Question 511352
Let 
Present age of Alvaro =a
Present age of Alyssa =s
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In 5 years means we have add 5 in their present age
Alvaro’s age in five years = a+5
Alyssa’s age in five years = s+5
Then the ratio of Alvaro’s age to Alyssa’s age will be 3:5
(a+5):( s+5) = 3:5
a+5 / s+5 = 3/5
5(a+5)=3(s+5)
5a+25=3s+15
5a=3s+15-25
5a=3s-10
a=(3s-10)/5.................(1)
In ten years, the ratio of Alvaro’s age to Alyssa’s age will be 2:3. Same procedure will be applied for 10 years
(a+10):( s+10) = 2:3
a+10 / s+10 = 2/3
3(a+10)= 2(s+10)
3a+30=2s+20
3a=2s+20-30
3a=2s-10
a=(2s-10)/3……………………(2)
Put the value of a from (1) to (2)
(3s-10)/5=(2s-10)/3
(3)(3s-10)=(5)(2s-10)
9s-30=10s-50
9s-10s=10s-50+30
-s=-20
Multiply by (-1) both sides
s=20
Put the value s in (1)
a=(3s-10)/5……………………..(1)
a=(3(20)-10)/5
a=(60-10)/5
a=50/5
a=10
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Present age of Alvaro =a=10 years old
Present age of Alyssa =s = 20 years old
Sum of their ages = 20+10 = 30
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Check
(a+5):( s+5) = 3:5
(10+5):(20+5)=3:5
15:25=3:5
3:5=3:5
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(a+10):( s+10) = 2:3
(10+10):(20+10) = 2:3
20:30=2:3
2:3=2:3