Question

find two number their mean proportion is 24 and their third proportion is 1536.

Answer 1

Let the two numbers be a and b.
Their mean proportion = 24
=>. a:24 = 24:b
ab= 24×24 =>a=24×24÷b.......(1 )

[Keep 24×24 as it is, Don't multiply]

Their third proportion =1536
=> a:b = b: 1536
b2= 1536a........(2)

Substituting a= 24×24÷b in equation (2),
b2=1536×24×24÷b
b3=1536×24×24
b3=64×24×24×24
${b}^{3} = {4}^{3} \times {24}^{3}$
b=96.
and, a=24×24÷96
a=6.

The two numbers are 6 and 96.

Hope it helps you!!!