Final answer:
Emma's statement is only partially correct. A positive fraction less than one divided by a positive number greater than one will result in a smaller positive fraction. However, if divided by a positive number (a fraction) less than one, the result could be greater than the original fraction.
Step-by-step explanation:
Emma claims that any positive fraction less than one divided by a positive number results in a quotient that is a positive fraction less than one. This statement can be both true and false, depending on the positive number we are dividing by. If the positive number is greater than one, then the quotient will indeed be a positive fraction less than the original fraction. If the denominator is 1, the fraction remains unchanged. However, if the positive number is itself a fraction less than one, then the quotient could be greater than the original fraction.
For example, let's take a positive fraction ½ and divide it by a positive number 2:
½ ÷ 2 = ¼ which is a positive fraction less than ½. In this case, Emma is correct. But if we divide ½ by ¼ (which is less than 1):
½ ÷ ¼ = 2, the result is not a fraction less than one.
To summarize, whether the quotient remains a positive fraction less than one depends on whether the divisor is greater or less than one.