The ratio of the price of gold in City A to City B is 5: 4. If the price in City A increases by 10% and the price in City B decreases by 5%, find the new ratio of their prices in simplest form.

Solution:
1. Let the original price of gold in City A be 5x and in City B be 4x.
2. The price in City A increases by 10%:
- New price in City A: 5x + 0.10 \cdot 5x = 5.5x
3. The price in City B decreases by 5%:
- New price in City B: 4x - 0.05 \cdot 4x = 3.8x
4. The new ratio of the prices in City A to City B is given by:
- \frac{5.5x}{3.8x} = \frac{5.5}{3.8}
5. Simplify the fraction:
- Multiply both numerator and denominator by 10 to get rid of decimals: \frac{55}{38}