Sure, let's go through each part step-by-step in detail.
### Part 1: Perform the Indicated Operations
#### (i) [tex]\(-1 + 3 - 6\)[/tex]
1. Start with [tex]\(-1\)[/tex].
2. Add [tex]\(3\)[/tex]: [tex]\(-1 + 3 = 2\)[/tex].
3. Subtract [tex]\(6\)[/tex]: [tex]\(2 - 6 = -4\)[/tex].
So, [tex]\(-1 + 3 - 6 = -4\)[/tex].
#### (ii) [tex]\(-4 + (2 \times -3)\)[/tex]
1. Start with [tex]\(2 \times -3\)[/tex].
2. Calculate the multiplication: [tex]\(2 \times -3 = -6\)[/tex].
3. Now, add [tex]\(-4\)[/tex]: [tex]\(-4 + (-6) = -4 - 6 = -10\)[/tex].
So, [tex]\(-4 + (2 \times -3) = -10\)[/tex].
### Part 2: Perform the Fraction Operations
#### (i) [tex]\(2 \frac{1}{5} + \frac{3}{10}\)[/tex]
1. Convert [tex]\(2 \frac{1}{5}\)[/tex] to an improper fraction.
- [tex]\(2 \frac{1}{5}\)[/tex] is the same as [tex]\(2 + \frac{1}{5}\)[/tex], which is [tex]\(2.2\)[/tex].
- In improper fraction form, [tex]\(2.2\)[/tex] is [tex]\(\frac{11}{5}\)[/tex] (since [tex]\(2 \frac{1}{5} = \frac{10}{5} + \frac{1}{5} = \frac{11}{5}\)[/tex]).
2. Add [tex]\(\frac{11}{5}\)[/tex] and [tex]\(\frac{3}{10}\)[/tex].
- Convert [tex]\(\frac{11}{5}\)[/tex] to have a common denominator with [tex]\(\frac{3}{10}\)[/tex].
- [tex]\(\frac{11}{5} = \frac{22}{10}\)[/tex] (by multiplying both numerator and denominator by 2).
- Now add: [tex]\(\frac{22}{10} + \frac{3}{10} = \frac{22 + 3}{10} = \frac{25}{10}\)[/tex].
- Simplify [tex]\(\frac{25}{10}\)[/tex] to [tex]\(2.5\)[/tex].
So, [tex]\(2 \frac{1}{5} + \frac{3}{10} = 2.5\)[/tex].
#### (ii) [tex]\(\frac{7}{8} \times \frac{2}{3} - \frac{1}{4}\)[/tex]
1. Multiply [tex]\(\frac{7}{8}\)[/tex] by [tex]\(\frac{2}{3}\)[/tex].
- [tex]\(\frac{7}{8} \times \frac{2}{3} = \frac{7 \times 2}{8 \times 3} = \frac{14}{24}\)[/tex].
- Simplify [tex]\(\frac{14}{24}\)[/tex]:
- The greatest common divisor (GCD) of 14 and 24 is 2.
- [tex]\(\frac{14}{24} = \frac{7}{12}\)[/tex] when divided by GCD.
2. Subtract [tex]\(\frac{1}{4}\)[/tex] from [tex]\(\frac{7}{12}\)[/tex].
- Convert [tex]\(\frac{1}{4}\)[/tex] to have a common denominator with [tex]\(\frac{7}{12}\)[/tex].
- [tex]\(\frac{1}{4} = \frac{3}{12}\)[/tex] (by multiplying both numerator and denominator by 3).
- Now subtract: [tex]\(\frac{7}{12} - \frac{3}{12} = \frac{7 - 3}{12} = \frac{4}{12}\)[/tex].
- Simplify [tex]\(\frac{4}{12}\)[/tex]:
- The GCD of 4 and 12 is 4.
- [tex]\(\frac{4}{12} = \frac{1}{3}\)[/tex].
So, [tex]\(\frac{7}{8} \times \frac{2}{3} - \frac{1}{4} = 0.33333333333333337\)[/tex] (or [tex]\(\frac{1}{3}\)[/tex] approximately).
### Summary
1. (i) [tex]\(-1 + 3 - 6 = -4\)[/tex]
(ii) [tex]\(-4 + (2 \times -3) = -10\)[/tex]
2. (i) [tex]\(2 \frac{1}{5} + \frac{3}{10} = 2.5\)[/tex]
(ii) [tex]\(\frac{7}{8} \times \frac{2}{3} - \frac{1}{4} = 0.33333333333333337\)[/tex]
