If h:k = 2:5, x:y=3:4 and 2h+x:k+2y = 1:2, find the ratio h-x: k-y.​

Given:

[tex]h:k=2:5,x:y=3:4,2h+x:k+2y=1:2[/tex]

To find:

[tex]h-x:k-y[/tex]

Solution:

We have, [tex]h:k=2:5[/tex] and [tex]x:y=3:4[/tex].

Let the values of h and k are 2a and 5a respectively.

Let the values of x and y are 3b and 4b respectively.

We have,

[tex]2h+x:k+2y=1:2[/tex]

It can be written as

[tex]\dfrac{2h+x}{k+2y}=\dfrac{1}{2}[/tex]

[tex]\dfrac{2(2a)+(3b)}{5a+2(4b)}=\dfrac{1}{2}[/tex]

[tex]\dfrac{4a+3b}{5a+8b}=\dfrac{1}{2}[/tex]

[tex]2(4a+3b)=1(5a+8b)[/tex]

On further simplification, we get

[tex]8a+6b=5a+8b[/tex]

[tex]8a-5a=8b-6b[/tex]

[tex]3a=2b[/tex]

[tex]a=\dfrac{2b}{3}[/tex]

Now,

[tex]h-x:k-y=\dfrac{h-x}{k-y}[/tex]

[tex]h-x:k-y=\dfrac{2-3b}{5a-4b}[/tex]

[tex]h-x:k-y=\dfrac{2\times \dfrac{2b}{3}-3b}{5\times \dfrac{2b}{3}-4b}[/tex]

[tex]h-x:k-y=\dfrac{\dfrac{4b}{3}-3b}{\dfrac{10b}{3}-4b}[/tex]

On further simplification, we get

[tex]h-x:k-y=\dfrac{\dfrac{4b-9b}{3}}{\dfrac{10b-12b}{3}}[/tex]

[tex]h-x:k-y=\dfrac{-5b}{-2b}[/tex]

[tex]h-x:k-y=\dfrac{5}{2}[/tex]

[tex]h-x:k-y=5:2[/tex]

Therefore, the ratio [tex]h-x:k-y[/tex] is [tex]5:2[/tex].

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