For any given test, the traditional item relational structure theory can be used for detecting the item relational structure of the students. However, we do not know whether each item of the test is efficient or not, for improving above-mentioned drawback, we can first use the Q-matrix theory to obtain a validate test with all items which are efficient. In this paper, efficient items of the multiplications of fractions were constructed accordingly. Using Liu's before-test item structure theory, we can construct the efficient before-test item structural of the test. After testing the students, we can use the traditional item relational structure theory to construct the efficient after-test item structural of the students, and then, using Liu's criterion related validity index, we can evaluate the item relational structure of the test, and the results could be useful for cognitive diagnosis and remedial instruction.