Can you explain this question?

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

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Mehdi Balti, if you are asking others to do your school assignments, please stop. It won't help you. Instead, we will help you to fail the exam.

Chew-Seong Cheong - 5 years, 11 months ago

Mrinmay Dhar - 5 years, 11 months ago

Thanks <3 you

Mehdi Balti - 5 years, 11 months ago

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

The answer is c.

Ikkyu San - 5 years, 11 months ago

Thank you man :)

Mehdi Balti - 5 years, 11 months ago

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

Chew-Seong Cheong - 5 years, 11 months ago

Thanks :)

Mehdi Balti - 5 years, 11 months ago

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

Raghav Arora - 5 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)

Raghav Arora - 5 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$