# SOLUTION: A princess is as old as the prince will be when the princess is twice the age that the prince was when the princess' age was half the sum of their present ages. How old are they?

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 Click here to see ALL problems on Age Word Problems Question 173075: A princess is as old as the prince will be when the princess is twice the age that the prince was when the princess' age was half the sum of their present ages. How old are they?Answer by Edwin McCravy(8908)   (Show Source): You can put this solution on YOUR website!A princess is as old as the prince will be when the princess is twice the age that the prince was when the princess' age was half the sum of their present ages. How old are they? ``` Let A = the present age of the princess. Let B = the present age of the prince. The problem talks about (1) a certain number of year in the future, and (2) a certain number of years in the past. Let x = the number of years in the future that is spoken of. Let y = the number of years in the past that is spoken of. A princess is (now) as old as the prince will be (in x years). Therefore A = B + x In x years, the princess will be twice the age that the prince was y years ago. Therefore, A+x = 2(B-y) And y years ago, the princess was half the sum of their present ages: We have 4 unknowns but only 3 equations. So we suspect that there may be more than one solution. Simplifying those A - B - x = 0 A - 2B + x + 2y = 0 A - B - 2y = 0 That give the matrix: [1 -1 -1 0 | 0] [1 -2 1 2 | 0] [1 -1 0 -2 | 0] which simplifies to rref form: [1 0 0 -8 | 0] [0 1 0 -6 | 0] [0 0 1 -2 | 0] A - 8y = 0 B - 6y = 0 x - 2y = 0 A = 8y B = 6y x = 2y (A, B, x) = (8y, 6y, 2y) The first solution that makes sense is when y=1 Then A=8, B=6, x=2, and y=1 The princess' present age is 8 The prince's present age is 6 The number of years in the future talked about is 2 The number of years in the past talked about is 1. Checking: The sum of their present ages is 14. Then half the sum of their present ages is 7. And indeed, the princess was 7 1 year ago. In x=2 years the Princess will be 10, and that is twice the age the Prince was y=1 year ago, for he was 5 then. The next answer is when y=2 (A, B, x) = (8y, 6y, 2y) The next solution that is when y=2 Then A=16, B=12, x=4, and y=2 The princess' present age is 16 The prince's present age is 12 The number of years in the future talked about is 4 The number of years in the past talked about is 2. Checking: The sum of their present ages is 28. Then half the sum of their present ages is 14. And indeed, the princess was 14 2 years ago. In x=4 years the Princess will be 20, and that is twice the age the Prince was y=2 years ago, for he was 10 then. There are many solutions, all found by substituting arbitrary values for y. Edwin```