Simplify the following :
`3-1""1/6-7/15`

Answer

To simplify the expression \(3 - 1 \frac{1}{6} - \frac{7}{15}\), we will follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \(1 \frac{1}{6}\) can be converted to an improper fraction.

To convert:
\[ 1 \frac{1}{6} = \frac{(1 \times 6) + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6} \]

### Step 2: Rewrite the expression
Now we can rewrite the expression with the improper fraction:
\[ 3 - \frac{7}{6} - \frac{7}{15} \]

### Step 3: Convert whole number to a fraction
We can express \(3\) as a fraction with a denominator of \(1\):
\[ 3 = \frac{3}{1} \]

### Step 4: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(1\), \(6\), and \(15\) to combine the fractions.

The prime factorization is:
- \(1 = 1\)
- \(6 = 2 \times 3\)
- \(15 = 3 \times 5\)

The LCM is:
\[ LCM = 2 \times 3 \times 5 = 30 \]

### Step 5: Convert each fraction to have the same denominator
Now, we will convert each fraction to have a denominator of \(30\):

1. For \(\frac{3}{1}\):
\[ \frac{3}{1} = \frac{3 \times 30}{1 \times 30} = \frac{90}{30} \]

2. For \(\frac{7}{6}\):
\[ \frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30} \]

3. For \(\frac{7}{15}\):
\[ \frac{7}{15} = \frac{7 \times 2}{15 \times 2} = \frac{14}{30} \]

### Step 6: Substitute back into the expression
Now we can substitute these fractions back into the expression:
\[ \frac{90}{30} - \frac{35}{30} - \frac{14}{30} \]

### Step 7: Combine the fractions
Since they all have the same denominator, we can combine them:
\[ \frac{90 - 35 - 14}{30} \]

### Step 8: Perform the subtraction in the numerator
Calculating the numerator:
\[ 90 - 35 = 55 \]
\[ 55 - 14 = 41 \]
So we have:
\[ \frac{41}{30} \]

### Final Answer
The simplified form of the expression \(3 - 1 \frac{1}{6} - \frac{7}{15}\) is:
\[ \frac{41}{30} \]

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