Question

Find two numbers such that the mea
proportional between them is 14 and the third
proportional to them is 112.​

Answer 1

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Let the required numbers be a and b.

Given:-

14 is the mean proportional between a and b

⠀⠀⠀⠀⠀⠀$⟹$ $\bold{a: 14 = 14: b}$

⠀⠀⠀⠀⠀⠀$⟹$$\bold{ ab = 196}$

⠀⠀⠀⠀⠀⠀$⟹$ a = $\dfrac{196}{b}$....(1)

Also given, third proportional to A and B

Also given, third proportional to A and B is 112.

⠀⠀⠀⠀⠀⠀$⟹$$\bold{ a: b = b: 112}$

⠀⠀⠀⠀⠀⠀$⟹$$\bold{b² = 112a....(2)}$

⠀⠀⠀⠀⠀⠀⠀Using (1) we have:

⠀⠀⠀⠀⠀⠀⠀⠀b² = 112 ×$\dfrac{196}{b}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\bold{b³ = (14)³ (2)³}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\bold{b = 28}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀from(1),

⠀⠀⠀⠀⠀⠀⠀⠀⠀a = $\dfrac{196}{28}$ = 7

$\boxed{\red{\tt{Thus,\: the\: two \:numbers \:are \:7 \:and \:28}}}$

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Answer 2

Solution :

Let's assume the required numbers be a and b.

Given, 14 is the mean proportional between a and b.

a: 14 = 14: b

ab = 196

a = 196/b .......... (1)

Also, given, third proportional to a and b is 112.

a: b = b: 112

b² = 112a ........... (2)

Using (1), we have:

b² = 112 x (196/b)

b³ = 14³ x 2³

b = 28

From (1),

a = 196/ 28 = 7

=> Therefore, the two numbers are 7 and 28.