# Can you explain this question?

The value of 2.136363636... is:
a-47/220
b-68/495
c-2(3/22)
d-None

Note by Mehdi Balti
2 years, 11 months ago

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- 2 years, 11 months ago

let x=2.136363636...

10x=21.363636.... 1000x=2136.3636

1000x-10x=990x 990x=(2136.3636..-21.3636...) 990x=2115

x=2215/990 =47/22 =2(3/22)

- 2 years, 11 months ago

$$2.1\dot{3}\dot{6} = 2.1 + 0.0\dot{3}\dot{6} = 2.1 + \dfrac{36}{990} = \dfrac{2.1\times 990 + 36}{990} = \dfrac{2079+36}{990} = \dfrac{2115}{990} = \dfrac{47}{22} = 2\frac{5}{22}$$

- 2 years, 11 months ago

2(5/22)? I thought it's 2(3/22)!

- 2 years, 11 months ago

Thanks :)

- 2 years, 11 months ago

$$\begin{eqnarray}\begin{split}2.1363636\ldots=&\ 2+0.1+0.036+0.00036+0.0000036+\cdots\\=&\ 2+\dfrac1{10}+\left(\dfrac{36}{10^3}+\dfrac{36}{10^5}+\dfrac{36}{10^7}+\cdots\right)\\=&\ 2+\dfrac1{10}+\dfrac{\frac{36}{10^3}}{1-\frac1{10^2}}\\=&\ 2+\dfrac1{10}+\dfrac{\frac{36}{1000}}{\frac{99}{100}}\\=&\ 2+\dfrac1{10}+\dfrac2{55}\\=&\ 2+\dfrac{11+4}{110}\\=&\ 2+\dfrac3{22}=\boxed{2\dfrac3{22}}\end{split}\end{eqnarray}$$

- 2 years, 11 months ago

Thank you man :)

- 2 years, 11 months ago

c

- 2 years, 11 months ago

Thanks <3 you

- 2 years, 11 months ago