Recurring Decimals
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Question 1 of 3
1. Question
Express as an infinite sum$$0.\dot{5}$$Write fractions as “a/b”- `S_∞=` (5/9)
Hint
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Excellent!
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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
2. Question
Express as an infinite sum$$0.4\dot{7}$$Write fractions as “a/b”- `S_∞=` (43/90)
Hint
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Incorrect
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
3. Question
Express as an infinite sum$$0.\dot{6}0\dot{3}$$Write fractions as “a/b”- `S_∞=` (67/111)
Hint
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Well Done!
Incorrect
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`