The smallest of the fractions `(3)/(4), (5)/(6), (7)/(12), (2)/(3)`

Options

A. `(2)/(3)`
B. `(3)/(4)
C. `(5)/(6)`
D. `(7)/(12)`

Answer

To find the smallest of the fractions \( \frac{3}{4}, \frac{5}{6}, \frac{7}{12}, \frac{2}{3} \), we can follow these steps:

### Step 1: Identify the fractions
We have the following fractions to compare:
- \( \frac{3}{4} \)
- \( \frac{5}{6} \)
- \( \frac{7}{12} \)
- \( \frac{2}{3} \)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are 4, 6, 12, and 3. We need to find the LCM of these numbers.

- The prime factorization of the denominators:
- \( 4 = 2^2 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 3 = 3^1 \)

The LCM is found by taking the highest power of each prime:
- For 2, the highest power is \( 2^2 \) (from 4 or 12).
- For 3, the highest power is \( 3^1 \) (from 6, 12, or 3).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of 12.

1. \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

2. \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]

3. \( \frac{7}{12} \):
\[ \frac{7}{12} = \frac{7 \times 1}{12 \times 1} = \frac{7}{12} \]

4. \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]

### Step 4: Compare the converted fractions
Now we have:
- \( \frac{9}{12} \)
- \( \frac{10}{12} \)
- \( \frac{7}{12} \)
- \( \frac{8}{12} \)

### Step 5: Identify the smallest fraction
Now we can compare the numerators since the denominators are the same:
- \( 9 \) (from \( \frac{9}{12} \))
- \( 10 \) (from \( \frac{10}{12} \))
- \( 7 \) (from \( \frac{7}{12} \))
- \( 8 \) (from \( \frac{8}{12} \))

The smallest numerator is \( 7 \), which corresponds to the fraction \( \frac{7}{12} \).

### Final Answer
The smallest fraction among \( \frac{3}{4}, \frac{5}{6}, \frac{7}{12}, \frac{2}{3} \) is:
\[ \frac{7}{12} \]

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