Certainly! Let's evaluate each of the given arithmetic expressions step-by-step to match them with their simplified equivalents.
1. Expression: [tex]\(5 / 6 - 3 / 4\)[/tex]
- Step 1: Find a common denominator for the fractions [tex]\(5 / 6\)[/tex] and [tex]\(3 / 4\)[/tex]. The least common denominator (LCD) of 6 and 4 is 12.
- Step 2: Convert each fraction to have the LCD:
[tex]\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \][/tex]
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
- Step 3: Subtract the fractions:
[tex]\[ \frac{10}{12} - \frac{9}{12} = \frac{10 - 9}{12} = \frac{1}{12} \][/tex]
So, [tex]\(5 / 6 - 3 / 4 = \frac{1}{12}\)[/tex].
2. Expression: [tex]\(3 / 5 + 1 / 3\)[/tex]
- Step 1: Find a common denominator for the fractions [tex]\(3 / 5\)[/tex] and [tex]\(1 / 3\)[/tex]. The least common denominator (LCD) of 5 and 3 is 15.
- Step 2: Convert each fraction to have the LCD:
[tex]\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \][/tex]
[tex]\[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \][/tex]
- Step 3: Add the fractions:
[tex]\[ \frac{9}{15} + \frac{5}{15} = \frac{9 + 5}{15} = \frac{14}{15} \][/tex]
[tex]\[ \frac{14}{15} \approx 0.933... \][/tex]
So, [tex]\(3 / 5 + 1 / 3 \approx 0.933\)[/tex], which matches closely with [tex]\(\frac{14}{15}\)[/tex].
3. Expression: [tex]\(6 / 1 \times 2 / 3\)[/tex]
- Step 1: Simplify the expression by multiplying the fractions:
[tex]\[ \frac{6}{1} \times \frac{2}{3} = \frac{6 \times 2}{1 \times 3} = \frac{12}{3} = 4 \][/tex]
So, [tex]\(6 / 1 \times 2 / 3 = 4\)[/tex].
Thus, the correct matchings of the expressions with their simplified equivalents are:
[tex]\[ 5 / 6 - 3 / 4 = \frac{1}{12} \][/tex]
[tex]\[ 3 / 5 + 1 / 3 = 0.933 \][/tex]
[tex]\[ 6 / 1 \times 2 / 3 = 4 \][/tex]
