Evaluate
(i) `(-8)/(15)+(2)/(-3)`
(ii) `(-7)/(10)+(13)/(-15)+(27)/(20)`
(iii) `-1+(7)/(-9)+ (11)/(12)`
(iv) `(-11)/(39)+ (5)/(26)+2`
(v) `2+(-1)/(2)+(-3)/(4)`
(vi) `(-9)/(11)+(2)/(3)+(-3)/(4)`

Answer

To evaluate the given rational number expressions step by step, let's solve each part one by one.

### (i) Evaluate `(-8)/(15) + (2)/(-3)`

1. Rewrite the expression:
\[ -\frac{8}{15} - \frac{2}{3} \]

2. Find a common denominator. The LCM of 15 and 3 is 15.

3. Rewrite \(-\frac{2}{3}\) with a denominator of 15:
\[ -\frac{2}{3} = -\frac{2 \times 5}{3 \times 5} = -\frac{10}{15} \]

4. Now, add the fractions:
\[ -\frac{8}{15} - \frac{10}{15} = -\frac{8 + 10}{15} = -\frac{18}{15} \]

5. Simplify:
\[ -\frac{18}{15} = -\frac{6}{5} \]

**Final Answer for (i)**: \(-\frac{6}{5}\)

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### (ii) Evaluate `(-7)/(10) + (13)/(-15) + (27)/(20)`

1. Rewrite the expression:
\[ -\frac{7}{10} - \frac{13}{15} + \frac{27}{20} \]

2. Find a common denominator. The LCM of 10, 15, and 20 is 60.

3. Rewrite each fraction:
\[ -\frac{7}{10} = -\frac{7 \times 6}{10 \times 6} = -\frac{42}{60} \]
\[ -\frac{13}{15} = -\frac{13 \times 4}{15 \times 4} = -\frac{52}{60} \]
\[ \frac{27}{20} = \frac{27 \times 3}{20 \times 3} = \frac{81}{60} \]

4. Now, combine the fractions:
\[ -\frac{42}{60} - \frac{52}{60} + \frac{81}{60} = \frac{-42 - 52 + 81}{60} = \frac{-94 + 81}{60} = \frac{-13}{60} \]

**Final Answer for (ii)**: \(-\frac{13}{60}\)

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### (iii) Evaluate `-1 + (7)/(-9) + (11)/(12)`

1. Rewrite the expression:
\[ -1 - \frac{7}{9} + \frac{11}{12} \]

2. Convert -1 to a fraction with a common denominator:
\[ -1 = -\frac{108}{108} \quad (\text{LCM of 9 and 12 is 36}) \]

3. Rewrite each fraction:
\[ -\frac{7}{9} = -\frac{28}{36} \]
\[ \frac{11}{12} = \frac{33}{36} \]

4. Combine:
\[ -\frac{108}{108} - \frac{28}{36} + \frac{33}{36} = \frac{-108 - 28 + 33}{36} = \frac{-103}{36} \]

**Final Answer for (iii)**: \(-\frac{103}{36}\)

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### (iv) Evaluate `(-11)/(39) + (5)/(26) + 2`

1. Rewrite the expression:
\[ -\frac{11}{39} + \frac{5}{26} + 2 \]

2. Convert 2 to a fraction:
\[ 2 = \frac{78}{39} \]

3. Find a common denominator for \(-\frac{11}{39}\) and \(\frac{5}{26}\). The LCM of 39 and 26 is 78.

4. Rewrite each fraction:
\[ -\frac{11}{39} = -\frac{22}{78} \]
\[ \frac{5}{26} = \frac{15}{78} \]

5. Combine:
\[ -\frac{22}{78} + \frac{15}{78} + \frac{78}{39} = \frac{-22 + 15 + 78}{78} = \frac{71}{78} \]

**Final Answer for (iv)**: \(\frac{71}{78}\)

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### (v) Evaluate `2 + (-1)/(2) + (-3)/(4)`

1. Rewrite the expression:
\[ 2 - \frac{1}{2} - \frac{3}{4} \]

2. Convert 2 to a fraction:
\[ 2 = \frac{8}{4} \]

3. Find a common denominator (which is 4):
\[ -\frac{1}{2} = -\frac{2}{4} \]

4. Combine:
\[ \frac{8}{4} - \frac{2}{4} - \frac{3}{4} = \frac{8 - 2 - 3}{4} = \frac{3}{4} \]

**Final Answer for (v)**: \(\frac{3}{4}\)

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### (vi) Evaluate `(-9)/(11) + (2)/(3) + (-3)/(4)`

1. Rewrite the expression:
\[ -\frac{9}{11} + \frac{2}{3} - \frac{3}{4} \]

2. Find a common denominator. The LCM of 11, 3, and 4 is 132.

3. Rewrite each fraction:
\[ -\frac{9}{11} = -\frac{108}{132} \]
\[ \frac{2}{3} = \frac{88}{132} \]
\[ -\frac{3}{4} = -\frac{99}{132} \]

4. Combine:
\[ -\frac{108}{132} + \frac{88}{132} - \frac{99}{132} = \frac{-108 + 88 - 99}{132} = \frac{-119}{132} \]

**Final Answer for (vi)**: \(-\frac{119}{132}\)

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