Number System

high importance~2 Q in Tier 135 formulas⚡ 18 shortcuts6 subtopics

Overview Learn (6)Formulas (35)Examples (5)Practice (97)

Fractions, decimals & recurring decimals

⏱ 3 min read🧩 5 question types🎯 13 practice Q

The idea in one minute

A fraction compares a part with the whole; a decimal writes the same idea in place value. Every terminating or repeating decimal is a fraction in disguise, and converting between the two forms follows two fixed rules.

Overview

A fraction like $\dfrac{3}{4}$ means three parts out of four equal parts. The top number is the numerator, the bottom is the denominator. Fractions, decimals and percentages are three languages for the same quantity: $\dfrac{3}{4} = 0.75 = 75\%$ .

Comparing fractions in one step

Two rules save long division. Cross-multiplication: $\dfrac{a}{b}$ versus $\dfrac{c}{d}$ , compare $ad$ with $bc$ ; the bigger product sits on the bigger fraction. Complement rule: if fractions sit close to $1$ , compare what is missing. $\dfrac{13}{17}$ misses $\dfrac{4}{17}$ , and $\dfrac{9}{11}$ misses $\dfrac{2}{11}$ ; since $\dfrac{4}{17} > \dfrac{2}{11}$ , the first fraction is smaller. Equal numerators: the smaller denominator wins, because the same cake is cut into fewer pieces.

Tip: Same numerator, smaller denominator: bigger fraction. Same denominator: bigger numerator wins.

Converting a fraction to a decimal

Divide, and watch the remainders. If a remainder repeats, the digits repeat too. When the denominator (in lowest terms) has only $2$ s and $5$ s, the division must end.

Rule: A fraction terminates only when its lowest-terms denominator equals $2^m \times 5^n$ . Otherwise it repeats forever.

$\dfrac{9}{50} = \dfrac{18}{100} = 0.18$ terminates. $\dfrac{8}{13}$ must repeat, because $13$ has a prime factor other than $2$ or $5$ .

Repeating decimal to fraction

Pure repeating: numerator is the repeating block, denominator is as many nines as the block length. $0.\overline{7} = \dfrac{7}{9}$ and $0.\overline{72} = \dfrac{72}{99} = \dfrac{8}{11}$ . Mixed, with non-repeating digits first: subtract. $0.4\overline{7} = \dfrac{47 - 4}{90} = \dfrac{43}{90}$ and $2.4\overline{7} = 2 + \dfrac{43}{90} = \dfrac{223}{90}$ .

Rule: All digits minus non-repeating digits, over as many nines as repeating digits followed by as many zeros as non-repeating digits.

Adding repeating decimals

Convert each to a fraction first, then add. $0.\overline{7} + 0.\overline{3} + 0.\overline{2} = \dfrac{7 + 3 + 2}{9} = \dfrac{12}{9} = \dfrac{4}{3}$ . With a common denominator of nine, the numerators simply add.

Watch: Never add repeating decimals digit by digit; the shift between them breaks place value.

Fraction of a quantity

"Two-fifths of a number" means multiply: $\dfrac{2}{5} \times N$ . When a fraction of a number is given, divide to recover the number. If $\dfrac{2}{5}$ of a number is $48$ , the number is $48 \times \dfrac{5}{2} = 120$ .

Example: $\dfrac{3}{5}$ of a number exceeds $\dfrac{1}{4}$ of it by $63$ . Then $\left(\dfrac{3}{5} - \dfrac{1}{4}\right)N = \dfrac{7}{20}N = 63$ , so $N = 180$ .

Continued fractions

A stack like $3 + \dfrac{1}{2 + \dfrac{1}{4}}$ unwinds from the bottom up: the innermost $\dfrac{1}{4}$ first, then the middle level, then add $3$ . Keep every level as a fraction; never convert to decimals half way.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Mixed repeating decimal to fraction

How to spot it:

A decimal with some non-repeating digits and then a repeating block must be written as a fraction.

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

Method

Count the digits that do not repeat ( $s$ ) and the digits that repeat ( $r$ ).
Numerator: all digits after the point, minus the non-repeating block.
Denominator: $r$ nines followed by $s$ zeros.
Reduce the fraction.

Why it works:

Multiplying by the denominator shifts one repeat cycle into place, and subtraction cancels the tail.

Try this

Convert 0.53 repeating on the 3 into a fraction.

Show solution

Non-repeating: $5$ (one digit). Repeating: $3$ (one digit).
Numerator: $53 - 5 = 48$ . Denominator: $90$ .
$\dfrac{48}{90} = \dfrac{8}{15}$ .

Answer

8/15

Type 2common2 practice Q

Sum of repeating decimals

How to spot it:

Two or more repeating decimals must be added, or their sum compared with a whole number.

0.\overline{a} + 0.\overline{b} = \dfrac{a + b}{9}

Method

Convert every decimal to a fraction.
Bring the fractions to a common denominator.
Add and reduce.

Why it works:

Fractions are exact; truncated decimals are not, so convert before adding.

Try this

Find the value of 0.7 repeating, plus 0.3 repeating, plus 0.2 repeating.

Show solution

$0.\overline{7} = \dfrac{7}{9}$ , $0.\overline{3} = \dfrac{3}{9}$ , $0.\overline{2} = \dfrac{2}{9}$ .
Sum: $\dfrac{7 + 3 + 2}{9} = \dfrac{12}{9}$ .
$= \dfrac{4}{3}$ .

Answer

4/3

Type 3common2 practice Q

Terminating or repeating test

How to spot it:

The question asks whether a given fraction terminates, or which of several fractions does.

q = 2^m 5^n \Rightarrow \text{terminates}

Method

Reduce the fraction to lowest terms.
Factorise the denominator.
If only $2$ s and $5$ s appear, it terminates; the decimal places equal the larger of $m$ and $n$ .

Why it works:

Dividing by 2s and 5s only ever shifts the decimal point; other primes leave a remainder cycle.

Try this

Does 27/150 terminate or repeat? If it terminates, write the decimal.

Show solution

Reduce: $\dfrac{27}{150} = \dfrac{9}{50}$ .
$50 = 2 \times 5^2$ , so it terminates.
$\dfrac{9}{50} = \dfrac{18}{100} = 0.18$ .

Answer

Terminating, 0.18

Type 4very common3 practice Q

Comparing fractions

How to spot it:

The question asks which fraction is greater, or to arrange fractions in order.

\dfrac{a}{b} > \dfrac{c}{d} \iff ad > bc

Method

Try the quick rules first: same numerator, or same denominator.
Otherwise cross-multiply and compare the products.
Near $1$ : compare the shortfalls from $1$ instead.

Why it works:

Cross-multiplication clears denominators in one step, with no division.

Try this

Which is greater: 8/13 or 13/21?

Show solution

Cross-multiply: $8 \times 21 = 168$ .
$13 \times 13 = 169$ .
$169 > 168$ , so $\dfrac{13}{21}$ is greater.

Answer

13/21

Type 5common2 practice Q

Fraction word problems

How to spot it:

A fraction of a number is given, or one fraction of a quantity exceeds another by a stated amount.

Method

Name the unknown number $N$ .
Translate each fraction phrase into multiplication by $N$ .
Build one equation from the given relation.
Solve for $N$ and answer what is asked.

Why it works:

'Two-fifths of' is exactly $\dfrac{2}{5} \times N$ , so sentences become equations line by line.

Try this

Two-fifths of a number is 48. What is three-quarters of the number?

Show solution

$\dfrac{2}{5} N = 48$ , so $N = 48 \times \dfrac{5}{2} = 120$ .
Three-quarters: $\dfrac{3}{4} \times 120$ .
$= 90$ .

Answer

Formula sheet

Pure repeating decimal

0.\overline{ab} = \dfrac{ab}{99}

Mixed repeating decimal

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

Terminating test

\dfrac{p}{q} \text{ terminates} \iff q = 2^m \times 5^n

q in lowest terms

Shortcuts that save time

⚡ Nines for pure repeats

One repeating digit gives 9 in the denominator, two give 99, three give 999.

Example

Write 0.\overline{45} as a fraction.

Show solution

Two repeating digits, so divide by $99$ .
$0.\overline{45} = \dfrac{45}{99}$ .
$= \dfrac{5}{11}$ .

Answer

5/11

⚡ Zeros after nines for mixed repeats

Denominator: one 9 per repeating digit, then one 0 per non-repeating digit. Numerator: all digits minus the non-repeating block.

Example

Convert 0.2\overline{45} to a fraction.

Show solution

Non-repeating digit: $2$ (one digit). Repeating block: $45$ (two digits).
Numerator: $245 - 2 = 243$ . Denominator: $990$ .
$\dfrac{243}{990} = \dfrac{27}{110}$ .

Answer

27/110

⚡ Compare what is missing

For fractions close to 1, compare the shortfalls instead of the fractions.

Example

Which is largest: 11/12, 13/14, 17/18, 19/21?

Show solution

Shortfalls: $\dfrac{1}{12}, \dfrac{1}{14}, \dfrac{1}{18}, \dfrac{2}{21}$ .
The smallest shortfall is $\dfrac{1}{18}$ .
So $\dfrac{17}{18}$ is the largest.

Answer

17/18

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

0.2\overline{45} = 245/999.

Mixed repeat subtracts: (245 - 2)/990 = 27/110.

Mistake 02

Cutting a repeating decimal short, 0.33 for 1/3, then adding.

Convert to an exact fraction first, then add.

Mistake 03

Testing termination on a reducible denominator, like 50 in 27/50 reduced wrongly.

Reduce the fraction first; only then check for 2s and 5s.

Mistake 04

Adding numerators and denominators: 1/2 + 1/3 = 2/5.

Use a common denominator: 1/2 + 1/3 = 5/6.

Quick revision

Read this the night before the exam.

Same numerator: smaller denominator means bigger fraction.
Cross-multiply to compare: $ad$ against $bc$ .
Terminates only when the lowest-terms denominator is $2^m 5^n$ .
Pure repeat: block over nines. Mixed: subtract, then nines and zeros.
Convert repeating decimals to fractions before adding.
'Of' means multiply; recover the number by dividing.

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

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

Continue learning