Find the missing number (triangle)

Find the missing number (triangle) Options: 9 8 7 10

Please write the logic also.

• Hello and welcome to PSE. Did you come up with this puzzle yourself or is it copied from elsewhere? If it's not your you should provide the source of the puzzle. – rhsquared Aug 26 '18 at 7:32
• Also are you sure your options for the answer are right? Did you get the options with the question? – nikki Aug 26 '18 at 8:01

$$?=10.$$



Explanation:

Suppose that

we have an equilateral triangle $\Delta \rm ABC$: Then,

the number in the middle of the triangle (say, $\rm M$) is created from the formula: $$\rm M = (I\times II) + III - 10.$$



Examples:

1. \begin{align} 33 &= (5\times 7) + 8 - 10 \\ &= 35 -2 \\ &= 33\;\color{green}{\checkmark} \tag1 \end{align}

1. \begin{align} 63 &= (7\times 9) + 10 - 10 \\ &= 63 + 0 \\ &= 63\;\color{green}{\checkmark}\tag2\end{align}

And last but not least...

1. \begin{align}132 &= (12\times 11)\:+\:? - 10 \\ &= 132\:+\:? - 10 \\ &= 122\:+\:? \\ &\Downarrow \\ ?&=132-122 \\ &= 10.\tag3\end{align}

• (+1) nice. As a challenge, seeing as you love mathjax perhaps you could try producing the triangle with it? ;) – TheSimpliFire Aug 26 '18 at 13:09
• @TheSimpliFire if you can actually do that, that would be pretty cool, because I have no idea, hahah :P – Mr Pie Aug 26 '18 at 13:23
• Found the $\LaTeX$ code: tex.stackexchange.com/questions/244376/drawing-a-triangle – TheSimpliFire Aug 26 '18 at 13:54
• Oh my... that is beautiful! But unfortunately, when I tried it, it does not work... Thus far, my best bet is the following: $$\LARGE\triangle$$ Since it contains 'document' commands, I am pretty sure this can only be done when writing an actual document (paper) using $\LaTeX$, but this theory is a little debunked because the user in the link can format it on the TeX site... perhaps Puzzling alone is culpable, because when I type it, there is an "unknown environment 'document'". – Mr Pie Aug 26 '18 at 21:50

Pattern:

The fractional part of $\frac{\text{number in triangle}}{\text{number left of triangle}}$ is less than $\frac12$.

Hence the answer is

$10$

as

$$\frac{132}{7}=18.85\cdots,\quad\frac{132}{8}=16.5,\quad\frac{132}{9}=14.66\cdots,\quad\frac{132}{10}=13.2$$

Examples:

$$\frac{33}8=4.125,\quad\frac{63}{10}=6.3$$

• Talking about $\LaTeX$, go here!! – Mr Pie Sep 1 '18 at 5:09
• @user477343 Nice! Feel free to check out my mathmatical fortnight challenge on my profile :) – TheSimpliFire Sep 1 '18 at 18:22
• Can that be found in your blog? Edit: Oh wait, nevermind. The $\LaTeX$ just needed to load, hahah :P – Mr Pie Sep 1 '18 at 18:24
• @user477343 THat'll have to wait until 16/09 (when it ends). But it's there on my SE profile – TheSimpliFire Sep 1 '18 at 18:24
• Hmm... let's get @Cleo :D – Mr Pie Sep 1 '18 at 18:25