Which rational number does not lie between 3/5 and 4/5?

Concept

To compare rational numbers, express them in equivalent forms with a common positive denominator. For numbers a < b, a value x lies between them only when a < x < b; multiplying numerator and denominator by the same positive number preserves a fraction and its order.

Application

1. The least common denominator of 5, 30, 3, and 10 is 30.

2. Rewrite the interval endpoints: 3/5 = 18/30 and 4/5 = 24/30. Therefore, a fraction lies between them only if its numerator over 30 is greater than 18 and less than 24.

3. Rewrite the offered values: 19/30 stays 19/30, 2/3 = 20/30, 7/10 = 21/30, and 16/30 stays 16/30.

4. The numerators 19, 20, and 21 are between 18 and 24, whereas 16 is below 18. Thus, 16/30 does not lie between 3/5 and 4/5.

Cross-check

In decimal form, 3/5 = 0.6 and 4/5 = 0.8. The value 16/30 is about 0.533, which is below 0.6, while the other offered values are greater than 0.6 and less than 0.8. Hence the required value is 16/30.