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Let's solve this step by step:

### Let:
- \( x \) be the number of units representing the adult tickets sold **before** the show.
- \( y \) be the number of units representing the child tickets sold **before** the show.

#### Information given:
- The ratio of adult to child tickets **before** the show is \( 9:4 \), which means \( \frac{x}{y} = \frac{9}{4} \), or \( x = \frac{9}{4}y \).
- After selling 20 more adult tickets, the new ratio of adult to child tickets becomes \( 8:3 \), meaning \( \frac{x + 20}{y} = \frac{8}{3} \).

We now have two key equations:
1. \( x = \frac{9}{4}y \)
2. \( \frac{x + 20}{y} = \frac{8}{3} \)

#### Step 1: Solve the second equation for \( x \)
From the second equation:
\[
\frac{x + 20}{y} = \frac{8}{3}
\]
Multiply both sides by \( y \):
\[
x + 20 = \frac{8}{3}y
\]
Now, substitute \( x = \frac{9}{4}y \) into this equation:
\[
\frac{9}{4}y + 20 = \frac{8}{3}y
\]

#### Step 2: Eliminate the fractions
Multiply the entire equation by 12 (the least common denominator of 4 and 3) to eliminate the fractions:
\[
12 \times \left( \frac{9}{4}y \right) + 12 \times 20 = 12 \times \left( \frac{8}{3}y \right)
\]
Simplifying:
\[
27y + 240 = 32y
\]

#### Step 3: Solve for \( y \)
Now subtract \( 27y \) from both sides:
\[
240 = 5y
\]
Divide by 5:
\[
y = 48
\]

#### Step 4: Solve for \( x \)
Now that we know \( y = 48 \), substitute it into \( x = \frac{9}{4}y \):
\[
x = \frac{9}{4} \times 48 = 108
\]

Thus, the number of adult tickets sold **before** the show was **108**.

---

Would you like more details or have any questions about this solution?

Here are five related questions:
1. How would the solution change if 30 extra adult tickets were sold instead of 20?
2. What if the ratio of adult to child tickets after the sale was \( 7:3 \) instead of \( 8:3 \)?
3. Can you find the total number of child tickets sold before the show?
4. What would be the impact if child tickets were sold instead of adult tickets after the initial sales?
5. How does solving for ratios differ when solving for fractions?

**Tip:** Always check ratios and proportions by setting up equations based on given conditions.

The day before a show, a theatre had sold adult and child tickets in the ratio 9:4. On the day of the show, the theatre sold 20 more adult tickets and no more child tickets. The ratio of adult to child tickets sold became 8:3. Work out how many adult tickets had been sold the day before the show.

Solve a problem involving ticket sales where adult and child tickets are sold in the ratio 9:4, and after selling 20 more adult tickets, the new ratio becomes 8:3. Step-by-step solution provided for understanding ratios and proportions. Suitable for students in Grades 8-10.

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