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Kevin is 2 times as old as Gabriela. 12 years ago, Kevin was 6 times as old as Gabriela. How old is Gabriela now?

Kevin is \(2\) times as old as Gabriela. \(12\) years ago, Kevin was \(6\) times as old as Gabriela.\newlineHow old is Gabriela now?

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Full solution

Q. Kevin is \(2\) times as old as Gabriela. \(12\) years ago, Kevin was \(6\) times as old as Gabriela.\newlineHow old is Gabriela now?
  1. Denoting Kevin and Gabriela's ages: Let's denote Kevin's current age as KK and Gabriela's current age as GG. According to the problem, Kevin is 22 times as old as Gabriela.\newlineFirst equation: K=2GK = 2G
  2. First equation: Kevin is 22 times as old as Gabriela: 1212 years ago, Kevin's age was K−12K - 12 and Gabriela's age was G−12G - 12. At that time, Kevin was 66 times as old as Gabriela.\newlineSecond equation: K−12=6(G−12)K - 12 = 6(G - 12)
  3. Second equation: 1212 years ago: Now we have a system of two equations:\newline11) K=2GK = 2G\newline22) K−12=6(G−12)K - 12 = 6(G - 12)\newlineWe can substitute the value of KK from the first equation into the second equation to find GG.
  4. System of two equations: Substituting K=2GK = 2G into the second equation:\newline2G−12=6(G−12)2G - 12 = 6(G - 12)\newlineExpanding the right side of the equation:\newline2G−12=6G−722G - 12 = 6G - 72
  5. Substituting K=2GK = 2G into the second equation: Now, let's solve for GG:2G−12−2G=6G−72−2G2G - 12 - 2G = 6G - 72 - 2G−12=4G−72-12 = 4G - 72Add 7272 to both sides:−12+72=4G−72+72-12 + 72 = 4G - 72 + 7260=4G60 = 4G
  6. Expanding the equation: Divide both sides by 44 to find GG:604=4G4\frac{60}{4} = \frac{4G}{4}15=G15 = G
  7. Solving for G: We found that Gabriela's current age is G=15G = 15 years old.

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