Which of the following fractions is greatest `(7)/(12), (11)/(16), (12)/(17), (13)/(18), (31)/(36)` ?

Options

A. `(13)/(18)`
B. `(12)/(17)`
C. `(31)/(36)`
D. `(7)/(12)`

Answer

To determine which of the given fractions is the greatest among \( \frac{7}{12}, \frac{11}{16}, \frac{12}{17}, \frac{13}{18}, \frac{31}{36} \), we can follow these steps:

### Step 1: Find a Common Denominator
To compare the fractions easily, we can find a common denominator. The least common multiple (LCM) of the denominators (12, 16, 17, 18, 36) will help us convert all fractions to have the same denominator.

### Step 2: Calculate the LCM
The LCM of the denominators can be calculated as follows:
- The prime factorization of each number:
- \( 12 = 2^2 \times 3 \)
- \( 16 = 2^4 \)
- \( 17 = 17^1 \)
- \( 18 = 2 \times 3^2 \)
- \( 36 = 2^2 \times 3^2 \)

Taking the highest power of each prime:
- \( 2^4 \) from 16
- \( 3^2 \) from 18 and 36
- \( 17^1 \) from 17

Thus, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 17 = 16 \times 9 \times 17 = 2448 \]

### Step 3: Convert Each Fraction
Now, we convert each fraction to have the common denominator of 2448:
1. \( \frac{7}{12} = \frac{7 \times 204}{12 \times 204} = \frac{1428}{2448} \)
2. \( \frac{11}{16} = \frac{11 \times 153}{16 \times 153} = \frac{1683}{2448} \)
3. \( \frac{12}{17} = \frac{12 \times 144}{17 \times 144} = \frac{1728}{2448} \)
4. \( \frac{13}{18} = \frac{13 \times 136}{18 \times 136} = \frac{1768}{2448} \)
5. \( \frac{31}{36} = \frac{31 \times 68}{36 \times 68} = \frac{2108}{2448} \)

### Step 4: Compare the Numerators
Now we compare the numerators:
- \( 1428 \) (for \( \frac{7}{12} \))
- \( 1683 \) (for \( \frac{11}{16} \))
- \( 1728 \) (for \( \frac{12}{17} \))
- \( 1768 \) (for \( \frac{13}{18} \))
- \( 2108 \) (for \( \frac{31}{36} \))

### Step 5: Identify the Greatest Fraction
From the numerators, we see that \( 2108 \) is the largest. Therefore, the greatest fraction is:
\[ \frac{31}{36} \]

### Final Answer
The greatest fraction among the given options is \( \frac{31}{36} \).

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