Name: Decimals Word Problems And Worked Solutions
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Not all decimal numbers behave in the same way. Some end after a few digits; others go on forever. Understanding the different types of decimal number is essential for GCSE Maths, as it underlies the distinction between rational and irrational numbers and the method for converting recurring decimals to fractions.

Prerequisite: This article assumes you are familiar with place value and the decimal point. For a reminder, see What Are Decimal Numbers?.

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Overview: The Two Main Categories

Type	Description	Examples
Terminating	Finite digits after the decimal point	0.5, 3.14, 7.625
Non-terminating recurring	Infinite digits, but a block repeats	0.333..., 1.272727...
Non-terminating non-recurring	Infinite digits, no repeating pattern	pi = 3.14159..., sqrt(2) = 1.41421...

Terminating Decimals

A terminating decimal is a number that has a finite number of digits after the decimal point.
It comes to an end — it doesn’t go on forever.

Examples:

$\text{[math]}$

All of these stop after a certain number of digits.

Worked Example

Convert the fraction seven-eighths into a decimal.

We divide 7 by 8:

$\text{[math]}$

The decimal ends after three places, so it is terminating.

Converting a recurring decimal to a fraction

The algebraic method uses the fact that multiplying by a power of 10 shifts the decimal by the length of the repeating block.

A terminating decimal has a finite number of digits after the decimal point. It comes to a definite end.

Number	Note
0.7	1 decimal place
3.14	2 decimal places
0.125	3 decimal places
12.5000	Trailing zeros do not change the value — this is still terminating

Which fractions give terminating decimals?

A fraction in its lowest terms produces a terminating decimal if and only if the denominator has no prime factors other than 2 and 5.

Fraction	Denominator factored	Decimal
1/2	Denominator 2 = 2¹	0.5 — terminating
3/4	Denominator 4 = 2²	0.75 — terminating
7/8	Denominator 8 = 2³	0.875 — terminating
1/5	Denominator 5 = 5¹	0.2 — terminating
3/20	Denominator 20 = 2² × 5	0.15 — terminating
1/3	Denominator 3 (prime ≠ 2, 5)	0.333... — recurring
2/7	Denominator 7 (prime ≠ 2, 5)	0.285714... — recurring

Worked Example: Is 9/40 a terminating or recurring decimal?

40 = 2^3 x 5. The denominator has only factors of 2 and 5, so 9/40 is a terminating decimal. Divide: 9 / 40 = 0.225 $\text{[math]}$

Non-Terminating Recurring Decimals

A recurring decimal has an infinite number of digits after the decimal point, but a block of one or more digits repeats indefinitely in a fixed pattern. Because they can be expressed as fractions, recurring decimals are rational numbers.

A non-terminating decimal goes on forever — it has an infinite number of digits after the decimal point. A recurring decimal has an infinite number of digits after the decimal point, but a block of one or more digits repeats indefinitely in a fixed pattern. Because they can be expressed as fractions, recurring decimals are rational numbers.

Notation

A bar (vinculum) is placed over the repeating block:

Written out	Bar notation	What repeats
0.333...	0.3̅ (bar over 3)	digit 3 repeats
0.272727...	0.2̅7̅ (bar over 27)	block 27 repeats
1.416666...	1.41̅6̅ (bar over 6 only)	mixed: 41 is non-repeating, then 6 repeats

These are of two types:

Recurring (Repeating)
Non-Recurring (Non-Repeating)

Recurring Decimal Numbers

A recurring decimal has a set of digits that repeat indefinitely in a fixed pattern.

Examples:

$\text{[math]}$
$\text{[math]}$

$\text{[math]}$

The bar (overline) indicates the repeating digits.

Worked Example 1 — Pure Recurring Decimal

Convert $\text{[math]}$ into a fraction.

Let:

$\text{[math]}$

Multiply both sides by 10:

$\text{[math]}$

Subtract the first equation from this:

$\text{[math]}$

Worked Example 2 — Mixed Recurring Decimal

Convert $\text{[math]}$ into a fraction.

Let:

$\text{[math]}$

Multiply by 1000 (to move the decimal just before the repeating part):

$\text{[math]}$

Multiply by 10 (for one repeat shift):

$\text{[math]}$

Subtract:

$\text{[math]}$

Non-Recurring Decimal Numbers (Irrational Numbers)

A non-terminating, non-recurring decimal has an infinite number of digits after the decimal point, and those digits never settle into a repeating pattern. These numbers cannot be written as fractions and are called irrational numbers.

Number	Decimal expansion	Note
$\text{[math]}$	3.14159265358979...	The digits are known to trillions of places; no block repeats
$\text{[math]}$	1.41421356237...	Irrational — proved by contradiction
$\text{[math]}$	1.73205080757...	Irrational
e (Euler's number)	2.71828182845...	Irrational

Show that $\text{[math]}$ is non-terminating and non-recurring.

1.41421356237... The digits continue without end, and no repeating block can be identified. A formal proof that $\text{[math]}$ is irrational (and therefore non-recurring) uses proof by contradiction: assume $\text{[math]}$ = $\text{[math]}$ in lowest terms, then $\text{[math]}$ , which forces both p and q to be even — contradicting the assumption that p/q is in lowest terms.

Worked Examples

Score:

Convert 0.777... to a fraction.

Solution

Let x = 0.777... $\text{[math]}$ Subtract: $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Convert 0.363636... to a fraction.

Solution

The repeating block (36) has 2 digits, so multiply by 100. Let x = 0.363636... $\text{[math]}$ Subtract: $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Convert 0.5166... (mixed recurring: 0.51666...) to a fraction.

Solution

The repeating block (6) has 1 digit. The non-repeating part (51) has 2 digits. Let x = 0.5166... $\text{[math]}$ (shift past the non-repeating part) $\text{[math]}$ Subtract: $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Rounding Decimal Numbers

Sometimes, we need to round decimals to a specific place value (e.g., 1 decimal place, 2 decimal places).