Illustrated Commentary on Nine Chapters, Vol. 1

Now suppose we have 1/3 and 2/5. The question asks: what is the sum?

Now suppose we have 2/3, 4/7, and 5/9. The question asks: what is the sum?

Now suppose we have 1/2, 2/3, 3/4, and 4/5. The question asks: what is the sum?

Combining Fractions

Your servant, Chunfeng referring to Li Chunfeng, the Tang dynasty mathematician and others, respectfully note: The combining of fractions is not a simple matter. Denominators lack a fixed ratio, and the numerators are varied and mutually intertwined. Because the denominators are inconsistent, their coarseness and fineness differ; thus, logic cannot follow a single path. Therefore, one must unify the various fractions and equalize their group denominators so that the denominators can be combined. Hence, it is called "Combining Fractions."

The Method says: Cross-multiply the denominators with the numerators and add them to form the dividend the total sum of numerators after conversion; multiply the denominators together to form the divisor the new common denominator. Cross-multiply the denominators with the numerators and reduce them.