The difference between the smallest and the largest fraction among `3/2 , 1/5 , 4/3 , 5/4` is

Options

A. `12//8`
B. `13//8`
C. `13//10`
D. `15//10`

Answer

To find the difference between the smallest and the largest fraction among \( \frac{3}{2}, \frac{1}{5}, \frac{4}{3}, \frac{5}{4} \), we will follow these steps:

### Step 1: Identify the fractions
The fractions we have are:
- \( \frac{3}{2} \)
- \( \frac{1}{5} \)
- \( \frac{4}{3} \)
- \( \frac{5}{4} \)

### Step 2: Convert fractions to a common denominator
To compare these fractions easily, we will convert them to a common denominator. The denominators are \( 2, 5, 3, \) and \( 4 \).

#### Finding the LCM of the denominators:
- The prime factorization of the denominators:
- \( 2 = 2^1 \)
- \( 5 = 5^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)

The LCM will take the highest power of each prime:
- \( 2^2 \) from \( 4 \)
- \( 3^1 \) from \( 3 \)
- \( 5^1 \) from \( 5 \)

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 3: Convert each fraction to have the common denominator of 60
Now we will convert each fraction:
- \( \frac{3}{2} = \frac{3 \times 30}{2 \times 30} = \frac{90}{60} \)
- \( \frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60} \)
- \( \frac{4}{3} = \frac{4 \times 20}{3 \times 20} = \frac{80}{60} \)
- \( \frac{5}{4} = \frac{5 \times 15}{4 \times 15} = \frac{75}{60} \)

### Step 4: Identify the smallest and largest fractions
Now we have:
- \( \frac{90}{60} \)
- \( \frac{12}{60} \)
- \( \frac{80}{60} \)
- \( \frac{75}{60} \)

From these, we can see:
- The largest fraction is \( \frac{90}{60} \) (which is \( \frac{3}{2} \))
- The smallest fraction is \( \frac{12}{60} \) (which is \( \frac{1}{5} \))

### Step 5: Calculate the difference
Now we calculate the difference between the largest and smallest fractions:
\[ \text{Difference} = \frac{90}{60} - \frac{12}{60} = \frac{90 - 12}{60} = \frac{78}{60} \]

### Step 6: Simplify the difference
Now we simplify \( \frac{78}{60} \):
\[ \frac{78}{60} = \frac{13}{10} \]

### Final Answer
The difference between the smallest and the largest fraction is \( \frac{13}{10} \).

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