To evaluate the expression
[tex]\[
\frac{3}{2} \cdot \left(\frac{5}{4} + \frac{5}{36}\right),
\][/tex]
we follow these steps:
1. Add the fractions inside the parentheses:
[tex]\[
\frac{5}{4} + \frac{5}{36}.
\][/tex]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 36 is 36. Convert both fractions to have this common denominator:
[tex]\[
\frac{5}{4} = \frac{5 \times 9}{4 \times 9} = \frac{45}{36}.
\][/tex]
The second fraction is already over 36:
[tex]\[
\frac{5}{36}.
\][/tex]
Now add the two fractions:
[tex]\[
\frac{45}{36} + \frac{5}{36} = \frac{45 + 5}{36} = \frac{50}{36}.
\][/tex]
Simplify [tex]\(\frac{50}{36}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{50 \div 2}{36 \div 2} = \frac{25}{18}.
\][/tex]
So,
[tex]\[
\frac{5}{4} + \frac{5}{36} = \frac{25}{18}.
\][/tex]
2. Multiply the result by the fraction outside the parentheses:
[tex]\[
\frac{3}{2} \cdot \frac{25}{18}.
\][/tex]
To multiply fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{3 \times 25}{2 \times 18} = \frac{75}{36}.
\][/tex]
Simplify [tex]\(\frac{75}{36}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3:
[tex]\[
\frac{75 \div 3}{36 \div 3} = \frac{25}{12}.
\][/tex]
So, the evaluated expression is:
[tex]\[
\frac{3}{2} \cdot \left(\frac{5}{4} + \frac{5}{36}\right) = \frac{25}{12}.
\][/tex]
Therefore, the simplified fraction result is [tex]\(\frac{25}{12}\)[/tex].
