Final answer:
The ratio of Joy's cupcakes to Lizzy's cupcakes is 67:89, which is already in its simplest form because 67 and 89 are prime numbers with no common factors other than 1.
Step-by-step explanation:
If Joy has 67 cupcakes and Lizzy has 89 cupcakes, to find the ratio of the two numbers, we simply compare the number of cupcakes each person has. A ratio can be written in several forms including as a fraction, with a colon, or with the word 'to'. In this case, the ratio of Joy's cupcakes to Lizzy's cupcakes can be expressed as 67:89, 67/89, or 67 to 89.
While we could simplify this ratio if both numbers shared a common factor, since 67 and 89 are both prime numbers and do not have any common factors other than 1, the ratio is already in its simplest form. Hence, the ratio of Joy's cupcakes to Lizzy's cupcakes is 67:89.