Question

The sum of two numbers is 25 and the geometric mean is 52% lower than twice their average . find the numbers
A) 17, 8
B) 10, 15
C) 16, 9
D) 12, 13​

Answer 1

Answer:

only 16,9 satisfies it.

Mark it the brainliest.

Answer 2

The two numbers are 16 and 9. (Option C is correct)

Given,

Sum of 2 numbers = 25

The geometric mean is 52% lower than twice their average

To Find,

The two numbers

Solution,

Let the two numbers be 'x' and 'y'

As we know,

Arithmetic Mean = $\frac{Sum of numbers}{Total numbers}$

Also, the average is the same as the arithmetic mean.

The total numbers are 2 and their sum is 25. Substituting this information in the equation we get -

Arithmetic Mean = $\frac{25}{2}$

The geometric mean of two numbers x and y would be $\sqrt{xy}$

According to the question, the geometric mean is 52% lower than twice their average.

Using this we get -

2 * (Arithmetic Mean) * (1 - $\frac{52}{100}$) = Geometric mean

2 * ($\frac{25}{2}$) * (1 - $\frac{52}{100}$) = $\sqrt{xy}$

25 * ( $\frac{100-52}{100}$) =  $\sqrt{xy}$

$\sqrt{xy}$ = 25 * ( $\frac{100-52}{100}$)

$\sqrt{xy}$ = 25 * ( $\frac{48}{100}$)

$\sqrt{xy}$ =  ($\frac{48}{4}$)

$\sqrt{xy}$ =  12

x*y = 12²

x*y = 144

Now, we have -

x + y = 25 → x = 25 - y

x*y = 144

y(25 - y) = 144

25y - y² = 144

y² - 25y + 144 = 0

Solving the quadratic equation by middle term splitting

y² - 25y + 144 = 0

y² - 16y - 9y + 144 = 0

y(y - 16) - 9(y - 16) = 0

(y - 16)(y - 9) = 0

y = 16 or 9

If y = 16,

x = 25 - y

x = 25 - 16

x = 9

If y = 9,

x = 25 - y

x = 25 - 9

x = 16

In both cases we get, the two numbers 16 and 9.

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