To determine the smallest number of acts that both Brennan and Audrey must have written, we start by setting up equations based on the information provided.
Let \( n \) be the number of plays Brennan has written, each having 10 acts. Therefore, the total number of acts written by Brennan is:
\[ 10n \]
Let \( m \) be the number of plays Audrey has written, each having 5 acts. Therefore, the total number of acts written by Audrey is:
\[ 5m \]
According to the problem, these two totals are equal:
\[ 10n = 5m \]
To simplify the equation, we can divide both sides by 5:
\[ 2n = m \]
This equation demonstrates that the number of plays written by Audrey is twice the number of plays written by Brennan. We can express \( m \) in terms of \( n \):
\[ m = 2n \]
Now, we need to find the smallest values of \( n \) and \( m \) such that both \( n \) and \( m \) are whole numbers (since they represent counts of plays).
Choosing the smallest integer \( n = 1 \):
- If \( n = 1 \):
\[ m = 2 \times 1 = 2 \]
This means:
- Brennan has written \( n = 1 \) play, which corresponds to:
\[ 10n = 10 \times 1 = 10 \text{ acts} \]
- Audrey has written \( m = 2 \) plays, which corresponds to:
\[ 5m = 5 \times 2 = 10 \text{ acts} \]
Thus, the total number of acts written by both playwrights is equal: 10 acts.
Therefore, the smallest number of acts that each must have written is:
\[ \boxed{10} \]
1 answer