the arithematic mean of two numbers is 15 and geometric meanis 12.Find the two numbers and find harmonic mean of these numbers.
To find the two numbers, let's call them "a" and "b".
We are given their arithmetic mean, which is equal to the sum of the numbers divided by 2. In this case, it is given that their arithmetic mean is 15. So, we have the equation:
(a + b) / 2 = 15
To find the geometric mean, we need to take the square root of the product of the two numbers. In this case, it is given that the geometric mean is 12. So, we have the equation:
√(a * b) = 12
To find the two numbers, we can solve these two equations simultaneously.
Let's start by solving the first equation for "b":
(a + b) / 2 = 15
a + b = 30
b = 30 - a
Now, substitute the value of "b" in the second equation:
√(a * (30 - a)) = 12
Now, square both sides of the equation to remove the square root:
a * (30 - a) = 12^2
a * (30 - a) = 144
30a - a^2 = 144
a^2 - 30a + 144 = 0
Now, solve this quadratic equation for "a". There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. Let's use factoring in this case.
(a - 12)(a - 12) = 0
a - 12 = 0
a = 12
Since we have found the value of "a" as 12, we can substitute it back into the equation for "b":
b = 30 - a
b = 30 - 12
b = 18
Therefore, the two numbers are 12 and 18.
Now, to find the harmonic mean of these numbers, we use the formula:
Harmonic Mean = 2 / ((1/a) + (1/b))
Substituting the values of "a" and "b":
Harmonic Mean = 2 / ((1/12) + (1/18))
Harmonic Mean = 2 / (3/36 + 2/36)
Harmonic Mean = 2 / (5/36)
Harmonic Mean = 2 * (36/5)
Harmonic Mean = 72/5
Harmonic Mean = 14.4
Therefore, the harmonic mean of 12 and 18 is 14.4.