At a certain regatta the number of races on each day varies jointly as the number of days from the beginning and end of the regatta up to and including the day in question. On three successive days there were respectively $6, 5$ and $3$ races. Which days were these, and how long did the regatta last?

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Solution

Let the regetta lasted $N$ days and the days in question are $(x - 1)^{t h}, x^{t h},$ and $(x + 1)^{t h}$ . Then

Number of races on $(x - 1)^{t h}$ day $= (x - 1) (N - (x - 2)) = 6 k . . . (1)$

Number of races on $x^{t h}$ day $= x (N - (x - 1)) = 5 k . . . (2)$

Number of races on $(x - 1)^{t h}$ day $= (x + 1) (N - x) = 3 k . . . (3)$

Subtracting (2) from (1), we get $2 x - 2 - N = k . . . (4)$

Subtracting (3) from (2), we get $2 x - N = 2 k . . . (5)$

Subtracting (4) from (5), we get $k = 2$

From (5), we have $2 x - N = 4 ⟹ N = 2 x - 4$

Substituting the value of $k$ and $N$ in (2), we get

$x (x - 3) = 10 ⟹ x^{2} - 3 x - 10 = 0 ⟹ (x - 5) (x + 2) = 0$

$∴ x = 5$ and $N = 6$

Hence, the regatta lasted for 6 days and the 3 successive days were 4th, 5th and 6th

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At a certain regatta the number of races on each day varies jointly as the number of days from the beginning and end of the regatta up to and including the day in question. On three successive days there were respectively 6,5 and 3 races. Which days were these, and how long did the regatta last?