Certainly! Let's break down the given expression step-by-step.
We need to solve the following expression:
[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 1: Calculate the Numerator Part 1
First, evaluate the term [tex]\(\frac{1}{6} + 0.1 + \frac{1}{15}\)[/tex]:
[tex]\[
\frac{1}{6} + 0.1 + \frac{1}{15} = 0.3333333333333333
\][/tex]
### Step 2: Calculate the Numerator Part 2
Next, evaluate the term [tex]\(\frac{1}{6} + 0.1 - \frac{1}{15}\)[/tex]:
[tex]\[
\frac{1}{6} + 0.1 - \frac{1}{15} = 0.2
\][/tex]
### Step 3: Compute the Entire Numerator
Now, divide the results from Steps 1 and 2, then multiply by 2.52:
[tex]\[
\left(\frac{0.3333333333333333}{0.2}\right) \cdot 2.52 = 4.199999999999999
\][/tex]
### Step 4: Calculate the Denominator Part 1
Next, evaluate the term [tex]\(0.5 - \frac{1}{3} + 0.25 - \frac{1}{5}\)[/tex]:
[tex]\[
0.5 - \frac{1}{3} + 0.25 - \frac{1}{5} = 0.21666666666666667
\][/tex]
### Step 5: Calculate the Denominator Part 2
Evaluate the term [tex]\(0.25 - \frac{1}{6}\)[/tex]:
[tex]\[
0.25 - \frac{1}{6} = 0.08333333333333334
\][/tex]
### Step 6: Compute the Entire Denominator
Now, divide the results from Steps 4 and 5, then multiply by [tex]\(\frac{7}{13}\)[/tex]:
[tex]\[
\left(\frac{0.21666666666666667}{0.08333333333333334}\right) \cdot \frac{7}{13} = 1.3999999999999997
\][/tex]
### Step 7: Calculate the Final Result
Finally, divide the numerator computed in Step 3 by the denominator computed in Step 6:
[tex]\[
\frac{4.199999999999999}{1.3999999999999997} = 3.0
\][/tex]
### Conclusion
After evaluating all the steps, the final result of the given expression is:
[tex]\[
\boxed{3.0}
\][/tex]
