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Say, I have a packet of milk and there is not much left in it. If I pour it in a cup, it might less than 1/4 filled.

Can I describe it like this

There is less than one fourth of a cup of milk in the packet.

Or in another context

The amount of juice in the box will be slightly more than three fourth of a cup.

Also, should I use a or the with cup? one fourth of a cup or one fourth of the cup?

1

Both of your statements are fine. (just change three fourth to three fourths)

You should use "a cup". cup in this context is a general reference to a measurement. It is the same for everyone.

He drove on for half a mile.

You could use the if you were referring to a specific cup, though.

There was enough juice to fill three fourths of the cup.

0

It's grammatically correct to say one fourth of the cup. We are using "the" to refer to a specific cup.

Similarly, it's OK to say three fourths, not fourth, of the cup. We can also use one quarter and three quarters instead.

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