Question

If X,y,z are in continued proportion
then show that.
X-y upon x-z is equal to y upon y+z

lesson ratio n proportion
class 9
state board ​

Answer 1

Solution :

x, y and z are in continued proportion .

x : y :: y : z

> xz = y² .

To show :

( x - y )/( x - z ) = y/( y + z)

> ( x - y)( y + z ) = y( x - z)

> xy + xz - y² - yz = xy - yz

xy and yz gets cancelled .

> xy - y² = 0

> y² = xz

> x : y :: y : z

Thus , they are in continued proportion.

Hence Shown .

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Additional Information :

(a + b)² = a² + 2ab + b²

(a + b)² = (a - b)² + 4ab

(a - b)² = a² - 2ab + b²

(a - b)² = (a + b)² - 4ab

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

2 (a² + b²) = (a + b)² + (a - b)²

4ab = (a + b)² - (a - b)²

ab = {(a + b)/2}² - {(a-b)/2}²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a + b)³ = a³ + 3a²b + 3ab² b³

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)( a² - ab + b² )

a³ + b³ = (a + b)³ - 3ab( a + b)

a³ - b³ = (a - b)( a² + ab + b²)

a³ - b³ = (a - b)³ + 3ab ( a - b )

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Answer 2

Question :-

If X, Y , Z are in continued proportion , then show that -

• $\sf \frac{x - y}{x - z} = \frac{y}{y + z}$

Given :-

X, Y, z are in continued proportion , it means that -

• $\sf \: x \ratio \: y \ratio \ratio \: y \ratio \:z$

SolutioN :-

$\sf \: x \ratio \: y \ratio\ratio y\ratio \: z \\ \\ \sf \: we \: can \: write \: it \: as \mapsto\\ \\ \sf \implies \: \frac{x}{y} = \frac{y}{z} \\ \\ \sf \implies \: x \: \times \: z = y \times y \\ \\ \sf \implies \: xz = {y}^{2} \\ \\ \sf \implies \: {y}^{2} = xz - - - - (i)$

Now,

$\sf \frac{x - y}{x - z} = \frac{y}{y + z} \\ \\ \implies \sf (x - y) \times (y + z) = y(x - z) \\ \\ \implies \sf \: xy + xz - {y}^{2} - yz = xy - yz \\ \\ \implies \sf \: \cancel{ xy} - \cancel{xy} + xz - \cancel{ yz }- {y}^{2} + \cancel{ yz} = 0 \\ \\ \implies \sf \: xz - {y}^{2} = 0\\ \\ \implies \sf \: {y}^{2} = xz \\ \\ \sf from \: equation \: (i) \mapsto \\ \\ \implies \sf \: xz = xz \\ \\ \implies \boxed{ \bold{lhs = rhs}}$

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Hope it will help you :)