Question

If the ratio of H.M. and G.M. of two quantities is 12 : 13, then the ratio of the numbers is

• A. 1 : 2
• B. 2 : 3
• C. 3 : 4
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Let x and y be the numbers

∴ H.M.$= \frac { 2 x y } { x + y }$, G.M. $= \sqrt { x y }$

∴ $\frac { \text { H.M. } } { \text { G.M. } } = \frac { 2 \sqrt { x y } } { x + y } = \frac { 2 \sqrt { x / y } } { \frac { x } { y } + 1 }$ ⇒ $\frac { 12 } { 13 } = \frac { 2 r } { r ^ { 2 } + 1 }$ , $r = \sqrt { \frac { x } { y } }$)

⇒ $12 r ^ { 2 } - 26 r + 12 = 0$ ⇒ $6 r ^ { 2 } - 13 r + 6 = 0$

∴ $r = \frac { 13 \pm \sqrt { 13 ^ { 2 } - 4.6 .6 } } { 2 \times 6 } = \frac { 13 \pm 5 } { 12 } = \frac { 18 } { 12 } , \frac { 8 } { 12 } = \frac { 3 } { 2 } , \frac { 2 } { 3 }$

∴ Ratio of numbers $= \frac { x } { y } = r ^ { 2 } : 1 = \frac { 9 } { 4 } : 1$ or $\frac { 4 } { 9 } : 1 = 9 : 4$ or 4 : 9