Question

1. a teacher and a student are decorating a bulletin board. the teacher decorates of the board, and the student decorates of the bulletin board. the student says they have decorated exactly of the bulletin board. which statement is true? a the student is correct because is equal to. b the student is correct because is equal to. c the student is incorrect because is less than. d the student is incorrect because is greater than.

Explanation:

Step 1: Convert \(\frac{1}{6}\) to twelfths

To add the fractions, we need a common denominator. The denominator of the teacher's fraction is 12, so we convert \(\frac{1}{6}\) to twelfths. Multiply numerator and denominator by 2: \(\frac{1\times2}{6\times2}=\frac{2}{12}\).

Step 2: Add the two fractions

Add the teacher's fraction \(\frac{5}{12}\) and the student's converted fraction \(\frac{2}{12}\): \(\frac{5}{12}+\frac{2}{12}=\frac{5 + 2}{12}=\frac{7}{12}\).

Step 3: Compare \(\frac{7}{12}\) with \(\frac{1}{2}\)

Convert \(\frac{1}{2}\) to twelfths: \(\frac{1}{2}=\frac{6}{12}\). Since \(\frac{7}{12}>\frac{6}{12}\), \(\frac{7}{12}>\frac{1}{2}\). So the student's claim that they decorated exactly \(\frac{1}{2}\) is incorrect, and the reason is that \(\frac{7}{12}\) is greater than \(\frac{1}{2}\), which matches option D.

Answer:

First, we need to find the total fraction of the bulletin board decorated by the teacher and the student. The teacher decorates \(\frac{5}{12}\) and the student decorates \(\frac{1}{6}\). We convert \(\frac{1}{6}\) to twelfths: \(\frac{1}{6}=\frac{1\times2}{6\times2}=\frac{2}{12}\). Then we add the two fractions: \(\frac{5}{12}+\frac{2}{12}=\frac{7}{12}\). Now we compare \(\frac{7}{12}\) with \(\frac{1}{2}\) (which is \(\frac{6}{12}\)). Since \(\frac{7}{12}>\frac{6}{12}\), the student is incorrect because \(\frac{7}{12}\) is greater than \(\frac{1}{2}\)? Wait, no, let's check the options again. Wait, maybe there's a typo in the original problem, but looking at the options, option D says "The student is incorrect because \(\frac{7}{12}\) is greater than \(\frac{1}{2}\)" (assuming the numerator is 7). Wait, but let's re-express: \(\frac{1}{2}=\frac{6}{12}\), and \(\frac{7}{12}>\frac{6}{12}\), so the total decorated is more than \(\frac{1}{2}\), so the student's claim that they decorated exactly \(\frac{1}{2}\) is wrong. But looking at the options, option D: "The student is incorrect because \(\frac{7}{12}\) is greater than \(\frac{1}{2}\)". So the correct answer is D. Wait, but let's check the options again. Wait, maybe the teacher's fraction was \(\frac{5}{12}\) and student's \(\frac{1}{6}\), total \(\frac{5 + 2}{12}=\frac{7}{12}\), which is more than \(\frac{6}{12}=\frac{1}{2}\). So the student is incorrect because \(\frac{7}{12}>\frac{1}{2}\), so option D.