$\LaTeX$ Storage Note

To store $\LaTeX$ codes that might be useful later

Strictly not for chatting and trying $\LaTeX$ codes :)

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.

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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.
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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

$\text{Important facts}$

$\text{1. The rpm of a gear is inversely proportional to its number of teeth} \\ \text{2. The gears connected at the top of one another move with same speed.}$

$\text{Visual explanation}$

<Picture Latex code here>

$\text{It is clear from the picture, that there is a/no problem in direction.} \\ \text{Hence, the system will/will not work.}$

$\text{Casework}$

$\text{Without loss of generality, let the leftmost gear in some color rotate at 2 rpm.}$

$\text{Up Setup}$

$\colorbox{silver}{\color{#333333}\text{Speed of Silver gear}} =\dfrac{50\times 2}{20}=5 \text{rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of brown gear}}=\dfrac{20\times 5}{10}=10 \text{ rpm}$

$\colorbox{silver}{\color{#333333}\text{Speed of Silver gear}} =\dfrac{50\times 2}{20}=5 \text{ rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of brown gear}}=\dfrac{20\times 5}{10}=10 \text{ rpm}$

$\colorbox{gold}{\color{#333333}\text{Speed of gold gear}} = \dfrac{20\times 5}{50}= 2\text{ rpm}$

$\colorbox{#815331}{\color{#FFFFFF}\text{Speed of Bronze gear}}=\dfrac{50\times 2}{25}=4 \text{ rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of inner brown gear}}=\dfrac{20\times 5}{25}=4 \text{ rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of brown gear}}=\dfrac{20\times 5}{25}=4 \text{ rpm}$

$\colorbox{#69047e}{\color{#FFFFFF}\text{Speed of Purple gear(A)}} =\dfrac{10\times 4}{20}=2\text{ rpm}$

$\text{Down Setup}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of brown gear}}=\dfrac{50\times 2}{10}=10 \text{ rpm}$

$\colorbox{silver}{\color{#333333}\text{Speed of silver gear}} = \dfrac{10\times 10}{20}= 5\text{ rpm}$

$\colorbox{gold}{\color{#333333}\text{Speed of gold gear}} = \dfrac{20\times 5}{50}= 2\text{ rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of inner brown gear}} \ \color{#333333}{=} \ \colorbox{gold}{\color{#333333}\text{Speed of gold gear}} = 2\text{ rpm}$

$\colorbox{gold}{\color{#333333}\text{Speed of gold gear}} =2\text{ rpm}$

$\colorbox{silver}{\color{#333333}\text{Speed of silver gear}} = \dfrac{20\times 2}{20}= 2\text{ rpm}$

$\colorbox{#624F41}{\color{#FFFFFF}\text{Speed of brown gear}}=\dfrac{20\times 2}{10}=4 \text{ rpm}$

$\colorbox{#3d99f6}{\color{#FFFFFF}\text{Speed of blue gear(B)}}=\dfrac{10\times 4}{20}=2 \text{ rpm}$

$\text{Answer}$

$\large{\color{#69047e}\text{Speed of Purple gear(A)} = \color{#3d99f6}\text{Speed of Blue gear(B)}=\color{#333333}2\text{ rpm}}$

Latex code for Gear Problems :)