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Two Cubes

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Rationals Between

What fractions can you find between the square roots of 56 and 58?

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Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Mediant

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2


If you take two tests and get $a$ marks out of a maximum $b$ in the first, scoring ${a\over b}$, and you do better with $c$ marks out of $d$ in the second, scoring ${c\over d}$, you expect the result of combining the marks for the two tests, the mediant ${{a+c}\over {b+d}}$, to lie between the results for the two tests separately. If 0 < ${a\over b}$ < ${c\over d} $ prove, with reference to the diagram, that $${a\over b} < {{a+c}\over {b+d}} < {c\over d}.$$