Let's solve each part of the given problem step-by-step.
### Part (i)
We need to simplify:
[tex]\[ \left(2 \frac{4}{5} + 1 \frac{3}{10}\right) \times 1 \frac{1}{2} \][/tex]
#### Step 1: Convert Mixed Numbers to Improper Fractions
1. [tex]\(2 \frac{4}{5}\)[/tex]:
[tex]\[ 2 \frac{4}{5} = 2 + \frac{4}{5} = \frac{10}{5} + \frac{4}{5} = \frac{14}{5} \][/tex]
2. [tex]\(1 \frac{3}{10}\)[/tex]:
[tex]\[ 1 \frac{3}{10} = 1 + \frac{3}{10} = \frac{10}{10} + \frac{3}{10} = \frac{13}{10} \][/tex]
3. [tex]\(1 \frac{1}{2}\)[/tex]:
[tex]\[ 1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \][/tex]
#### Step 2: Sum the First Two Fractions
[tex]\[ \frac{14}{5} + \frac{13}{10} \][/tex]
First, find a common denominator, which is 10:
[tex]\[ \frac{14 \times 2}{5 \times 2} + \frac{13}{10} = \frac{28}{10} + \frac{13}{10} = \frac{41}{10} \][/tex]
#### Step 3: Multiply the Sum by the Third Fraction
[tex]\[ \left(\frac{41}{10}\right) \times \left(\frac{3}{2}\right) \][/tex]
Multiply the numerators and the denominators:
[tex]\[ \frac{41 \times 3}{10 \times 2} = \frac{123}{20} \][/tex]
This fraction simplifies to a decimal:
[tex]\[ \frac{123}{20} = 6.15 \][/tex]
So the answer for part (i) is:
[tex]\[ 6.15 \][/tex]
### Part (ii)
We need to simplify:
[tex]\[ \left(1 \frac{5}{6} \times 2 \frac{3}{4}\right) + \left(1 \frac{5}{6} \right) \][/tex]
#### Step 1: Convert Mixed Numbers to Improper Fractions
1. [tex]\(1 \frac{5}{6}\)[/tex]:
[tex]\[ 1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6} \][/tex]
2. [tex]\(2 \frac{3}{4}\)[/tex]:
[tex]\[ 2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \][/tex]
#### Step 2: Multiply the Fractions
[tex]\[ \left(\frac{11}{6}\right) \times \left(\frac{11}{4}\right) \][/tex]
Multiply the numerators and the denominators:
[tex]\[ \frac{11 \times 11}{6 \times 4} = \frac{121}{24} \][/tex]
#### Step 3: Convert Back to Mixed Numbers (if necessary)
[tex]\[ \frac{121}{24} \approx 5.04 \][/tex]
#### Step 4: Sum the Result and the First Fraction
So we have:
[tex]\[ 5.04 + \frac{11}{6} \approx 5.04 + 1.83 = 6.875 \][/tex]
So the answer for part (ii) is:
[tex]\[ 6.875 \][/tex]
### Summary of Results
- Part (i): [tex]\( 6.15 \)[/tex]
- Part (ii): [tex]\( 6.875 \)[/tex]
These are the simplified results for the given problems.
