Look at the Denominator and You Know How the Decimal Ends
Learn to convert a fraction into its decimal form, predict from the denominator whether it terminates, find the repeating block of one seventh and one thirteenth, and turn any repeating decimal back into a fraction.
Can you tell whether a fraction's decimal ends without dividing?
Divide by and the division stops: , remainder zero. Divide by and it never stops: with the same remainder returning for ever.
What decides which happens? Only the denominator, once the fraction is in lowest terms.
The rule turns out to be this simple: **if the denominator in lowest terms is built only from s and s, the decimal terminates. Otherwise it repeats for ever in a fixed cycle. Nothing else can happen to a fraction — no fraction produces a decimal that runs on without a pattern.
That last point matters more than it sounds. It means the decimal expansion is a test** for rationality: terminating or repeating means rational, and anything else means irrational. The previous part of this chapter proved irrational by contradiction; this part supplies the practical version.
This page covers the third part of the CBSE Class 9 Mathematics chapter on the world of numbers — converting fractions to decimals, predicting the outcome from the denominator, finding repeating blocks, and converting decimals back into the form.
What decides which happens? Only the denominator, once the fraction is in lowest terms.
The rule turns out to be this simple: **if the denominator in lowest terms is built only from s and s, the decimal terminates. Otherwise it repeats for ever in a fixed cycle. Nothing else can happen to a fraction — no fraction produces a decimal that runs on without a pattern.
That last point matters more than it sounds. It means the decimal expansion is a test** for rationality: terminating or repeating means rational, and anything else means irrational. The previous part of this chapter proved irrational by contradiction; this part supplies the practical version.
This page covers the third part of the CBSE Class 9 Mathematics chapter on the world of numbers — converting fractions to decimals, predicting the outcome from the denominator, finding repeating blocks, and converting decimals back into the form.
How do you convert a fraction into its decimal form?
Divide the numerator by the denominator, carrying zeros, and watch the remainders. The remainders tell you when to stop and what repeats.
Worked example 1 — a terminating expansion. Convert .
Dividing by : into goes times with remainder ; into goes with remainder ; into goes with remainder .
The remainder reached zero, so the division ended. Check by multiplying back: .
Worked example 2 — another terminating case. Convert .
Multiplying top and bottom to reach a power of ten is faster than long division whenever it is possible — and it is possible exactly when the expansion terminates.
Worked example 3 — a repeating expansion. Convert .
into goes times with remainder — and the remainder is where we started, so the digit repeats for ever:
The bar notation marks the repeating block.
Worked example 4 — a mixed case. Convert .
into goes with remainder ; into goes with remainder ; into goes with remainder — and the remainder has now repeated.
So two digits settle first and then one digit repeats. Check: should give , and , closing on as more digits are taken.
Why a fraction can never produce a patternless decimal. Dividing by , every remainder is one of — at most possibilities. So within steps either a remainder is zero, ending the division, or a remainder repeats, and once a remainder repeats the digits after it must repeat too, because the same division is being done again.
That argument also bounds the block length. For denominator the repeating block can be at most digits long, since only non-zero remainders exist. The next sections use both halves of this fact.
Worked example 1 — a terminating expansion. Convert .
Dividing by : into goes times with remainder ; into goes with remainder ; into goes with remainder .
The remainder reached zero, so the division ended. Check by multiplying back: .
Worked example 2 — another terminating case. Convert .
Multiplying top and bottom to reach a power of ten is faster than long division whenever it is possible — and it is possible exactly when the expansion terminates.
Worked example 3 — a repeating expansion. Convert .
into goes times with remainder — and the remainder is where we started, so the digit repeats for ever:
The bar notation marks the repeating block.
Worked example 4 — a mixed case. Convert .
into goes with remainder ; into goes with remainder ; into goes with remainder — and the remainder has now repeated.
So two digits settle first and then one digit repeats. Check: should give , and , closing on as more digits are taken.
Why a fraction can never produce a patternless decimal. Dividing by , every remainder is one of — at most possibilities. So within steps either a remainder is zero, ending the division, or a remainder repeats, and once a remainder repeats the digits after it must repeat too, because the same division is being done again.
That argument also bounds the block length. For denominator the repeating block can be at most digits long, since only non-zero remainders exist. The next sections use both halves of this fact.
Formula
What does the denominator tell you about whether the decimal ends?
Put the fraction in lowest terms and factorise the denominator. Then:
Any other prime factor in — a , a , an — forces the expansion to repeat.
Worked predictions.
- : . Only s, so it terminates. Indeed
- : . Terminates. Indeed
- : . Terminates. Indeed
- : . Terminates, as
- : is neither nor . Repeats
- : . The decides it — repeats
- : . Repeats, as
Why the rule works. A terminating decimal is a fraction with a power of ten underneath: . And
so a denominator can be turned into a power of ten by multiplying only if it already consists of s and s. For one extra does it. For no multiplier works, because no power of ten has as a factor.
Lowest terms is not optional. Test . As written, and the rule seems to predict repeating. But
which terminates. The cancelled against the numerator, so it was never really in the denominator. Always cancel first, and then apply the test — this is the most frequent error in prediction questions.
How many decimal places a terminating expansion has. It is the larger of and in . For that is places, and has three. For it is places, and has two. Check on : four places, and has four.
Any other prime factor in — a , a , an — forces the expansion to repeat.
Worked predictions.
- : . Only s, so it terminates. Indeed
- : . Terminates. Indeed
- : . Terminates. Indeed
- : . Terminates, as
- : is neither nor . Repeats
- : . The decides it — repeats
- : . Repeats, as
Why the rule works. A terminating decimal is a fraction with a power of ten underneath: . And
so a denominator can be turned into a power of ten by multiplying only if it already consists of s and s. For one extra does it. For no multiplier works, because no power of ten has as a factor.
Lowest terms is not optional. Test . As written, and the rule seems to predict repeating. But
which terminates. The cancelled against the numerator, so it was never really in the denominator. Always cancel first, and then apply the test — this is the most frequent error in prediction questions.
How many decimal places a terminating expansion has. It is the larger of and in . For that is places, and has three. For it is places, and has two. Check on : four places, and has four.
How long is the repeating block of one seventh?
Divide, and keep dividing until a remainder comes back. The number of steps between one appearance of a remainder and the next is the block length.
**Worked example 1 — .** Dividing by and recording each remainder:
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder — back to the start
The block is and its **length is . Check**: , and , closing on .
Six is the largest possible length here, because dividing by has only the six non-zero remainders to and all six appeared.
**Worked example 2 — .**
- : digit , remainder
- : digit , remainder — back to the start
**Length ** — far shorter than the maximum of . Check: .
**Worked example 3 — .** The remainders run , six steps, giving
**Length **, again well short of the maximum of . Check: , so exactly.
Worked example 4 — the other sevenths. Dividing by uses the same cycle of remainders entered at a different point:
The digits appear rotated rather than rearranged — which is exactly what "the same cycle of remainders" means.
**The block length is at most , and is often much less.** For and it is ; for it is ; for it is . The maximum is a ceiling, not a prediction, and the only reliable method is to divide until a remainder returns.
Where the repeating starts also depends on the denominator. For the cycle begins two places in, because contains a as well as the — the s produce a terminating opening and the produces the tail. **A denominator of s and s only gives all terminating; nothing but other primes gives immediate repetition; a mixture gives a mixture.**
**Worked example 1 — .** Dividing by and recording each remainder:
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder
- : digit , remainder — back to the start
The block is and its **length is . Check**: , and , closing on .
Six is the largest possible length here, because dividing by has only the six non-zero remainders to and all six appeared.
**Worked example 2 — .**
- : digit , remainder
- : digit , remainder — back to the start
**Length ** — far shorter than the maximum of . Check: .
**Worked example 3 — .** The remainders run , six steps, giving
**Length **, again well short of the maximum of . Check: , so exactly.
Worked example 4 — the other sevenths. Dividing by uses the same cycle of remainders entered at a different point:
The digits appear rotated rather than rearranged — which is exactly what "the same cycle of remainders" means.
**The block length is at most , and is often much less.** For and it is ; for it is ; for it is . The maximum is a ceiling, not a prediction, and the only reliable method is to divide until a remainder returns.
Where the repeating starts also depends on the denominator. For the cycle begins two places in, because contains a as well as the — the s produce a terminating opening and the produces the tail. **A denominator of s and s only gives all terminating; nothing but other primes gives immediate repetition; a mixture gives a mixture.**
How do you turn a repeating decimal back into a fraction?
Multiply by a power of ten to shift the repeating block, then subtract so the tails cancel.
Worked example 1 — a terminating decimal. Convert . Count the places and use that power of ten:
No algebra needed — that is what terminating means.
Worked example 2 — one repeating digit. Convert . Let . Multiplying by shifts one place:
Worked example 3 — a two-digit block. Convert . The block has two digits, so multiply by . Let :
Check by dividing back: . Correct.
Worked example 4 — repeating, but not from the start. Convert , meaning . Two multipliers are needed. Let :
Check: . Correct — and the denominator explains the shape, one settling digit then one repeating.
Why the subtraction works. Multiplying by , where is the block length, moves the decimal point by exactly one whole block. The two numbers then have identical infinite tails, so subtracting removes the tails completely and leaves a whole number — and a whole number over a whole number is a fraction.
Choose the multiplier from the block length, not the number of digits shown. For the block is two digits, so . For one digit settles before a one-digit block, so and . Picking the wrong power leaves a decimal tail behind and the method collapses.
Every terminating or repeating decimal is therefore rational, and combined with the previous section — every fraction terminates or repeats — the two statements close the loop: the rationals are exactly the terminating and repeating decimals. Every remaining point on the line, such as or , has an expansion that neither ends nor cycles. Each real number corresponds to one point on the number line and each point to one real number, and the decimal expansion is the address that locates it: each further digit narrows the interval by a factor of ten, closing in on a single point.
Worked example 1 — a terminating decimal. Convert . Count the places and use that power of ten:
No algebra needed — that is what terminating means.
Worked example 2 — one repeating digit. Convert . Let . Multiplying by shifts one place:
Worked example 3 — a two-digit block. Convert . The block has two digits, so multiply by . Let :
Check by dividing back: . Correct.
Worked example 4 — repeating, but not from the start. Convert , meaning . Two multipliers are needed. Let :
Check: . Correct — and the denominator explains the shape, one settling digit then one repeating.
Why the subtraction works. Multiplying by , where is the block length, moves the decimal point by exactly one whole block. The two numbers then have identical infinite tails, so subtracting removes the tails completely and leaves a whole number — and a whole number over a whole number is a fraction.
Choose the multiplier from the block length, not the number of digits shown. For the block is two digits, so . For one digit settles before a one-digit block, so and . Picking the wrong power leaves a decimal tail behind and the method collapses.
Every terminating or repeating decimal is therefore rational, and combined with the previous section — every fraction terminates or repeats — the two statements close the loop: the rationals are exactly the terminating and repeating decimals. Every remaining point on the line, such as or , has an expansion that neither ends nor cycles. Each real number corresponds to one point on the number line and each point to one real number, and the decimal expansion is the address that locates it: each further digit narrows the interval by a factor of ten, closing in on a single point.
Exam tip
Exam tip: cancel first, then factorise the denominator
Reduce to lowest terms before predicting. looks like it repeats, but it equals — the cancels.
Then factorise: only s and s means terminating; any other prime means repeating. Write the factorisation in the answer — — because that is the justification.
The number of decimal places in a terminating expansion is the larger index in : gives four places, .
Use bar notation correctly: , , . The bar covers the repeating block only.
Find a block by tracking remainders, not digits — the block closes when a remainder returns. For the remainders are , so the length is .
**The block length is at most ** but often far less — has length , not . Never quote the maximum as the answer.
For conversion, the multiplier comes from the block length: for one digit, for two. If digits settle first, use two multipliers — needs .
Always check by dividing back: confirms the conversion in one line.
And for classification, remember the full statement: terminating or repeating means rational; neither means irrational.
Then factorise: only s and s means terminating; any other prime means repeating. Write the factorisation in the answer — — because that is the justification.
The number of decimal places in a terminating expansion is the larger index in : gives four places, .
Use bar notation correctly: , , . The bar covers the repeating block only.
Find a block by tracking remainders, not digits — the block closes when a remainder returns. For the remainders are , so the length is .
**The block length is at most ** but often far less — has length , not . Never quote the maximum as the answer.
For conversion, the multiplier comes from the block length: for one digit, for two. If digits settle first, use two multipliers — needs .
Always check by dividing back: confirms the conversion in one line.
And for classification, remember the full statement: terminating or repeating means rational; neither means irrational.
Did you know
Why one seventh and one thirteenth repeat for the same length
repeats in blocks of . also repeats in blocks of . But repeats in blocks of , not .
The reason is hiding in a single line of arithmetic. Because , multiplying the block by gives
six nines. And for :
the same six nines. So both denominators divide , and that is what fixes the block length at six. Likewise , two nines — so divides and the block is two digits long.
The block length of is therefore the number of nines in the smallest string of nines that divides exactly. For it is one nine, giving length . For and it is six. For it is two.
This is why the maximum is only a ceiling. Dividing by visits at most non-zero remainders before repeating, so the block cannot be longer — but it can be shorter, and whether it is depends on which string of nines happens to divide.
There is a neat consequence. Since and both divide , so does — and also repeats in blocks of : . Check: .
So the question "how long is the block" is really the question "which repunit of nines does this denominator divide". A long division is the slow way to find out, and the multiplication above is the check that you found it.
The reason is hiding in a single line of arithmetic. Because , multiplying the block by gives
six nines. And for :
the same six nines. So both denominators divide , and that is what fixes the block length at six. Likewise , two nines — so divides and the block is two digits long.
The block length of is therefore the number of nines in the smallest string of nines that divides exactly. For it is one nine, giving length . For and it is six. For it is two.
This is why the maximum is only a ceiling. Dividing by visits at most non-zero remainders before repeating, so the block cannot be longer — but it can be shorter, and whether it is depends on which string of nines happens to divide.
There is a neat consequence. Since and both divide , so does — and also repeats in blocks of : . Check: .
So the question "how long is the block" is really the question "which repunit of nines does this denominator divide". A long division is the slow way to find out, and the multiplication above is the check that you found it.
Exam relevance
How do decimal expansions feed into JEE Main?
Because the repeating decimal is an infinite series in disguise, and the classification it provides is assumed knowledge in every chapter about real numbers.
This is the foundation for Class 11 Mathematics Sequences and Series, examined in JEE Main. A repeating decimal is precisely an infinite geometric series: means
with first term and common ratio , summing by to . **That is the same answer the subtraction gives**, and the subtraction is the Class 9 shortcut for the Class 11 formula. A student who can convert by hand already understands why the geometric sum converges when .
The completeness of the reals is the other thread. The idea developed here — that each further decimal digit narrows the interval by a factor of ten, closing on exactly one point — is the intuition behind limits in Class 11 Limits and Derivatives, and behind the nested-interval picture that makes rather than something slightly less. Questions exploiting that equality appear in competitive papers precisely because it feels wrong.
Number-theoretic flavour. The block-length question connects to modular arithmetic and to properties of primes, which is the territory of olympiad problems and of the tougher JEE Advanced number questions. The observation that divides a string of nines is the accessible form of a much-used result.
Where classification gets used. Class 11 Sets asks for and to be handled as sets, and questions on whether a given expression is rational rely on the decimal test from this page — most often on the fact that an irrational plus a rational is always irrational.
What the questions look like. For board work, expect convert to decimal and classify, predict from the denominator with justification, find the repeating block, and **convert a repeating decimal to — all short and fully self-contained. For JEE Main, the direct appearance is summing a repeating decimal as a geometric series, or deciding the rationality of a constructed expression.
How board and competitive emphasis differ. A board paper wants the long division shown and the factorisation written out as justification. A competitive paper wants the answer in seconds, and the geometric-series route is usually faster than long division.
The single trap that costs the most marks.** Applying the test before cancelling. has a in its denominator as written and terminates anyway, because the fraction was not in lowest terms. Questions are set with exactly that shape, and the one-line habit of cancelling first is what defeats them.
This is the foundation for Class 11 Mathematics Sequences and Series, examined in JEE Main. A repeating decimal is precisely an infinite geometric series: means
with first term and common ratio , summing by to . **That is the same answer the subtraction gives**, and the subtraction is the Class 9 shortcut for the Class 11 formula. A student who can convert by hand already understands why the geometric sum converges when .
The completeness of the reals is the other thread. The idea developed here — that each further decimal digit narrows the interval by a factor of ten, closing on exactly one point — is the intuition behind limits in Class 11 Limits and Derivatives, and behind the nested-interval picture that makes rather than something slightly less. Questions exploiting that equality appear in competitive papers precisely because it feels wrong.
Number-theoretic flavour. The block-length question connects to modular arithmetic and to properties of primes, which is the territory of olympiad problems and of the tougher JEE Advanced number questions. The observation that divides a string of nines is the accessible form of a much-used result.
Where classification gets used. Class 11 Sets asks for and to be handled as sets, and questions on whether a given expression is rational rely on the decimal test from this page — most often on the fact that an irrational plus a rational is always irrational.
What the questions look like. For board work, expect convert to decimal and classify, predict from the denominator with justification, find the repeating block, and **convert a repeating decimal to — all short and fully self-contained. For JEE Main, the direct appearance is summing a repeating decimal as a geometric series, or deciding the rationality of a constructed expression.
How board and competitive emphasis differ. A board paper wants the long division shown and the factorisation written out as justification. A competitive paper wants the answer in seconds, and the geometric-series route is usually faster than long division.
The single trap that costs the most marks.** Applying the test before cancelling. has a in its denominator as written and terminates anyway, because the fraction was not in lowest terms. Questions are set with exactly that shape, and the one-line habit of cancelling first is what defeats them.
Key takeaways
Terminating and repeating decimals: quick revision
- Divide numerator by denominator and watch the remainders: remainder ends the division, a repeated remainder starts a cycle.
- Only these two things can happen, because dividing by has at most possible remainders.
- **, , , — all terminating.
- , , — all repeating.
- The test: in lowest terms, the expansion terminates exactly when** . Any other prime forces repetition.
- Why: , so only s and s can be built up into a power of ten.
- Cancel first. appears to fail the test but equals .
- Number of places in a terminating expansion is the larger of and : gives four.
- ****, block length (remainders ).
- ****, length . ****, length .
- — the same cycle entered at a different point, so the digits are rotated.
- **Block length is at most **, often much less — a ceiling, not a prediction.
- A denominator mixing s or s with other primes gives digits that settle then repeat: .
- To convert back: directly.
- : , so .
- : , so .
- : , so and .
- The multiplier comes from the block length, and digits that settle first need a second multiplier.
- The rationals are exactly the terminating and repeating decimals; anything else, such as or , is irrational.
- Every real number is one point on the number line and every point one real number — each extra digit narrows the interval tenfold onto that point.
Pick a denominator nobody set you — , say — and divide until a remainder comes back, then multiply your block by and see how many nines you get.
- Only these two things can happen, because dividing by has at most possible remainders.
- **, , , — all terminating.
- , , — all repeating.
- The test: in lowest terms, the expansion terminates exactly when** . Any other prime forces repetition.
- Why: , so only s and s can be built up into a power of ten.
- Cancel first. appears to fail the test but equals .
- Number of places in a terminating expansion is the larger of and : gives four.
- ****, block length (remainders ).
- ****, length . ****, length .
- — the same cycle entered at a different point, so the digits are rotated.
- **Block length is at most **, often much less — a ceiling, not a prediction.
- A denominator mixing s or s with other primes gives digits that settle then repeat: .
- To convert back: directly.
- : , so .
- : , so .
- : , so and .
- The multiplier comes from the block length, and digits that settle first need a second multiplier.
- The rationals are exactly the terminating and repeating decimals; anything else, such as or , is irrational.
- Every real number is one point on the number line and every point one real number — each extra digit narrows the interval tenfold onto that point.
Pick a denominator nobody set you — , say — and divide until a remainder comes back, then multiply your block by and see how many nines you get.