Sister's age?

The difference between the two data points is 4 years.

Sister's current age = s

Convert the data points into fractions with equivalent denominators.

$\large \boxed{4} = \frac{8(s+2)}{5} - 4(s-2) = \frac{8s+16}{5}-\frac{4s-8}{1} = \boxed{\frac{8s+16}{5}-\frac{20s-40}{5}}$

Multiply by 5, and simplify.

$\boxed{20} = (8s+16)-(20s-40) = \boxed{56-12s}$

Subtract 20, and add 12s.

$\boxed{12s}=\boxed{36}$

Divide by 12.

$\large\boxed{\color{#20A900}s=3}$

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We can verify $4 = \frac{8(s+2)}{5} - 4(s-2)$ by knowing that the brother aged 4 years.

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$4 = (b+2) - (b-2)$

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If

$\large \frac{b+2}{s+2} = \frac{8}{5}$

Then

$b+2 = \large\frac{8(s+2)}{5}$

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And if

$\large \frac{b-2}{s-2} = 4$

Then

$b-2 = 4(s-2)$

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So

$\large 4 = (b+2) - (b-2) = \frac{8(s+2)}{5} - 4(s-2)$

The brother's present age is x and sister's present age is y

the two equations are,

(x - 2 ) : ( y - 2) = 4 : 1

or, $\frac{x - 2}{y - 2}$ = $\frac{4}{1}$

or , x - 2 = 4y - 8

or , x - 4y = 8 + 2

or , x - 4y = -6.........................................(1)

Again,

( x + 2) : ( y + 2 ) = 8 : 5

or , $\frac{x + 2}{ y + 2}$ = $\frac{8}{5}$

or , 5x + 10 = 8y + 16

or , 5x - 8y = 16 - 10

or , 5x - 8y = 6........................................(2)

Now ,

x - 4y = -6.........................................(1)

5x - 8y = 6........................................(2)

Multiplying equation (1) with 2 .

 2x - 8y = - 12

 5x - 8y = 6

( - ) ( + ) ( - )

-3 = -18

or, x = $\frac{-18}{-3}$

x = 6

Substitute the value of x in equation (1)

6 - 4y = -6

or, -4y = -6 - 6

or, -4y = -12

y = $\frac{-12}{-4}$

y = 3

Therefore sister's present age is 3 years

suppose, two years ago, the brothers age was= x and sisters age was =y

so, the ratio was , $\frac {x}{y}$ = $\frac {4}{1}$

or, $x=4y$ ......................[1]

so, after 2 years from now, it will be, $\frac {x+4}{y+4}$ = $\frac {8}{5}$

or, $5x+20=8y+32$ ........................[cross multiplication]

or, $5x=8y+12$

or, $20y=8y+12$ ...........................[x=4y.........equation 1]

or. $12y=12$

or, $y=1$

so, the sisters current age is= $1+2=3$ years old

let $S$ be the present age of the sister and $B$ be the present age of the brother

$2$ years ago, the ratio of the brother's age to the sister's age

$\dfrac{B-2}{S-2}=\dfrac{4}{1}$

Simplifying further, we get

$B=4S-6$ $\color{#D61F06}(1)$

$2$ years later, the ratio of the brother's age to the sister's age

$\dfrac{B+2}{S+2}=\dfrac{8}{5}$

Simplifying further, we get

$5B=8S+6$ $\color{#D61F06}(2)$

Substitute $\color{#D61F06}(1)$ in $\color{#D61F06}(2)$ , we obtain

$5(4S-6)=8s+6$

$20S-30=8s+6$

$12S=36$

$\color{#3D99F6}\large\boxed{S=3}$ $\boxed{\text{answer}}$