Find two numbers such that their mean proportional is 6 and third proportional is 20.25.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4 and 9
For two numbers \(a\) and \(b\), the mean proportional is \(\sqrt{ab}\) (since \(a:x=x:b\)) and the third proportional is \(\frac{b^{2}}{a}\) (since \(a:b=b:x\)).
Mean proportional 6: \(ab=36\) (1)
Third proportional 20.25: \(b^{2}=20.25a\) (2)
Multiplying (1) and (2), \(a\) cancels: \(b^{3}=36\times20.25=729\), so \(b=9\)
From (1): \(a=36\div9=4\)
Shortcut: Test the options; only the pair whose product is 36 can have mean proportional 6.
4 and 9: \(4\times9=36\) and \(\frac{81}{4}=20.25\), both conditions hold
2 and 18 also give 36, but \(\frac{18^{2}}{2}=162\); the other two pairs do not give 36