Four years ago, the ages of Ritu and Reena were in the ratio 5 : 6. Eight years from now, the ratio of their ages will be 8 : 9. What is the sum of their ages at present?

Let the present ages of Ritu and Reena be x and y. Four years ago their ratio was 5:6, giving 6(x - 4) = 5(y - 4) or 6x - 5y = 4. Eight years from now their ratio will be 8:9, giving 9(x + 8) = 8(y + 8) or 9x - 8y = -8; multiplying the first equation by 8 and the second by 5, then subtracting yields 3x = 72, so x = 24. Substituting x into the first equation gives 6(24) - 5y = 4, meaning 5y = 140 and y = 28, so the sum of their present ages is 52 years.

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