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