Some students planned a picnic.

Solution:

Let x students planned a picnic.

Then, the share of each student $=\frac{500}{x}$

According to question, 5 students fail to go picnic, then remaining students $=(x-5)$.

Therefore, new share of each student $=\frac{500}{x-5}$

It is given that

$\frac{500}{x-5}-\frac{500}{x}=5$

$\frac{500 x-500(x-5)}{(x-5) x}=5$

$\frac{2500}{(x-5) x}=5$

$5\left(x^{2}-5 x\right)=2500$

$\left(x^{2}-5 x\right)=500$

$x^{2}-5 x-500=0$

 

$x^{2}+20 x-25 x-500=0$

$x(x+20)-25(x+20)=0$

$(x+20)(x-25)=0$

Because x cannot be negative.

Thus, the total numbers of students attend a picnic

$=x-5$

$=25-5$

$=20$

Therefore, the total numbers of students attend a picnic be $x=20$