Question

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

Answer 1

Given:

`h:k=2:5,x:y=3:4,2h+x:k+2y=1:2`

To find:

`h-x:k-y`

Solution:

We have,
`h:k=2:5` and
`x:y=3:4`.

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,

`2h+x:k+2y=1:2`

It can be written as

`(2h+x)/(k+2y)=(1)/(2)`

`(2(2a)+(3b))/(5a+2(4b))=(1)/(2)`

`(4a+3b)/(5a+8b)=(1)/(2)`

`2(4a+3b)=1(5a+8b)`

On further simplification, we get

`8a+6b=5a+8b`

`8a-5a=8b-6b`

`3a=2b`

`a=(2b)/(3)`

Now,

`h-x:k-y=(h-x)/(k-y)`

`h-x:k-y=(2-3b)/(5a-4b)`

`h-x:k-y=(2* (2b)/(3)-3b)/(5* (2b)/(3)-4b)`

`h-x:k-y=((4b)/(3)-3b)/((10b)/(3)-4b)`

On further simplification, we get

`h-x:k-y=((4b-9b)/(3))/((10b-12b)/(3))`

`h-x:k-y=(-5b)/(-2b)`

`h-x:k-y=(5)/(2)`

`h-x:k-y=5:2`

Therefore, the ratio
`h-x:k-y` is
`5:2`.