Which one is the largest among the fractions
`((5)/(113)), ((7)/(120)), ((13)/(145)) and ((17)/(160)) `?

Options

A. `(5)/(113)`
B. `(7)/(120)`
C. `(13)/(145)`
D. `(17)/(160)`

Answer

To determine which of the fractions \( \frac{5}{113}, \frac{7}{120}, \frac{13}{145}, \) and \( \frac{17}{160} \) is the largest, we can use cross multiplication to compare the fractions without finding a common denominator. Here’s a step-by-step solution:

### Step 1: Compare \( \frac{5}{113} \) and \( \frac{7}{120} \)
We will cross-multiply:
\[ 5 \times 120 \quad \text{and} \quad 7 \times 113 \]
Calculating these:
\[ 5 \times 120 = 600 \]
\[ 7 \times 113 = 791 \]
Since \( 791 > 600 \), we conclude:
\[ \frac{7}{120} > \frac{5}{113} \]

### Step 2: Compare \( \frac{7}{120} \) and \( \frac{13}{145} \)
Next, we compare \( \frac{7}{120} \) with \( \frac{13}{145} \) using cross multiplication:
\[ 7 \times 145 \quad \text{and} \quad 13 \times 120 \]
Calculating these:
\[ 7 \times 145 = 1015 \]
\[ 13 \times 120 = 1560 \]
Since \( 1560 > 1015 \), we conclude:
\[ \frac{13}{145} > \frac{7}{120} \]

### Step 3: Compare \( \frac{13}{145} \) and \( \frac{17}{160} \)
Now, we compare \( \frac{13}{145} \) with \( \frac{17}{160} \):
\[ 13 \times 160 \quad \text{and} \quad 17 \times 145 \]
Calculating these:
\[ 13 \times 160 = 2080 \]
\[ 17 \times 145 = 2465 \]
Since \( 2465 > 2080 \), we conclude:
\[ \frac{17}{160} > \frac{13}{145} \]

### Conclusion
From the comparisons, we have:
1. \( \frac{7}{120} > \frac{5}{113} \)
2. \( \frac{13}{145} > \frac{7}{120} \)
3. \( \frac{17}{160} > \frac{13}{145} \)

Thus, the largest fraction among the given options is:
\[ \frac{17}{160} \]

### Final Answer
The largest fraction is \( \frac{17}{160} \).

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