Oooh! That was an amazing event!

Although, both are same method, basically one can approach in $2$ ways.

Approach 1: Let the expected number of people be $x$ . As per the question, there was a increase of $25\%$ people. So,

$125\%\text{ of} \, x=800$ $\Rightarrow \frac{125}{100}\times x=800$ $\Rightarrow x=800\times\frac{100}{125}=\color{#0C6AC7} {\boxed {640}}$

Approach 2: There was a increase of $\frac x4$ people. So,

$x+\frac x4=800$ $\Rightarrow \frac{5x}{4}=800$ $\Rightarrow x=800\times\frac{4}{5}=\color{#0C6AC7} {\boxed {640}}$

Assume that the number of people he expected is x. $800 = \frac54 \times x$ $x = 640$

• $25\%$ increase $\implies$ $125\%x = 800$
• $\frac{5}{4} x =800$
• $\therefore x = \boxed{660}$

Let n be the expected number of the organizers. Based on the problem, we can generate an equation which is expected number + increase = 800 attendees.

Then, we can equate it algebraically as

n + (n x 1/4) = 800

factoring the common out, we have

n ( 1 + 0.25) = 800 since 1/4 = 0.25

n (1.25) = 800

1.25n = 800

n = 640 (by dividing both sides by the coefficient of n to have its coefficient only 1)