If for two numbers the ratio of their H.M. to G.M. is 20:29, then the

1.	If for two numbers the ratio of their H.M. to G.M. is 20:29, then the numbers are in the ratio(a) 3 : 40 (b) 4 : 25 (c) 1 : 22 (d) 2 : 27
Answer» (b) 4 : 25 Let the two numbers be a and b. Given, \(\frac{\text{H.M}}{\text{G.M}}\) = \(\frac{20}{29}\) ⇒ \(\frac{\frac{2ab}{a+b}}{\sqrt{ab}}\) = \(\frac{20}{29}\) ⇒ \(\frac{2\sqrt{ab}}{a+b}\) = \(\frac{20}{29}\) ⇒ 58\(\sqrt{ab}\) = 20(a+b) ⇒ 20a - 58\(\sqrt{ab}\) + 20b = 0 ⇒ 20\(\frac{a}{b}-58\)\(\sqrt{\frac{a}{b}}\) + 20 = 0 (Dividing all terms by b) ⇒ 20x² – 58x + 20 = 0 (where x = \(\sqrt{\frac{a}{b}}\)) ⇒ 20\(x\)² – 50\(x\) – 8\(x\) + 20 = 0 ⇒ 10\(x\) (2\(x\) – 5) – 4 (2\(x\) – 5) = 0 ⇒ (10\(x\) – 4) (2\(x\) – 5) = 0 ⇒ \(x\) = \(\frac{2}{5}\) or \(\frac{5}{2}\) ⇒ \(\sqrt{\frac{a}{b}}\) = \(\frac{2}{5}\) or \(\frac{5}{2}\) ⇒ \({\frac{a}{b}}\) = \(\frac{4}{25}\) or \(\frac{25}{4}\) Thus from the given options, the two numbers are in the ratio 4 : 25.

If for two numbers the ratio of their H.M. to G.M. is 20:29, then the numbers are in the ratio(a) 3 : 40 (b) 4 : 25 (c) 1 : 22 (d) 2 : 27

Answer»

(b) 4 : 25

Let the two numbers be a and b.

Given, \(\frac{\text{H.M}}{\text{G.M}}\) = \(\frac{20}{29}\) ⇒ \(\frac{\frac{2ab}{a+b}}{\sqrt{ab}}\) = \(\frac{20}{29}\)

⇒ \(\frac{2\sqrt{ab}}{a+b}\) = \(\frac{20}{29}\) ⇒ 58\(\sqrt{ab}\) = 20(a+b)

⇒ 20a - 58\(\sqrt{ab}\) + 20b = 0

⇒ 20\(\frac{a}{b}-58\)\(\sqrt{\frac{a}{b}}\) + 20 = 0 (Dividing all terms by b)

⇒ 20x² – 58x + 20 = 0 (where x = \(\sqrt{\frac{a}{b}}\))

⇒ 20\(x\)² – 50\(x\) – 8\(x\) + 20 = 0

⇒ 10\(x\) (2\(x\) – 5) – 4 (2\(x\) – 5) = 0

⇒ (10\(x\) – 4) (2\(x\) – 5) = 0

⇒ \(x\) = \(\frac{2}{5}\) or \(\frac{5}{2}\)

⇒ \(\sqrt{\frac{a}{b}}\) = \(\frac{2}{5}\) or \(\frac{5}{2}\)

⇒ \({\frac{a}{b}}\) = \(\frac{4}{25}\) or \(\frac{25}{4}\)

Thus from the given options, the two numbers are in the ratio 4 : 25.