Question

Find the third proportion to $4\frac{1}{6}$, $5$.

Answer 1

Find the third proportion of the given terms:

Let the third proportion to $4\frac{1}{6}$ and $5$ be $x$.

If three quantities $a$, $b$ and $c$ are in continued proportion, that is, if $a:b::b:c$, then $b^2=ac$.

This is as same as when four quantities are in proportion, then the product of extremes is equal to the product of means.

When it is continued proportion, the product of means is basically the square of the mean/middle proportion.

So if $4\frac{1}{6}:5::5:x$, then $5^2=4\frac{1}{6}\times x$

$\Rightarrow$$5^2=\frac{25}{6}\times x$ [ The mixed fraction $4\frac{1}{6}$ can be converted to an improper fraction as $\frac{(6\times 4)+1}{6}$which is equal to $\frac{25}{6}$]

$\Rightarrow$$25=\frac{25}{6}\times x$

$\Rightarrow$ $x=\frac{25\times 6}{25}$

$\Rightarrow$ $x=6$

Hence, the third proportion to $4\frac{1}{6}$and $5$ is $6$.