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]
