Nine Chapters on Mathematical Art, Vol. 1

When fractions are of the same type, there is no sense of distance; when they are of different types, there is no sense of nearness. Yet, through unification, we know that even if they occupy different positions, they can be made to follow one another. Though they may appear near yet differ in form, we know that even if they are in the same list, they may contradict each other. Thus, the method of balancing and unifying is essential. To master the measurement of numbers and to apply them is to reach the goal. It is like using an ivory pick to untie a knot; there is nowhere the principle does not reach. Multiply to distribute, reduce to collect, balance and unify to make them compatible; is this not the framework of arithmetic? As for the single method, one may let the denominator divide to find the rate multiply the numerator to balance. If the dividend is divided by the divisor and there is a remainder, name the remainder using the divisor. Now, seeking the dividend, we balance the numerators and unify the denominators. When the division by the denominator is complete, the remainder is reduced by the common factor to obtain the denominator (the divisor) and the remainder (the numerator); all follow this rule. The remainder is the numerator. If the denominators are the same, simply combine them.

There is 8/9 minus 1/5. Question: What is the remainder?

There is 3/4 minus 1/3. Question: What is the remainder?