Happy Town had a circus performance during the school holidays. $20\%$ of the people were performers and the rest were audience. Female performers were twice the number of male performers. $60\%$ of the audience were male and there were $1200$ more male audience than female audience. After some males left the circus, $20\%$ of the remaining people in the circus were male. How many males left the circus?

Full solution

Q. Happy Town had a circus performance during the school holidays.

20\%

of the people were performers and the rest were audience. Female performers were twice the number of male performers.

60\%

of the audience were male and there were

1200

more male audience than female audience. After some males left the circus,

20\%

of the remaining people in the circus were male. How many males left the circus?

Denote Total People: Let's denote the total number of people in the circus as $T$ . According to the problem, $20\%$ of the people were performers. Therefore, the number of audience members is $80\%$ of $T$ .
Number of audience $= 0.8 \times T$
Calculate Male and Female Audience: We are given that $60\%$ of the audience were male and there were $1200$ more male audience than female audience. Let's denote the number of male audience as $M_a$ and the number of female audience as $F_a$ . We can set up the following equation: $\newline$ $M_a = 0.6 \times (\text{Number of audience})$ $\newline$ $F_a = M_a - 1200$ $\newline$ Substituting the expression for the number of audience from Step $1$ , we get: $\newline$ $M_a = 0.6 \times (0.8 \times T)$ $\newline$ $F_a = 0.6 \times (0.8 \times T) - 1200$
Equation for Male and Female Audience: Since $M_a$ is $1200$ more than $F_a$ , we can write: $\newline$ $0.6 \times (0.8 \times T) = (0.6 \times (0.8 \times T) - 1200) + 1200$ $\newline$ This simplifies to: $\newline$ $0.6 \times (0.8 \times T) = 0.6 \times (0.8 \times T)$ $\newline$ This is a tautology and does not help us find $T$ . We need to find another way to express the relationship between $M_a$ and $F_a$ .