If the Arithmetic mean between two numbers is 64 and the Geometric mean between them is 16. TheHarmonic Mean between them is ________.(a) 64(b) 4(c) 16(d) 4

To find the harmonic mean between two numbers, we need to use the formula:

Harmonic Mean = 2 / ((1/number1) + (1/number2))

Given that the arithmetic mean between the two numbers is 64, we can set up the equation:

(arithmetic mean) = (number1 + number2) / 2 64 = (number1 + number2) / 2

Simplifying the equation, we get: number1 + number2 = 128

Next, we are given that the geometric mean between the two numbers is 16. The formula for the geometric mean is:

Geometric Mean = sqrt(number1 * number2)

Setting up the equation using the given information, we have: 16 = sqrt(number1 * number2)

Squaring both sides of the equation, we get: 256 = number1 * number2

Now, we can solve the system of equations to find the values of number1 and number2.

From the equation number1 + number2 = 128, we can solve for number1: number1 = 128 - number2

Substituting this value into the equation 256 = number1 * number2, we have: 256 = (128 - number2) * number2

Expanding and rearranging the equation, we get: 256 = 128 * number2 - number2^2 0 = number2^2 - 128 * number2 + 256

This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring the equation, we have: 0 = (number2 - 64)^2

This means that number2 = 64.

Substituting this value back into the equation number1 + number2 = 128, we have: number1 + 64 = 128 number1 = 64

So, the two numbers are 64 and 64.

Now, we can calculate the harmonic mean using the formula:

Harmonic Mean = 2 / ((1/number1) + (1/number2)) Harmonic Mean = 2 / ((1/64) + (1/64)) Harmonic Mean = 2 / (2/64) Harmonic Mean = 2 / (1/32) Harmonic Mean = 2 * 32 Harmonic Mean = 64

Therefore, the harmonic mean between the two numbers is 64.

The correct answer is (a) 64.