. If you have shown that 0.dot4dot5 = 5/11, then 0.0dot4dot5 is one tenth of it, so it is 5/110 = 1/22. Do not start again from scratch. Two fractions in their simplest form are equal only when their numerators and denominators match, so simplify before you compare or conclude.

Method: 1. Convert each recurring decimal into a fraction, keeping any whole number separate. 2. Combine the fractions using a common denominator, or divide by 10 if the earlier result allows it. 3. Simplify the final fraction. 4. Compare or conclude in a sentence if the question asks for it.