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In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

##### Age 14 to 16 Challenge Level:

Kyle got 75% on section A of his maths exam, but he only got 35% on section B.
He says this means he should score 55% overall, but his teacher said he only scored 50% overall! How can this be?

Once you have had a chance to think about it, click below to see what he scored on each section.

In section A, there were 12 questions and Kyle got 9 correct.
In section B, there were 20 questions and Kyle got 7 correct.
Overall, there were 32 questions and Kyle got 16 correct.

As section A had fewer questions than section B, the score in section B is given more weighting.
The difference in overall scores is because Kyle worked out the average of his two scores, but his teacher worked out the mediant
Given two fractions $\frac{a}{c}$ and $\frac{b}{d}$, the mediant is defined as $\frac{a+b}{c+d}$.
Do you agree that Kyle only deserved 50%?

Here is an interactive diagram for you to explore the properties of the mediant of two fractions.

Can you find some values of $a, b, c$ and $d$ so that Kyle and his teacher both get the same value for the overall score (that is, the average and the mediant are the same)?

The pass mark for an exam is 50%. If I scored 25% on the first $n$ questions, under what circumstances can I still pass the exam?

Is it true that the mediant always lies between the two fractions? How do you know?