The shaded part for 1 single triangle is $4/9$ths of the total area of the triangle. If this was considered to be 4 units, then the unshaded $5/9$ths would be 5 units. Thus the total area of the whole figure is 14 units and so $4/14$ths or $2/7$ths are shaded.

I believe that's the right answer, but can someone tell me is there a more efficient or a method that utilises geometry?


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    $\begingroup$ Your answer took less than three lines, I'm not sure how you expect something more efficient than that! :) $\endgroup$
    – David
    Jan 22, 2018 at 23:40
  • $\begingroup$ @David i mean like, is there a way to do it by looking at the figure or using proportions? I feel like i'm missing something. $\endgroup$
    – salman
    Jan 22, 2018 at 23:42
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    $\begingroup$ Your proof is great, I don't think you're missing anything. Also I suspect your proof is exactly what the question-setter intended. $\endgroup$
    – David
    Jan 22, 2018 at 23:44
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    $\begingroup$ I don't think there will be a more efficient or elegant solution using geometry, since the geometry of the problem is irrelevant. It would work the same for a circle, square, or any other shape you care to name. $\endgroup$
    – John Doe
    Jan 22, 2018 at 23:47
  • $\begingroup$ @David well rather embarrassingly, I came across this in a 10-11 year old's school entrance exam I am helping my younger sibling with. So I feel like there is some assumed knowledge that I'm omitting. But if this is the correct rubric then that's great, thanks! $\endgroup$
    – salman
    Jan 22, 2018 at 23:54

2 Answers 2


Here’s a quick visual way of thinking that I would expect a 10- or 11-year-old to be capable of.

No arithmetic beyond counting nor a single word nor any spare space is necessary. (I presume expect that students are permitted to draw on or annotate their examination papers.)

(Each dot represents what we would call a unit of area.)

  • $\begingroup$ @salman Within each triangle, $4$ dots of $9$ (i.e., $4/9$ of the area of a single triangle) fall in the shaded region. When considering both triangles, $4$ of $14$ dogs (i.e., $2/7$ of the area of the total figure) fall in the shaded region. $\endgroup$ Jan 23, 2018 at 1:37

Do not worry. This is without doubt the easiest and fastest approach to solve this question.

So I feel like there is some assumed knowledge that I'm omitting

Mathematics is full of known results and equations, but you can't have known results for literally everything! :)

  • $\begingroup$ I dispute that the method the OP describes is “without a doubt the easiest and fastest.” See my answer ;) $\endgroup$ Jan 23, 2018 at 1:27
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    $\begingroup$ @ChaseRyanTaylor I only summed up what the commentors said in a community wiki, though I find your approach Interesting! :) $\endgroup$ Jan 23, 2018 at 1:29

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