Question

Question

how would you solve this problem
the ratio of *name*'s age to his brother's is 2:3. In ten years, the ratio would be 4:5. How old is *name* and his brother now and in 10 years?
also how would you solve it if the ratio changed to something different, ex: 1:3 to 3:5 in 10 years

Answer 1

*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