Question 1199409
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Yvonne is 4 years younger than Paul. 
Flip this around to say that Paul is 4 years older than Yvonne.


x = Yvonne's current age
x+4 = Paul's current age



The ratio of their current ages is 2:3
We can treat that ratio as a fraction and form the equation below so we can solve for x.


x/(x+4) = 2/3
3x = 2(x+4)
3x = 2x+8
3x-2x = 8
x = 8
Yvonne is currently 8 years old.
Paul is currently x+4 = 8+4 = 12 years old.


Note that 8/12 = 2/3 to fit the ratio 2:3 (of Yvonne's current age to Paul's current age).
It's the same idea as saying the ratio 8:12 reduces to 2:3.


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Let y be the number of years that elapse so that their future ages (Yvonne to Paul) form the ratio 4:5.


Fast forwarding that amount into the future yields:
8+y = Yvonne's future age
12+y = Paul's future age


Divide these items and set the result equal to 4/5.
Solve for y.


(8+y)/(12+y) = 4/5
5(8+y) = 4(12+y)
40+5y = 48+4y
5y-4y = 48-40
y = <font color=red>8</font>


We see that <font color=red>8 years</font> must elapse.


Yvonne will be 8+y = 8+8 = 16 years old.
Paul will be 12+y = 12+8 = 20 years old.
The ratio 16:20 fully reduces to 4:5 after dividing both parts by the GCF 4.
The answer is confirmed.


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Answer: <font color=red>8 years</font>
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