Let's solve the given mathematical expression step-by-step:
[tex]\[ 31 \cdot \left(7 \frac{4}{5} - 2 \frac{1}{3} + 3 \frac{1}{2}\right) \div 2 \frac{3}{5} \][/tex]
### Step 1: Convert Mixed Numbers to Improper Fractions
First, convert the mixed numbers to improper fractions:
1. [tex]\( 7 \frac{4}{5} \)[/tex]:
[tex]\[ 7 \frac{4}{5} = 7 + \frac{4}{5} = \frac{35}{5} + \frac{4}{5} = \frac{39}{5} \][/tex]
In decimal form, this is approximately [tex]\( 7.8 \)[/tex].
2. [tex]\( 2 \frac{1}{3} \)[/tex]:
[tex]\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \][/tex]
In decimal form, this is approximately [tex]\( 2.333 \)[/tex].
3. [tex]\( 3 \frac{1}{2} \)[/tex]:
[tex]\[ 3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} \][/tex]
In decimal form, this is approximately [tex]\( 3.5 \)[/tex].
4. [tex]\( 2 \frac{3}{5} \)[/tex]:
[tex]\[ 2 \frac{3}{5} = 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} \][/tex]
In decimal form, this is approximately [tex]\( 2.6 \)[/tex].
### Step 2: Perform Operations Inside the Parentheses
[tex]\[ 7 \frac{4}{5} - 2 \frac{1}{3} + 3 \frac{1}{2} \][/tex]
Using the decimal forms:
[tex]\[ 7.8 - 2.333 + 3.5 \][/tex]
Calculate step-by-step:
[tex]\[ 7.8 - 2.333 = 5.467 \][/tex]
[tex]\[ 5.467 + 3.5 = 8.967 \][/tex]
### Step 3: Multiply by 31
[tex]\[ 31 \cdot 8.967 \approx 277.967 \][/tex]
### Step 4: Divide by 2 \frac{3}{5}
[tex]\[ \frac{277.967}{2.6} \approx 106.910 \][/tex]
### Final Answer
The value of the expression [tex]\( 31 \cdot \left(7 \frac{4}{5} - 2 \frac{1}{3} + 3 \frac{1}{2}\right) \div 2 \frac{3}{5} \)[/tex] is approximately [tex]\( 106.910 \)[/tex].
