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This problem trains for: SAT-I, AMC-8, GMAT.

When passing through the Great Riddle Gate, Atreyu has to rapidly find an answer to the riddle: "Xayide has plotted to usurp the throne of the Childlike Empress for one eighth of her life. If Bastian does not help her become empress, she will die ten years from now, plotting and scheming. By then, she would have spent one fourth of her life plotting and scheming. How long would Xayide have lived?"

(characters from Michael Ende's "The Neverending Story")

figure

Denote Xayide's lifespan by x.

Right now, Xayide is x-10 years old.

The time she has been plotting up to now is:

equation

The time she would have been plotting by the end of her life (10 years from now) is:

equation

The difference between the span of time she plotted in total and the span of time she plotted until the present day is 10 years:

equation

Solve for x to find her total lifespan:

equation

equation

equation