Answer:
*name* is 10 years old now and brother is 15 years now
In 10 years they would be 20 and 25 respectively
Explanation:
Let x = *name*'s age now
let y = brother's age now
The first relationship is that the ratio of their ages now is 2:3 or, in fraction form 2/3
That means x/y = 2/3
Cross multiply => 3x = 2y =>
3x - 2y = 0 [1]
10 years from now x would become x + 10 years, y would become y + 10 years and the new ratio is 4/5
So (x + 10)/(y+10 = 4/5
Cross-multiply
5(x + 10) = 4(y + 10)
5x + 50 = 4y + 40
Subtract 4y from both sides
5x + 50 -4y = 40
and subtract 50 from both sides:
5x - 4y = 40 - 50
5x -4y = -10 [2]
Eliminate y terms by multiplying [1] by 2 and subtracting [1] from that
[1] x 2
=> 2(3x - 2y) = 2 x 0
=> 6x - 4y = 0 [3]
[3] - [2]
6x - 4y - (5x -4y) = 0 - (-10)
6x - 4y -5x + 4y = 10
x = 10
Using equation [1] we get
x/y = 2/3
=> 10/y = 2/3
10 x 3/2 = y
5 x 3 = y
y = 15
So their ages are 10 and 15 now.
In 10 years they would be 20 and 25
If the ratios change we would have a different set of equations to solve for
For example if the current ratio is 1/3
Then we would get x/y = 1/3
or 3x - y = 0 [4]
and if 10 years from now the ratio is 3/5 then
(x + 10)/(y + 10) = 3/5
=> 5(x+10) = 3(y + 10)
=> 5x + 50 = 3y + 30
=> 5x - 3y = -20 [5]
And we would solve these equations to get x and y in a similar mannter
FYI the answers would be
x = 5, y = 15