Question

If the arithmetic mean of two numbers is 13/2 and geometric mean 6 what are the numbers?

Answer 1

Answer:

The numbers are 9 and 4.

Step-by-step explanation:

Let the two numbers be b and c.

The arithmetic means of any of the two given numbers is calculated using the following formula:

Arithmetic mean = $\frac{b + c}{2}$

The geometric mean of the given two numbers b and c is calculated as follows:

Geometric mean = $\sqrt{bc}$

Now we have an arithmetic mean to be $\frac{13}{2}$and geometric mean to be 6 in the given question therefore we have,

$\frac{b + c}{2} = \frac{13}{2}$  and $\sqrt{bc} = 6$

⇒b + c = 13 and b×c = $6^{2} = 36$

We know that,

$(b-c)^{2} = (b+c)^{2} - 4bc$

⇒$(b-c)^{2} = 13^{2} - 4 (36)$

⇒$(b-c)^{2} = 169 - 144$

⇒$(b-c)^{2} = 25$

⇒$(b-c) = 5$

Therefore we have b + c = 13 and b - c = 5

We solve both equations and get b = 9 and c = 4.