Simplify the following :
(i) `7""3/4-3""5/6+7/8" (ii) " 2""3/5+1""7/10-3""2/15`

Answer

Let's simplify the given expressions step by step.

### Part (i): Simplify `7 3/4 - 3 5/6 + 7/8`

**Step 1: Convert mixed numbers to improper fractions.**
- For `7 3/4`:
\[ 7 \times 4 + 3 = 28 + 3 = 31 \quad \Rightarrow \quad \frac{31}{4} \]
- For `3 5/6`:
\[ 3 \times 6 + 5 = 18 + 5 = 23 \quad \Rightarrow \quad \frac{23}{6} \]

Now we have:
\[ \frac{31}{4} - \frac{23}{6} + \frac{7}{8} \]

**Step 2: Find the LCM of the denominators (4, 6, and 8).**
- The LCM of 4, 6, and 8 is 24.

**Step 3: Convert each fraction to have the common denominator of 24.**
- For `31/4`:
\[ \frac{31}{4} = \frac{31 \times 6}{4 \times 6} = \frac{186}{24} \]
- For `23/6`:
\[ \frac{23}{6} = \frac{23 \times 4}{6 \times 4} = \frac{92}{24} \]
- For `7/8`:
\[ \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \]

Now we have:
\[ \frac{186}{24} - \frac{92}{24} + \frac{21}{24} \]

**Step 4: Combine the fractions.**
\[ \frac{186 - 92 + 21}{24} = \frac{115}{24} \]

**Step 5: Convert the improper fraction back to a mixed number.**
\[ 115 \div 24 = 4 \quad \text{(remainder 23)} \quad \Rightarrow \quad 4 \frac{23}{24} \]

### Final Answer for Part (i):
\[ 4 \frac{23}{24} \]

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### Part (ii): Simplify `2 3/5 + 1 7/10 - 3 2/15`

**Step 1: Convert mixed numbers to improper fractions.**
- For `2 3/5`:
\[ 2 \times 5 + 3 = 10 + 3 = 13 \quad \Rightarrow \quad \frac{13}{5} \]
- For `1 7/10`:
\[ 1 \times 10 + 7 = 10 + 7 = 17 \quad \Rightarrow \quad \frac{17}{10} \]
- For `3 2/15`:
\[ 3 \times 15 + 2 = 45 + 2 = 47 \quad \Rightarrow \quad \frac{47}{15} \]

Now we have:
\[ \frac{13}{5} + \frac{17}{10} - \frac{47}{15} \]

**Step 2: Find the LCM of the denominators (5, 10, and 15).**
- The LCM of 5, 10, and 15 is 30.

**Step 3: Convert each fraction to have the common denominator of 30.**
- For `13/5`:
\[ \frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30} \]
- For `17/10`:
\[ \frac{17}{10} = \frac{17 \times 3}{10 \times 3} = \frac{51}{30} \]
- For `47/15`:
\[ \frac{47}{15} = \frac{47 \times 2}{15 \times 2} = \frac{94}{30} \]

Now we have:
\[ \frac{78}{30} + \frac{51}{30} - \frac{94}{30} \]

**Step 4: Combine the fractions.**
\[ \frac{78 + 51 - 94}{30} = \frac{35}{30} \]

**Step 5: Simplify the fraction.**
\[ \frac{35}{30} = \frac{7}{6} \quad \Rightarrow \quad 1 \frac{1}{6} \]

### Final Answer for Part (ii):
\[ 1 \frac{1}{6} \]

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