Question

Three positive numbers are in continued proportion. The middle number is 18. The first number is 4/9 times the third number. Find the third number .

Answer 1

Answer:

Required third number is 12.

Step-by-step explanation:

Let,

First number be a

Third number be 4a / 9, since third number is 4 / 9 of the first number.

From the properties of ratio and proportion : If a, b and c are in continued proportion, then

= > a / b = b / c

= > a x c = b x b

= > ac = b^2 , where b is the middle term & a and c are the remaining terms.

Here,

Middle terms is 18 , and the remaining terms are a and 4a / 9.

On the basis of the identity given above :

= > 18^2 = a x ( 4a / 9 )

= > 18^2 = a x a x 4 / 9

= > 18^2 = a^2 x ( 2 / 3 )^2

= > 18^2 ( 2a / 3 )^2

= > 18 = 2a / 3

= > 18 x 3 / 2 = a

= > 27 = a

Hence,

= > Third number is : 4 / 9 x 27

= > Third number is : 4 x 3

= > Third number is : 12

Hence the required third number is 12.

Answer 2

» Three positive numbers are in continued proportion.

• Let three positive integers be M, J, and N.

Means..

=> $\dfrac{M}{J}$ = $\dfrac{J}{N}$

Cross-multiply them

=> MN = J²

» Middle number = 18

» The first number is 4/9 times the third number.

• Let first number be M.

A.T.Q.

=> Third number = $\dfrac{4M}{9}$

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We have middle number (second number) 18 and the left numbers are M and $\dfrac{4M}{9}$

=> (18)² = M × $\dfrac{4M}{9}$

=> (18)² = $\dfrac{ {4M}^{2} }{9}$

=> 18 = $\sqrt{ \dfrac{ {4M}^{2} }{9} }$

=> 18 = $\dfrac{2M}{3}$

=> 18 × 3 = 2M

=> 2M = 54

=> M = 27

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» We have to find the third number.

So..

Third number is $\dfrac{4(27)}{9}$

=> $\dfrac{108}{9}$

=> 12

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$\textbf{Third number is 12}$

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