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 The royal sisters (Posted on 2018-04-09)
the Queen of Ruritania went shopping with her younger sister.
Each bought as many lengths of silk as she is years old and as many yards of each length as she bought lengths and paid for it as many francs per yard as she bought yards of each length.

The Queen spent 5803 francs more than her sister.
How old is the Queen?

Source : The weekend problems book; collected by Hubert Phillips UK 1932

 No Solution Yet Submitted by Ady TZIDON No Rating

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 Analytic solution | Comment 2 of 3 |
Queen age x, bought x lengths of, x yards at x per yard.  Cost = x^3
Likewise sister age y, cost = y^3.

x^3-y^3=5803
(x-y)(x^2+xy+y^2)=7*829
It seems reasonable that x-y=7 so

x^3 - (x-7)^3 = 5803
21x^2 - 147x + 343 = 5803
x^2 - 7x - 260 =0

which is quadratic with positive solution x=20.

Checking: 20^3=8000, 13^3=2197, 8000-2197=5803

Note: putting y=x-7 into (x^2+xy+y^2)=829 yields the same quadratic

 Posted by Jer on 2018-04-10 09:49:36

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