Recurring Decimals and Geometric Series (HSC SSCE Mathematics Advanced): Revision Notes

Worked Example: Converting $0.\dot{2}\dot{7}$ to a Fraction

Let's convert $0.\dot{2}\dot{7}$ to a fraction using the geometric series method.

Step 1: Expand the decimal to see the pattern

$0.\dot{2}\dot{7} = 0.272727...$

Step 2: Write it as a sum

$= 0.27 + 0.0027 + 0.000027 + ...$

Step 3: Identify the first term and common ratio

This is an infinite geometric progression with:

• First term: $a = 0.27$
• Common ratio: $r = 0.01$

Note that each term is multiplied by $0.01$ to get the next term.

Step 4: Apply the limiting sum formula

$0.\dot{2}\dot{7} = \frac{a}{1-r}$

$= \frac{0.27}{1-0.01}$

$= \frac{0.27}{0.99}$

Step 5: Convert to a fraction and simplify

$= \frac{27}{99}$

$= \frac{3}{11}$

Therefore, $0.\dot{2}\dot{7} = \frac{3}{11}$.

Worked Example: Converting $2.6\dot{4}\dot{5}$ to a Fraction

Step 1: Expand the decimal

$2.6\dot{4}\dot{5} = 2.645454545...$

Notice that only the digits 45 repeat, while 2.6 does not recur.

Step 2: Separate the non-recurring and recurring parts

$= 2.6 + (0.045 + 0.00045 + 0.0000045 + ...)$

The non-recurring part ($2.6$) stays separate, and we add the infinite geometric series.

Step 3: Identify the geometric series parameters

For the recurring part:

• First term: $a = 0.045$
• Common ratio: $r = 0.01$

Step 4: Apply the limiting sum formula to the recurring part

$2.6\dot{4}\dot{5} = 2.6 + \frac{0.045}{1-0.01}$

$= 2.6 + \frac{0.045}{0.99}$

Step 5: Convert to fractions and combine

$= \frac{26}{10} + \frac{45}{990}$

Find a common denominator (110):

$= \frac{286}{110} + \frac{5}{110}$

$= \frac{291}{110}$

Therefore, $2.6\dot{4}\dot{5} = \frac{291}{110}$.