Question

The am of two numbers is 6.5 and their gm is 6.the two numbers are

Answer 1

We need to recall the following formulas.

For the two numbers $a$ and $b$

• $AM=\frac{a+b}{2}$
• $GM=\sqrt{ab}$

Given:

Arithmetic mean $=6.5$

Geometric mean $=6$

Let consider $\alpha$ and $\beta$ are the two numbers.

Then,

$\frac{\alpha +\beta }{2}=6.5$

$\alpha +\beta =13$                  .......$(1)$

And

$\sqrt{\alpha \beta } =6$

$\alpha\beta =36$

$\alpha =\frac{36}{\beta }$                         ........$(2)$

Substitute equation $(2)$ in $(1)$.

$\frac{36}{\beta } +\beta =13$

$\beta ^{2}+36=13\beta$

$\beta ^2-13\beta +36=0$

$(\beta -9)(\beta -4)=0$

⇒ $\beta =9$  or  $\beta =4$

⇒ $\alpha =4$  or  $\alpha =9$                     .....From eq. $(1)$

Hence, the numbers are $4$ and $9$.

Answer 2

Given - Arithmetic mean and geometric mean

Find - Numbers

Solution - Let the numbers be a and b.

Arithmetic mean = a + b/2

Geometric mean = ✓ab

Keep the values in formula

6 = ✓ab

Taking square

ab = 6²

Taking square

a = 36/b

Keep the value of a in arithmetic mean

$6.5 = \frac{\frac{36}{b} + b}{2}$

Shifting 2 to other side of equation and solving the numerator part.

$6.5 \times 2 = \frac{36 + {b}^{2} }{b}$

Shifting b to other side of equation.

$13b = 36 + {b}^{2}$

Solving the equation now

${b}^{2} - 13b + 36 = 0$

b² - 9b - 4b + 36 = 0

b(b - 9) - 4 (b - 9) = 0

(b - 9) (b - 4) = 0

b = 9, 4

If value of b = 9, then

Value of a = 36/b

Value of a = 36/9

Value of a = 4

If value of b = 4, then

Value of a = 36/4

Value of a = 9

So, the two numbers are 4 and 9.