Procedure

Evaluate a nested fraction from the inside out

Algebra • Expressions • Simplification

Statement

A nested fraction is evaluated outwards from the innermost level: at each level combine the whole-number part with the fraction over a common denominator, then invert the single fraction so formed, using

\frac{1}{\frac{p}{q}} = \frac{q}{p}

for non-zero

p

, and repeat until one fraction is left.

How to use it

1Never begin at the outside: the outer level is not a single number until the levels inside it have been collapsed.
2Keep everything as fractions rather than decimals, so that the inversions stay exact and the numbers stay small.
3Check the size as you go - each level's value should lie between its whole-number part and one more than it.

Worked example

Evaluate

3 + \frac{1}{1 + \frac{1}{4}}

Step-by-step

1
$1 + \frac{1}{4} = \frac{5}{4}$
Why: Collapse the innermost level first.
2
$\frac{1}{\frac{5}{4}} = \frac{4}{5}$
Why: Invert the single fraction just formed.
3
$3 + \frac{4}{5} = \frac{19}{5}$

Answer

\frac{19}{5}

Watch out

The levels must be taken in order:

3 + \frac{1}{1 + \frac{1}{4}}

is not

3 + \frac{1}{1} + 4

, and a mis-nesting changes the value completely. A level whose value is

0

cannot be inverted.

Also known as

continued fraction, nested fraction, complex fraction, fraction of fractions, reciprocal of a fraction

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (I)

2019 (II)

2019 (I)

2018 (II)

2017 (I)

2016 (II)