Yes, half is bigger than one third.
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Three fourths is bigger than one half.
When it represents a fraction of something, rather than absolute numbers. 1/3 of a pound is bigger than 1/2 of an ounce, for example.
The answer is one half, think of it like this, take two apples cut one into three equal pieces (thirds) and the other into two (halves), half an apple is bigger than a third.
Three fifths is bigger than one half. Two fifths is smaller than one half.
Nope... one half is bigger than two fifths.