To find which crown had a greater ratio of red gems to purple gems, we'll calculate the ratio for each crown.
King's Crown:
- Red gems = 5
- Purple gems = 8
The ratio of red gems to purple gems for the king's crown is: \[ \text{Ratio (King's Crown)} = \frac{5 \text{ red gems}}{8 \text{ purple gems}} = \frac{5}{8} \]
Prince's Crown:
- Red gems = 4
- Purple gems = 5
The ratio of red gems to purple gems for the prince's crown is: \[ \text{Ratio (Prince's Crown)} = \frac{4 \text{ red gems}}{5 \text{ purple gems}} = \frac{4}{5} \]
Now, we will compare the two fractions \(\frac{5}{8}\) and \(\frac{4}{5}\).
To compare the two fractions, we can find a common denominator. The least common multiple of 8 and 5 is 40.
Convert \(\frac{5}{8}\) to a fraction with a denominator of 40: \[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \]
Convert \(\frac{4}{5}\) to a fraction with a denominator of 40: \[ \frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40} \]
Now we can compare \(\frac{25}{40}\) and \(\frac{32}{40}\).
Since \(32 > 25\), we see that: \[ \frac{32}{40} > \frac{25}{40} \]
Thus, the prince's crown has a greater ratio of red gems to purple gems.
Therefore, the crown with the greater ratio of red gems to purple gems is the prince's crown.