Which fraction is larger, two thirds or three fifths?

Two thirds is the larger of the pair. Comparing fractions with different denominators means putting them on a common footing, and the easiest common denominator here is 15, because 3 and 5 both divide into it. Two thirds becomes ten fifteenths and three fifths becomes nine fifteenths, so two thirds wins by one fifteenth. A quicker method is cross multiplication: multiply the top of each fraction by the bottom of the other, giving 2 times 5 equals 10 against 3 times 3 equals 9, and the bigger product sits above the bigger fraction. Converting to decimals also works, since two thirds is about 0.667 and three fifths is exactly 0.6. What you must not do is compare numerators or denominators on their own, because a bigger denominator means the whole is cut into more and therefore smaller pieces, which is why one fifth is less than one third despite five being larger than three.

The fraction that sits between them, made by adding the tops and bottoms to get five eighths, always falls between the two originals.