The two numbers whose mean proportion is 12 and the third proportiona…

The correct answer is Option (2) → 4 and 36

For two numbers \(a\) and \(b\), the mean proportional is \(\sqrt{ab}\), and the third proportional is the number \(c\) with \(a:b=b:c\), that is \(c=\frac{b^{2}}{a}\).

• The mean proportional is 12: \(ab=144\)        (1)

• The third proportional is 324: \(\displaystyle\frac{b^{2}}{a}=324\)        (2)

• Multiplying (1) and (2), \(a\) cancels: \(b^{3}=144\times324=46656\), so \(b=36\)

• From (1): \(a=\frac{144}{36}=4\)

Shortcut:

• The product of the two numbers must be \(12^{2}=144\)

• The products in the options are 192, 144, 72 and 192, so only 4 and 36 can be the pair