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In 2 years a boy will be 3/4 as old as his sister. 2 years ago he was 2/3 as old as she. How old are they?

1 Answer

Sep 10, 2015

The boy is $10$ $10$
and his sister is $14$ $14$

Explanation:

Define
$XXX b$ $color(white)("XXX")b$ : boy's current age
$XXX g$ $color(white)("XXX")g$ : his sister's current age

We are told
[1] $XXX b + 2 = \frac{3}{4} (g + 2)$ $color(white)("XXX")b+2 = 3/4(g+2)$
and
[2] $XXX b - 2 = \frac{2}{3} (g - 2)$ $color(white)("XXX")b-2= 2/3(g-2)$

[1] can be simplified into standard form as
[3] $XXX 4 b - 3 g = - 2$ $color(white)("XXX")4b-3g=-2$

[2] can be simplified into standard form as
[4] $XXX 3 b - 2 g = 2$ $color(white)("XXX")3b-2g=2$

Multiply [3] by 2 and [4] by 3
[5] $XXX 8 b - 6 g = - 4$ $color(white)("XXX")8b-6g=-4$
[6] $XXX 9 b - 6 g = 6$ $color(white)("XXX")9b-6g=6$

Subtract [5] from [6]
[7] $XXX b = 10$ $color(white)("XXX")b=10$

Substitute $10$ $10$ for $b$ $b$ in [3]
[8] $XXX 4 (10) - 3 g = - 2$ $color(white)("XXX")4(10)-3g=-2$

[9] $XXX 3 g = 42$ $color(white)("XXX")3g = 42$

[10] $XXX g = 14$ $color(white)("XXX")g=14$

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