Compute:

[tex]\[
\frac{\left(\frac{1}{6}+0.1+\frac{1}{15}\right) \div \left(\frac{1}{6}+0.1-\frac{1}{15}\right) \cdot 2.52}{\left(0.5-\frac{1}{3}+0.25-\frac{1}{5}\right) \div \left(0.25-\frac{1}{6}\right) \cdot \frac{7}{13}}
\][/tex]



Answer :

Sure, let's solve the given mathematical expression step-by-step.

We need to compute:
[tex]\[ \frac{\left(\frac{1}{6}+0.1+\frac{1}{15}\right) \div\left(\frac{1}{6}+0.1-\frac{1}{15}\right) \cdot 2.52}{\left(0.5-\frac{1}{3}+0.25-\frac{1}{5}\right) \div\left(0.25-\frac{1}{6}\right) \cdot \frac{7}{13}} \][/tex]

### Step-by-Step Solution:

#### Step 1: Calculate the Numerator

1. Calculate [tex]\(\frac{1}{6} + 0.1 + \frac{1}{15}\)[/tex]:

[tex]\[ \frac{1}{6} = 0.1667 \][/tex]
[tex]\[ \frac{1}{15} = 0.0667 \][/tex]
So,
[tex]\[ \frac{1}{6} + 0.1 + \frac{1}{15} = 0.1667 + 0.1 + 0.0667 = 0.3333 \][/tex]

2. Calculate [tex]\(\frac{1}{6} + 0.1 - \frac{1}{15}\)[/tex]:

[tex]\[ \frac{1}{6} = 0.1667 \][/tex]
[tex]\[ \frac{1}{15} = 0.0667 \][/tex]
So,
[tex]\[ \frac{1}{6} + 0.1 - \frac{1}{15} = 0.1667 + 0.1 - 0.0667 = 0.2 \][/tex]

3. Divide the results from above calculations:

[tex]\[ \frac{0.3333}{0.2} = 1.6667 \][/tex]

4. Multiply by 2.52:

[tex]\[ 1.6667 \cdot 2.52 = 4.2 \][/tex]

#### Step 2: Calculate the Denominator

1. Calculate [tex]\(0.5 - \frac{1}{3} + 0.25 - \frac{1}{5}\)[/tex]:

[tex]\[ \frac{1}{3} = 0.3333 \][/tex]
[tex]\[ \frac{1}{5} = 0.2 \][/tex]
So,
[tex]\[ 0.5 - 0.3333 + 0.25 - 0.2 = 0.2167 \][/tex]

2. Calculate [tex]\(0.25 - \frac{1}{6}\)[/tex]:

[tex]\[ \frac{1}{6} = 0.1667 \][/tex]
So,
[tex]\[ 0.25 - 0.1667 = 0.0833 \][/tex]

3. Divide the results from these calculations:

[tex]\[ \frac{0.2167}{0.0833} = 2.6 \][/tex]

4. Multiply by [tex]\(\frac{7}{13}\)[/tex]:

[tex]\[ \frac{7}{13} = 0.5385 \][/tex]
So,
[tex]\[ 2.6 \cdot 0.5385 = 1.4 \][/tex]

#### Step 3: Compute the Final Result

Finally,
[tex]\[ \frac{4.2}{1.4} = 3 \][/tex]

Thus, the solution to the given expression is:
[tex]\[ \boxed{3} \][/tex]