The harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G 2= 27, then the two numbers are
6 and 3
This problem involves finding two numbers given their harmonic mean and a specific relationship between their arithmetic mean (A) and geometric mean (G).
Understanding Mean Definitions
Let the two numbers be $x$ and $y$. The definitions for the means are:
- Arithmetic Mean (A): $A = \frac{x+y}{2}$
- Geometric Mean (G): $G = \sqrt{xy}$
- Harmonic Mean (HM): $HM = \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x+y}$
A fundamental relationship between these means is $G^2 = A \times HM$. Also, note that $G^2 = xy$ and $x+y = 2A$.
Using the Given Information
We are given:
- Harmonic Mean (HM) = 4
- The relationship: $2A + G^2 = 27$
Step-by-Step Calculation
- Relate G2, A, and HM: Using the relationship $G^2 = A \times HM$, we substitute the given HM = 4: $$G^2 = 4A$$
- Substitute into the Given Equation: Now substitute $G^2 = 4A$ into the provided equation $2A + G^2 = 27$: $$2A + (4A) = 27$$
- Solve for A: Combine the terms involving A: $$6A = 27$$ Divide by 6 to find A: $$A = \frac{27}{6} = \frac{9}{2} = 4.5$$
- Solve for G2: Use the relation $G^2 = 4A$: $$G^2 = 4 \times 4.5 = 18$$
- Find the Sum and Product of the Numbers: We know that $x+y = 2A$ and $xy = G^2$.
- Sum: $x+y = 2 \times 4.5 = 9$
- Product: $xy = 18$
- Find the Two Numbers: We need two numbers whose sum is 9 and product is 18. We can set up a quadratic equation where the roots are the two numbers ($x$ and $y$): $$t^2 - (\text{sum of roots})t + (\text{product of roots}) = 0$$ $$t^2 - (x+y)t + xy = 0$$ Substitute the values: $$t^2 - 9t + 18 = 0$$ Factor the quadratic equation: $$(t - 6)(t - 3) = 0$$ The roots are $t = 6$ and $t = 3$.
Conclusion
Therefore, the two numbers are 6 and 3.
Verification
Let's check if these numbers satisfy the given conditions:
- Numbers: 6 and 3
- Arithmetic Mean (A): $A = \frac{6+3}{2} = \frac{9}{2} = 4.5$
- Geometric Mean (G): $G = \sqrt{6 \times 3} = \sqrt{18}$
- Harmonic Mean (HM): $HM = \frac{2 \times 6 \times 3}{6+3} = \frac{36}{9} = 4$ (Matches the given HM)
- Check the relation $2A + G^2 = 27$: $2(4.5) + (\sqrt{18})^2 = 9 + 18 = 27$ (Matches the given relation)
Both conditions are satisfied.
Important Questions from Relations between AM, GM, HM
If the product of n positive numbers is unity, then their sum is?
If p = tan2 x + cot2 x, then which one of the following is correct?
If \(a_1, a_2, a_3,...,a_n\) are positive real numbers whose product is a fixed number C, then the minimum value of \(a_1+a_2+...+a_n\) is
In an acute angled ΔABC, the least value of sec A + sec B + sec C is:
Let x be the HM and y be the GM of two positive numbers m and n. If 5x = 4y, then which one of the following is correct?
