Question

The Arithmetic mean of two numbers is 20, while their Geometric mean is 12. Calculate the two numbers.​

Answer 1

Answer:

The two numbers are 4 and 36

Step-by-step explanation:

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Answer 2

SOLUTION

GIVEN

The Arithmetic mean of two numbers is 20, while their Geometric mean is 12

TO DETERMINE

The two numbers

EVALUATION

Let the required numbers are a and b

Now it is given that the Arithmetic mean of two numbers is 20

Thus we have

$\sf{ \dfrac{a + b}{2} = 20}$

$\sf{ \implies \: a + b = 40 \: \: \: - - - - (1)}$

Again the Geometric mean is 12

So we get

$\sf{ \: \sqrt{ab} = 12\: }$

$\sf{ \implies \: ab = 144\: \: \: - - - - (2)}$

Now we have

$\sf{ {(a - b)}^{2} = {(a + b)}^{2} - 4ab}$

$\sf{ \implies \: {(a - b)}^{2} = {(40)}^{2} - 4 \times 144}$

$\sf{ \implies \: {(a - b)}^{2} = 1600 - 576}$

$\sf{ \implies \: {(a - b)}^{2} = 1024 \: \: }$

$\sf{ \implies \: {(a - b)}^{} = 32 \: \: \: \: - - - - (2)}$

Adding Equation 1 and Equation 2 we get

2a = 72

⇒ a = 36

From Equation 1 we get

b = 40 - 36 = 4

FINAL ANSWER

Hence the required two numbers are

36 and 4

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