Q:

Products Problem with 3 variables

Hi need some help with the following Question please
I write down three positive numbers a, b and c.
The product of a and b is 2.
The product of b and c is 24.
The product of c and a is 3.
What is the sum of all three numbers?

A:

$$ab=2\implies b=\frac{2}{a}$$
$$bc=24\implies b=\frac{24}{c}$$
$$\frac2a=\frac{24}c\implies c=12a$$
$$ac=a(12a)=12a^2=3\implies a=\frac{1}{2}$$
$$c=12\left(\frac12\right)=6$$
$$b=\frac2{1/2}=4$$
$$\therefore a+b+c=\frac12+4+6=\frac{21}{2}$$

