Question

Find the third proportion to $5\frac{1}{2},16\frac{1}{2}$

Answer 1

The third proportion of two numbers a and b is defined to be that number c such that $a:b::b:c$

Firstly we will simplify the two numbers

$5\frac{1}{2}=\frac{(5\times 2)+1}{2}=\frac{11}{2}$

$16\frac{1}{2}=\frac{(16\times 2)+1}{2}=\frac{33}{2}$

Let the third proportion be $x$

Therefore third proportion of $\frac{11}{2},\frac{33}{2}$ will be given as

$\frac{11}{2}:\frac{33}{2}::\frac{33}{2}:x$

$\Rightarrow \frac{\frac{11}{2}}{\frac{33}{2}}=\frac{\frac{33}{2}}{x}$

On simplifying

$\Rightarrow \frac{11\times 2}{33\times 2}=\frac{33}{2\times x}$

$\Rightarrow \frac{1}{3}=\frac{33}{2\times x}$

Cross multiplying we get,

$\Rightarrow 2\times x=33\times 3$

On simplifying

$\Rightarrow x=\frac{99}{2}$