Answer :
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]
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]
