Using the rearrangement property find the sum:
(i) `(4)/(3)+(3)/(5)+(-2)/(3)+(-11)/(5)`
(ii) `(-8)/(3)+(-1)/(4)+(-11)/(6)+(3)/(8)`
(iii) `(-13)/(20)+(11)/(14)+(-5)/(7)+(7)/(10)`
(iv) `(-6)/(7) +(-5)/(6)+(-4)/(9)+(-15)/(7)`

Answer

To solve the given problems using the rearrangement property, we will follow these steps for each part:

### (i) Find the sum: \( \frac{4}{3} + \frac{3}{5} - \frac{2}{3} - \frac{11}{5} \)

**Step 1:** Group the positive and negative fractions.
- Positive: \( \frac{4}{3} + \frac{3}{5} \)
- Negative: \( -\frac{2}{3} - \frac{11}{5} \)

**Step 2:** Calculate the sum of the positive fractions.
- LCM of denominators \(3\) and \(5\) is \(15\).
- Convert fractions:
\[ \frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \]
\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]
- Sum:
\[ \frac{20}{15} + \frac{9}{15} = \frac{29}{15} \]

**Step 3:** Calculate the sum of the negative fractions.
- Convert fractions:
\[ -\frac{2}{3} = -\frac{2 \times 5}{3 \times 5} = -\frac{10}{15} \]
\[ -\frac{11}{5} = -\frac{11 \times 3}{5 \times 3} = -\frac{33}{15} \]
- Sum:
\[ -\frac{10}{15} - \frac{33}{15} = -\frac{43}{15} \]

**Step 4:** Combine the results.
\[ \frac{29}{15} - \frac{43}{15} = \frac{29 - 43}{15} = -\frac{14}{15} \]

### Final Answer for (i):
\(-\frac{14}{15}\)

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### (ii) Find the sum: \( -\frac{8}{3} - \frac{1}{4} - \frac{11}{6} + \frac{3}{8} \)

**Step 1:** Group the positive and negative fractions.
- Positive: \( \frac{3}{8} \)
- Negative: \( -\frac{8}{3} - \frac{1}{4} - \frac{11}{6} \)

**Step 2:** Calculate the sum of the negative fractions.
- LCM of \(3\), \(4\), and \(6\) is \(12\).
- Convert fractions:
\[ -\frac{8}{3} = -\frac{32}{12}, \quad -\frac{1}{4} = -\frac{3}{12}, \quad -\frac{11}{6} = -\frac{22}{12} \]
- Sum:
\[ -\frac{32}{12} - \frac{3}{12} - \frac{22}{12} = -\frac{57}{12} \]

**Step 3:** Combine with the positive fraction.
\[ -\frac{57}{12} + \frac{3}{8} \]
- LCM of \(12\) and \(8\) is \(24\).
- Convert:
\[ -\frac{57}{12} = -\frac{114}{24}, \quad \frac{3}{8} = \frac{9}{24} \]
- Combine:
\[ -\frac{114}{24} + \frac{9}{24} = -\frac{105}{24} \]

### Final Answer for (ii):
\(-\frac{105}{24}\)

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### (iii) Find the sum: \( -\frac{13}{20} + \frac{11}{14} - \frac{5}{7} + \frac{7}{10} \)

**Step 1:** Group the positive and negative fractions.
- Positive: \( \frac{11}{14} + \frac{7}{10} \)
- Negative: \( -\frac{13}{20} - \frac{5}{7} \)

**Step 2:** Calculate the sum of the positive fractions.
- LCM of \(14\) and \(10\) is \(70\).
- Convert fractions:
\[ \frac{11}{14} = \frac{55}{70}, \quad \frac{7}{10} = \frac{49}{70} \]
- Sum:
\[ \frac{55}{70} + \frac{49}{70} = \frac{104}{70} \]

**Step 3:** Calculate the sum of the negative fractions.
- LCM of \(20\) and \(7\) is \(140\).
- Convert fractions:
\[ -\frac{13}{20} = -\frac{91}{140}, \quad -\frac{5}{7} = -\frac{100}{140} \]
- Sum:
\[ -\frac{91}{140} - \frac{100}{140} = -\frac{191}{140} \]

**Step 4:** Combine the results.
\[ \frac{104}{70} - \frac{191}{140} \]
- LCM of \(70\) and \(140\) is \(140\).
- Convert:
\[ \frac{104}{70} = \frac{208}{140} \]
- Combine:
\[ \frac{208}{140} - \frac{191}{140} = \frac{17}{140} \]

### Final Answer for (iii):
\(\frac{17}{140}\)

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### (iv) Find the sum: \( -\frac{6}{7} - \frac{5}{6} - \frac{4}{9} - \frac{15}{7} \)

**Step 1:** Group the fractions.
- Positive: None
- Negative: \( -\frac{6}{7} - \frac{5}{6} - \frac{4}{9} - \frac{15}{7} \)

**Step 2:** Combine the negative fractions.
- LCM of \(7\), \(6\), and \(9\) is \(126\).
- Convert fractions:
\[ -\frac{6}{7} = -\frac{108}{126}, \quad -\frac{5}{6} = -\frac{105}{126}, \quad -\frac{4}{9} = -\frac{56}{126}, \quad -\frac{15}{7} = -\frac{270}{126} \]
- Sum:
\[ -\frac{108 + 105 + 56 + 270}{126} = -\frac{539}{126} \]

### Final Answer for (iv):
\(-\frac{539}{126}\)

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