the number 1.45450450450450450450450450...... how can be written as an ordinary fraction?
+26396 the number 1.45450450450450450450450450...... how can be written as an ordinary fraction ?
\\ \small{\text{ I. $ 1.45450450450450450450450450\dots = 1.4545 + 0.0000\overline{045} = \frac{14545} {10000} + 0.0000\overline{045} =\frac{2909}{2000} + 0.0000\overline{045} $ }}$\\\\$ \small{\text{ II. $ 0.0000\overline{045} = 045*10^{-7} + 045*10^{-10} + 045*10^{-13} + 045*10^{-16} + \dots $ }} $\\$ \small{\text{ $ 0.0000\overline{045} = (\underbrace{045*10^{-7}}_{=a}) + (045*10^{-7}) *10^{-3} + (045*10^{-7})*10^{-6} + (045*10^{-7})*10^{-9} + \dots $ \textcolor[rgb]{1,0,0}{ this is a geometric series. } }} $\\$ \small{\text{ The sum of the geometric series is $ s=\frac{a}{1-r}$ we have $a=045*10^{-7}$ and $ r = \frac{ (045*10^{-7}) *10^{-3} }{ 045*10^{-7} } = \frac{ (045*10^{-7})*10^{-6} }{ (045*10^{-7}) *10^{-3} } = ... = 10^{-3} $ }} $\\$ \small{\text{ $ \begin{array}{rcl} s &=& \frac{ 045*10^{-7} }{ 1- 10^{-3} } \\ \\ &=& \frac{10^3}{10^3}*\frac{ 045*10^{-7} }{ 1- 10^{-3} } \\\\ &=& \frac{45}{10^4(10^3-1)} \\\\ &=& \frac{45}{ 9990000 } \end{Array} $ }}$\\\\$ \small{\text{ III. $ 1.45450450450450450450450450\dots = 1.4545 + 0.0000\overline{045} = \frac{14545} {10000} + 0.0000\overline{045} =\frac{2909}{2000} + \frac{45}{ 9990000 } $ }}$\\\\$ \small{\text{ $ 1.45450450450450450450450450\dots = \frac{2909}{2000} + \frac{45}{ 9990000 } = \frac{2909}{2000} + \frac{1}{ 2220000 } = \frac{3229}{2220} $ }}
+130466 I seem to do this a little differently from everybody else on here
Forget the whole number part for a moment
Write the non-repeating part and the repeating part of the decimal = 45450
Subtract the non-repeating part from this....= [45450 - 45 ] = 45405
Put this number over a fraction consisting of 9's and 0's.....the number of leading 9's equaling the number of digits in the repeating part (3), and the number of trailing 0's equaling the number of digits in the non-repating part (2)..so we have
45405 / 99900 = 1009 / 2220
Add back the whole number part.....so we have
1 + 1009/2220 = [2220 + 1009] / 2220 = 3229/2220 ...verify for yourself that this is the same as what we started with...!!!
+26396 the number 1.45450450450450450450450450...... how can be written as an ordinary fraction ?
\\ \small{\text{ I. $ 1.45450450450450450450450450\dots = 1.4545 + 0.0000\overline{045} = \frac{14545} {10000} + 0.0000\overline{045} =\frac{2909}{2000} + 0.0000\overline{045} $ }}$\\\\$ \small{\text{ II. $ 0.0000\overline{045} = 045*10^{-7} + 045*10^{-10} + 045*10^{-13} + 045*10^{-16} + \dots $ }} $\\$ \small{\text{ $ 0.0000\overline{045} = (\underbrace{045*10^{-7}}_{=a}) + (045*10^{-7}) *10^{-3} + (045*10^{-7})*10^{-6} + (045*10^{-7})*10^{-9} + \dots $ \textcolor[rgb]{1,0,0}{ this is a geometric series. } }} $\\$ \small{\text{ The sum of the geometric series is $ s=\frac{a}{1-r}$ we have $a=045*10^{-7}$ and $ r = \frac{ (045*10^{-7}) *10^{-3} }{ 045*10^{-7} } = \frac{ (045*10^{-7})*10^{-6} }{ (045*10^{-7}) *10^{-3} } = ... = 10^{-3} $ }} $\\$ \small{\text{ $ \begin{array}{rcl} s &=& \frac{ 045*10^{-7} }{ 1- 10^{-3} } \\ \\ &=& \frac{10^3}{10^3}*\frac{ 045*10^{-7} }{ 1- 10^{-3} } \\\\ &=& \frac{45}{10^4(10^3-1)} \\\\ &=& \frac{45}{ 9990000 } \end{Array} $ }}$\\\\$ \small{\text{ III. $ 1.45450450450450450450450450\dots = 1.4545 + 0.0000\overline{045} = \frac{14545} {10000} + 0.0000\overline{045} =\frac{2909}{2000} + \frac{45}{ 9990000 } $ }}$\\\\$ \small{\text{ $ 1.45450450450450450450450450\dots = \frac{2909}{2000} + \frac{45}{ 9990000 } = \frac{2909}{2000} + \frac{1}{ 2220000 } = \frac{3229}{2220} $ }}
