Question

the sum of two numbers is 45 and the mean proportional between them is 18 . find ​

Answer 1

Question :-

The sum of two numbers is 45 and the mean proportional between them is 18. The number is ...?

Answer:-

→ The required numbers are 36 and 9.

Step - by - step explanation :-

According to the question,

Sum of two numbers = 45,

Let the number are x and y ,

Then,

$\implies \: \bf{x + y = 45 }\\ \\ \implies \: \bf{ y = 45 - x} \: ........(1)$

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Given that mean proportion between them is 18.

Therefore,

$\bf{mean \: proportion \: = \sqrt{xy} = 18} \\ \\ \bf{ squaring \: on \: both \: sides \:} \\ \\ \implies \: \bf{xy = {(18)}^{2} } \\ \\ \implies \: \bf{xy = 324}$

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From equation (1)

$\implies \: \bf{ x(45 - x) = 324} \\ \\ \implies \small \: \bf{ {x}^{2} - 45x + 320 = 0} \\ \\ solve \: this \: quadratic \: equation \\ \\ \: \implies \: \bf{ {x}^{2} - 36x - 9x + 320 = 0} \\ \\ \implies \: \bf{ x(x - 36) - 9(x - 36) = 0} \\ \\ \implies \: \bf{(x - 9)(x - 36) = 0}$

Hence either ,

Case (1)

→ x - 9 = 0

→ x = 9

i.e,

Case (2)

→ x - 36 = 0

→ x = 36

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According to the question,

→ x + y = 45

When x = 9 ,

therefore,

→ 9 + y = 45

→ y = 36

When x = 36

therefore,

→ 36 + y = 45

→ y = 9

Hence numbers are → 36 and 9.

Answer 2

Answer:

The number are 36 & 9 (36,9)

Let number be x and y

Given

x+y=45

y=45−x ............ (eq. 1)

&

√(xy)=18

xy=18^2

xy = 324

xy = 324

substituting value of y from eq 1

x(45−x)=324

45x-x^2=324

x^2−45x+324=0

x^2−36x −9x+324=0

x(x−36) −9(x−36)=0

(x−36)(x −9)=0

(x−36)=0 or (x-9)=0

x= 36 or x= 9

when x= 36

then y= 45−x

45−36

y=9

When x= 9

then y=45−x

45−9

y=36

Numbers are 36 & 9