Alright! Let's break down each given expression step-by-step to arrive at the solutions.
1. Expression 1:
[tex]\[
10 \cdot 5 \times (10 + 10 - 5) \times \frac{21}{4}
\][/tex]
First, simplify inside the parentheses:
[tex]\[
10 + 10 - 5 = 15
\][/tex]
Now multiply:
[tex]\[
10 \cdot 5 = 50
\][/tex]
Next, plug in the value from the parentheses:
[tex]\[
50 \times 15 = 750
\][/tex]
Finally, multiply by \(\frac{21}{4}\):
[tex]\[
750 \times \frac{21}{4} = 750 \times 5.25 = 3937.5
\][/tex]
So, the result is:
[tex]\[
\boxed{3937.5}
\][/tex]
2. Expression 2:
[tex]\[
-(-3) \times \frac{1}{3} + \frac{1}{2}
\][/tex]
Distribute the negative sign:
[tex]\[
-(-3) = 3
\][/tex]
Now multiply:
[tex]\[
3 \times \frac{1}{3} = 1
\][/tex]
Add \(\frac{1}{2}\):
[tex]\[
1 + \frac{1}{2} = 1.5
\][/tex]
So, the result is:
[tex]\[
\boxed{1.5}
\][/tex]
3. Expression 3:
[tex]\[
0.2222 \times 10.001 - 100
\][/tex]
First, perform the multiplication:
[tex]\[
0.2222 \times 10.001 = 2.222222
\][/tex]
Then subtract 100:
[tex]\[
2.222222 - 100 = -97.777778
\][/tex]
So, the result is:
[tex]\[
\boxed{-97.777778}
\][/tex]
4. Expression 4:
[tex]\[
\frac{41}{3} + 31 - 4\frac{1}{5} + \frac{1}{3}
\][/tex]
Begin by simplifying the mixed fraction:
[tex]\[
4\frac{1}{5} = 4 + \frac{1}{5} = 4.2
\][/tex]
Convert \(\frac{41}{3}\) and \(\frac{1}{3}\) to decimal form:
[tex]\[
\frac{41}{3} = 13.\overline{6}
\][/tex]
[tex]\[
\frac{1}{3} = 0.\overline{3}
\][/tex]
Perform the addition and subtraction:
[tex]\[
13.\overline{6} + 31 - 4.2 + 0.\overline{3} = 13.6666 + 31 - 4.2 + 0.3333
\][/tex]
[tex]\[
= 44.9999 - 4.2
\][/tex]
[tex]\[
= 40.8
\][/tex]
So, the result is:
[tex]\[
\boxed{40.8}
\][/tex]
In conclusion, the results of each expression are:
1. \(\boxed{3937.5}\)
2. \(\boxed{1.5}\)
3. \(\boxed{-97.777778}\)
4. [tex]\(\boxed{40.8}\)[/tex]
