Recurring Decimals

Question 1 of 3

Express as an infinite sum

$0.\dot{5}$

Write fractions as “a/b”

$S_∞=$ (5/9)

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Limiting Sum Formula

$\color{#9a00c7}{S_∞}=\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\text(where) -1$

$<$

$r$

$<$

$1$

Common Ratio Formula

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

The recurring decimal can be written as a series

$0.5+0.05+0.005+0.0005+0.00005…$

First, solve for the value of $r$ .

$\color{#00880A}{r}$	$=$	$\frac{U_2}{U_1}$

	$=$	$\frac{0.05}{0.5}$	Substitute the first and second term

	$=$	$0.1$

Next, substitute the known values to the limiting sum formula

$\text(First term)$ $[a]$	$=$	$0.5$
$\text(Common Ratio)$ $[r]$	$=$	$0.1$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{0.5}}{1-\color{#00880A}{0.1}}$	Substitute known values

	$=$	$0.5/0.9$	Evaluate

	$=$	$5/9$	Multiply both value by $10$ to make them a whole number

$S_∞=5/9$

Question 2 of 3

Express as an infinite sum

$0.4\dot{7}$

Write fractions as “a/b”

$S_∞=$ (43/90)

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Limiting Sum Formula

$\color{#9a00c7}{S_∞}=\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\text(where) -1$

$<$

$r$

$<$

$1$

Common Ratio Formula

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

The recurring decimal can be written as a series

$0.4+0.07+0.007+0.0007+0.00007…$

First, solve for the value of $r$ .

Note that

$U_1$ will be the first recurring value, which is

$0.07$

$\color{#00880A}{r}$	$=$	$\frac{U_2}{U_1}$

	$=$	$\frac{0.007}{0.07}$	Substitute the first and second term

	$=$	$0.1$

Next, substitute the known values to the limiting sum formula

$\text(First term)$ $[a]$	$=$	$0.07$
$\text(Common Ratio)$ $[r]$	$=$	$0.1$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{0.07}}{1-\color{#00880A}{0.1}}$	Substitute known values

	$=$	$0.07/0.9$	Evaluate

	$=$	$7/90$	Multiply both value by $100$ to make them a whole number

Finally, add the value

$0.4$ to get the infinite sum value

$0.4+7/90$	$=$	$4/10+7/90$	$0.4=4/10$

	$=$	$(36+7)/90$	Apply the rule of adding fractions

	$=$	$(43)/(90)$	Evaluate

$S_∞=43/90$

Question 3 of 3

Express as an infinite sum

$0.\dot{6}0\dot{3}$

Write fractions as “a/b”

$S_∞=$ (67/111)

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Limiting Sum Formula

$\color{#9a00c7}{S_∞}=\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\text(where) -1$

$<$

$r$

$<$

$1$

Common Ratio Formula

$\color{#00880A}{r}=\frac{U_2}{U_1}=\frac{U_3}{U_2}$

The recurring decimal can be written as a series

$0.603+0.000603+0.000000603+0.000000000603…$

First, solve for the value of $r$ .

$\color{#00880A}{r}$	$=$	$\frac{U_2}{U_1}$

	$=$	$\frac{0.000603}{0.603}$	Substitute the first and second term

	$=$	$0.001$

Next, substitute the known values to the limiting sum formula

$\text(First term)$ $[a]$	$=$	$0.603$
$\text(Common Ratio)$ $[r]$	$=$	$0.001$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{a}}{1-\color{#00880A}{r}}$

$\color{#9a00c7}{S_∞}$	$=$	$\frac{\color{#e65021}{0.603}}{1-\color{#00880A}{0.001}}$	Substitute known values

	$=$	$0.603/0.999$	Evaluate

	$=$	$603/999$	Multiply both values by $1000$ to make them a whole number

	$=$	$67/111$	Simplify

$S_∞=67/111$

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Quiz summary

Information

1. Question

Hint

Excellent!

Limiting Sum Formula

Common Ratio Formula

2. Question

Hint

Nice Job!

Limiting Sum Formula

Common Ratio Formula

3. Question

Hint

Well Done!

Limiting Sum Formula

Common Ratio Formula

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