Sure, let's simplify the given mathematical expression step-by-step:
The expression to simplify is:
[tex]\[
-\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}
\][/tex]
### Step 1: Simplify [tex]\(-\frac{2}{3} \times \frac{3}{5}\)[/tex]
When multiplying fractions, multiply the numerators together and the denominators together:
[tex]\[
-\frac{2}{3} \times \frac{3}{5} = \frac{(-2) \times 3}{3 \times 5} = \frac{-6}{15} = -\frac{6}{15}
\][/tex]
Simplifying [tex]\(-\frac{6}{15}\)[/tex]:
[tex]\[
-\frac{6}{15} = -\frac{2}{5}
\][/tex]
Hence, the first term is [tex]\(-\frac{2}{5}\)[/tex].
### Step 2: Simplify [tex]\(\frac{5}{2}\)[/tex]
The second term is already in its simplest form:
[tex]\[
\frac{5}{2}
\][/tex]
### Step 3: Simplify [tex]\(\frac{3}{5} \times \frac{1}{6}\)[/tex]
Again, when multiplying fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{3}{5} \times \frac{1}{6} = \frac{3 \times 1}{5 \times 6} = \frac{3}{30} = \frac{1}{10}
\][/tex]
So, the third term is [tex]\(\frac{1}{10}\)[/tex].
### Step 4: Combine the terms
Now, we have:
[tex]\[
-\frac{2}{5} + \frac{5}{2} - \frac{1}{10}
\][/tex]
To add and subtract these fractions, we'll convert them to a common denominator. The common denominator for 5, 2, and 10 is 10.
[tex]\[
-\frac{2}{5} = -\frac{2 \times 2}{5 \times 2} = -\frac{4}{10}
\][/tex]
[tex]\[
\frac{5}{2} = \frac{5 \times 5}{2 \times 5} = \frac{25}{10}
\][/tex]
[tex]\(\frac{1}{10}\)[/tex] is already in terms of the denominator 10.
Now, combining these:
[tex]\[
-\frac{4}{10} + \frac{25}{10} - \frac{1}{10}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{-4 + 25 - 1}{10} = \frac{20}{10} = 2
\][/tex]
Therefore, the simplified result of the expression is:
[tex]\[
2
\][/tex]
