Number System

high importance~2 Q in Tier 133 formulas⚡ 18 shortcuts6 subtopics

Overview Learn (6)Formulas (33)Examples (5)Practice (97)

Subtopic 6 of 6·← Factors, prime factorisation & trailing zeros

Fractions, decimals & recurring decimals

Terminating or not? Reduce $\frac{p}{q}$ to lowest terms. It terminates only if $q$ has no prime factor other than 2 and 5.

Recurring decimal → fraction

Pure recurring: repeating digits over as many 9s. $0.\overline{36} = \frac{36}{99} = \frac{4}{11}$
Mixed recurring: (all digits after the point − non-repeating part) over (one 9 per repeating digit followed by one 0 per non-repeating digit). $0.2\overline{45} = \frac{245-2}{990}$
With a whole number: $2.4\overline{7} = 2 + \frac{47-4}{90}$

Comparing fractions: cross-multiply two at a time, convert to decimals, or compare the gap from 1 (complement) when all fractions are close to 1.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

Fractions in one line

A fraction $\frac{p}{q}$ means $p$ parts out of $q$ equal parts. $p$ is the numerator (top), $q$ the denominator (bottom, never 0). $\frac{3}{4}$ of ₹200 is ₹150.

Proper fraction: top < bottom ( $\frac{3}{7}$ ). Improper: top ≥ bottom ( $\frac{9}{4}$ ). Mixed: $2\frac{1}{4} = \frac{9}{4}$ .
Lowest terms: divide top and bottom by their HCF. $\frac{21}{168} = \frac{1}{8}$ .

Decimals: terminating and recurring

Divide the top by the bottom. Two things can happen:

The division stops: $\frac{7}{80} = 0.0875$ → a terminating decimal.
The digits repeat forever: $\frac{1}{3} = 0.333\dots = 0.\overline{3}$ → a recurring decimal. The bar shows the repeating block.

Terminating test. First reduce the fraction to lowest terms. It terminates only if the denominator has no prime factor other than 2 and 5.

$\frac{7}{80}$ : $80 = 2^4 \times 5$ → terminates.
$\frac{11}{48}$ : $48 = 2^4 \times 3$ → recurring (because of 3).
$\frac{21}{168}$ looks like it has 3 and 7 in the bottom, but it reduces to $\frac{1}{8}$ → terminates. Always reduce first!

Recurring decimal to fraction

Pure recurring (repeats right after the point): write the repeating digits over as many 9s. $0.\overline{7} = \frac{7}{9}$ , $0.\overline{72} = \frac{72}{99} = \frac{8}{11}$ , $0.\overline{142857} = \frac{142857}{999999} = \frac{1}{7}$ .

Mixed recurring (some digits do not repeat): numerator = (all digits after the point) − (non-repeating digits); denominator = one 9 for each repeating digit, then one 0 for each non-repeating digit. $0.4\overline{7} = \frac{47 - 4}{90} = \frac{43}{90}$ , $0.2\overline{45} = \frac{245 - 2}{990} = \frac{243}{990}$ .

With a whole number: handle the whole part separately. $3.1\overline{45} = 3 + \frac{145 - 1}{990} = 3 + \frac{8}{55} = \frac{173}{55}$ .

Why the 9s? $x = 0.\overline{7}$ → $10x = 7.\overline{7}$ → $10x - x = 7$ → $x = \frac{7}{9}$ .

Adding recurring decimals

Convert each to a fraction, add, and convert back if the options are decimals. $0.\overline{6} + 0.\overline{7} + 0.\overline{8} = \frac{6+7+8}{9} = \frac{21}{9} = 2.\overline{3}$ — not 2.1! Adding digits directly is the classic trap.

Comparing fractions

Cross-multiplication for two fractions: $\frac{5}{8}$ vs $\frac{7}{11}$ → $5 \times 11 = 55$ vs $8 \times 7 = 56$ → $\frac{7}{11}$ is bigger.
Decimal method: convert each to 3 decimal places.
Complement method: when all fractions are close to 1, compare the gap from 1. $\frac{13}{17}$ is $\frac{4}{17}$ short of 1; $\frac{9}{11}$ is $\frac{2}{11}$ short. The bigger the gap, the smaller the fraction.
Same numerators: the smaller denominator gives the bigger fraction.

Fraction word problems

" $\frac{3}{5}$ of a number is 63 more than $\frac{1}{4}$ of it" → $(\frac{3}{5} - \frac{1}{4})x = 63$ → $\frac{7}{20}x = 63$ → $x = 180$ . For "spends $\frac{1}{3}$ , then $\frac{1}{4}$ of the remaining" questions, take fractions of what is left, not of the whole.

Quick revision

Terminating ⇔ reduced denominator has only 2s and 5s.
Pure recurring: digits / 9s. Mixed: (all − non-repeating) / (9s then 0s).
Convert to fractions before adding recurring decimals.
Compare by cross-multiplying, decimals or complements.
"Of the remaining" → multiply fractions step by step.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Recurring decimal to fractionvery common2 practice Q

How to spot it:

A decimal with a bar, like $0.\overline{72}$ or $3.1\overline{45}$ , and fractions in the options.

0.a\overline{bc} = \frac{abc - a}{990}

Separate the whole number part.
Numerator: all digits after the point minus the non-repeating digits.
Denominator: one 9 per repeating digit, then one 0 per non-repeating digit.
Simplify and add the whole part back.

Why: multiplying by powers of 10 and subtracting cancels the endless repeating tail.

Example: Write $0.3\overline{8}$ as a fraction.

$\frac{38 - 3}{90} = \frac{35}{90} = \frac{7}{18}$ .

Type 2: Sum or difference of recurring decimalscommon2 practice Q

How to spot it:

Two or three barred decimals added or subtracted, e.g. $0.\overline{6} + 0.\overline{7} + 0.\overline{8}$ .

Convert each decimal to a fraction (use a common denominator like 9, 90, 99).
Add or subtract the fractions.
Convert back to the form used in the options.

Why: recurring decimals cannot be added digit by digit because the carry keeps coming from the endless tail.

Example: Find $0.\overline{5} + 0.\overline{4}$ .

$\frac{5}{9} + \frac{4}{9} = \frac{9}{9} = 1$ . (Not 0.9!)

Type 3: Terminating or non-terminating decimal testcommon2 practice Q

How to spot it:

'Which of the following fractions will give a terminating decimal?' (or non-terminating).

\frac{p}{q} \text{ (lowest terms) terminates} \iff q = 2^m 5^n

Reduce every fraction to lowest terms.
Factorise the denominator.
Only 2s and 5s → terminating; any other prime → recurring.

Why: $10 = 2 \times 5$ , so only such denominators can be turned into a power of 10.

Example: Does $\frac{27}{120}$ terminate?

$\frac{27}{120} = \frac{9}{40}$ and $40 = 2^3 \times 5$ → yes, it terminates ( $0.225$ ).

Type 4: Comparing and ordering fractionsvery common3 practice Q

How to spot it:

'Which is the largest/smallest fraction?' or 'arrange in ascending order'.

\frac{a}{b} > \frac{c}{d} \iff ad > bc \ (b, d > 0)

Two at a time: cross-multiply.
Many fractions: convert to decimals (3 places) or use the complement from 1 when all are close to 1.
Check the order asked (ascending = smallest first).

Why: cross-multiplying is the same as putting both fractions over the common denominator $bd$ .

Example: Which is larger: $\frac{7}{9}$ or $\frac{11}{14}$ ?

$7 \times 14 = 98$ , $9 \times 11 = 99$ . $98 < 99$ → $\frac{11}{14}$ is larger.

Type 5: Fraction of a quantity (word problems)common2 practice Q

How to spot it:

'3/5 of a number exceeds 1/4 of it by 63', 'spends 1/3 on rent and 1/4 of the remaining on food …'.

\text{part} = \text{fraction} \times \text{whole}

Let the whole be $x$ .
Translate each sentence into a fraction of $x$ (fractions of the remaining are multiplied).
Equate the known part to its value and solve.

Why: every part is a fixed fraction of the whole, so one known part fixes the whole.

Example: $\frac{2}{3}$ of a number is 48. What is $\frac{3}{4}$ of the number?

Number $= 48 \times \frac{3}{2} = 72$ . $\frac{3}{4} \times 72 = 54$ .

Formulas

Pure recurring

0.\overline{ab} = \frac{ab}{99},\quad 0.\overline{abc} = \frac{abc}{999}

Mixed recurring

0.a\overline{bc} = \frac{abc - a}{990}

Terminating test

\frac{p}{q} \text{ (lowest terms) terminates} \iff q = 2^m \, 5^n

Cross-multiplication

\frac{a}{b} > \frac{c}{d} \iff ad > bc \quad (b, d > 0)

Shortcut tricks

⚡ 9s-and-0s rule

Numerator: whole block after the point minus the non-repeating part. Denominator: one 9 per repeating digit, then one 0 per non-repeating digit.

Example: Convert $0.2\overline{45}$ into a fraction.

$\frac{245 - 2}{990} = \frac{243}{990} = \frac{27}{110}$ .

⚡ Complement comparison

When fractions are close to 1, compare $1 - \text{fraction}$ : the smallest gap gives the largest fraction.

Example: Which is the largest: $\frac{11}{12}, \frac{13}{14}, \frac{17}{18}, \frac{19}{21}$ ?

Gaps: $\frac{1}{12}, \frac{1}{14}, \frac{1}{18}, \frac{2}{21}$ . Smallest gap $\frac{1}{18}$ ⇒ $\frac{17}{18}$ is largest.

⚡ Terminating check in one glance

Cancel first, then look only at the denominator's primes.

Example: Which of $\frac{7}{80}, \frac{9}{42}, \frac{11}{75}, \frac{13}{120}$ is a terminating decimal?

$80 = 2^4 \times 5$ ✓. $\frac{9}{42} = \frac{3}{14}$ (7) ✗, $75 = 3 \times 5^2$ ✗, $120 = 2^3 \times 3 \times 5$ ✗. Answer: $\frac{7}{80}$ .

Where students lose marks

Writing $0.2\overline{45}$ as $\frac{245}{999}$ — ignoring the non-repeating digit.
Testing the denominator for 2s and 5s before reducing the fraction.
Assuming the fraction with the larger numerator is larger.

Practice sets — 13 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 13 questions

Suggested time 10 min · wrong answers go to your mistake notebook automatically.