Question
Find the third proportion to the following :
$(x - y)$ and $m (x - y)$
$(x - y)$ and $m (x - y)$
✓
Answer
Let z be lhe third proportion
$(x - y) : m (x - y) : : m (x - y) : z$
$\Rightarrow (x -y) x z - m(x - y) x m (x - y)$
$\Rightarrow (x - y) z = m^2 (x - y)^2$
$\Rightarrow z = m^2 (x - y)$
The third proportion is $m^2 (x - y) . $
$(x - y) : m (x - y) : : m (x - y) : z$
$\Rightarrow (x -y) x z - m(x - y) x m (x - y)$
$\Rightarrow (x - y) z = m^2 (x - y)^2$
$\Rightarrow z = m^2 (x - y)$
The third proportion is $m^2 (x - y) . $
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