Question

find two number such that the mean proportion is 12 and the third proportion is 96

Answer 1

Let the two numbers be a and b.
Their mean proportion = 12
a:12=12:b
ab =12×12 => a=12×12÷b........(1)
[Keep 12×12 as it is, Don't multiply]

Their third proportion=96
a:b =b:96
b2= 96a........(2)
Substituting a =12×12÷b in equation (2),

${b}^{2} = 96 \times {12}^{2} \div b$
${b}^{3} = 96 \times {12}^{2}$
${b}^{3} = {2}^{3} \times {12}^{3}$
$b = 24$
and, a=12×12÷24
a=6

The two numbers are 6 and 24.

Hope it helps you!!!