Simplify the following :
(i) `2""5/12-1""19/60+2""11/40" (ii) " 15""3/25+4""4/5-3""3/10-16""3/5`.

Answer

Let's simplify the given expressions step by step.

### Part (i): Simplify \( 2 \frac{5}{12} - 1 \frac{19}{60} + 2 \frac{11}{40} \)

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 2 \frac{5}{12} = \frac{2 \times 12 + 5}{12} = \frac{24 + 5}{12} = \frac{29}{12} \)
- \( 1 \frac{19}{60} = \frac{1 \times 60 + 19}{60} = \frac{60 + 19}{60} = \frac{79}{60} \)
- \( 2 \frac{11}{40} = \frac{2 \times 40 + 11}{40} = \frac{80 + 11}{40} = \frac{91}{40} \)

2. **Find the LCM of the Denominators**:
- The denominators are 12, 60, and 40. The LCM of these numbers is 120.

3. **Convert Each Fraction to Have the Same Denominator**:
- For \( \frac{29}{12} \):
\[ \frac{29}{12} = \frac{29 \times 10}{12 \times 10} = \frac{290}{120} \]
- For \( \frac{79}{60} \):
\[ \frac{79}{60} = \frac{79 \times 2}{60 \times 2} = \frac{158}{120} \]
- For \( \frac{91}{40} \):
\[ \frac{91}{40} = \frac{91 \times 3}{40 \times 3} = \frac{273}{120} \]

4. **Combine the Fractions**:
- Now substitute back into the expression:
\[ \frac{290}{120} - \frac{158}{120} + \frac{273}{120} = \frac{290 - 158 + 273}{120} = \frac{405}{120} \]

5. **Simplify the Result**:
- To simplify \( \frac{405}{120} \), find the GCD of 405 and 120, which is 15:
\[ \frac{405 \div 15}{120 \div 15} = \frac{27}{8} \]
- Convert to mixed number:
\[ 27 \div 8 = 3 \text{ remainder } 3 \Rightarrow 3 \frac{3}{8} \]

### Final Answer for Part (i):
\[ 3 \frac{3}{8} \]

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### Part (ii): Simplify \( 15 \frac{3}{25} + 4 \frac{4}{5} - 3 \frac{3}{10} - 16 \frac{3}{5} \)

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 15 \frac{3}{25} = \frac{15 \times 25 + 3}{25} = \frac{375 + 3}{25} = \frac{378}{25} \)
- \( 4 \frac{4}{5} = \frac{4 \times 5 + 4}{5} = \frac{20 + 4}{5} = \frac{24}{5} \)
- \( 3 \frac{3}{10} = \frac{3 \times 10 + 3}{10} = \frac{30 + 3}{10} = \frac{33}{10} \)
- \( 16 \frac{3}{5} = \frac{16 \times 5 + 3}{5} = \frac{80 + 3}{5} = \frac{83}{5} \)

2. **Find the LCM of the Denominators**:
- The denominators are 25, 5, 10, and 5. The LCM is 50.

3. **Convert Each Fraction to Have the Same Denominator**:
- For \( \frac{378}{25} \):
\[ \frac{378}{25} = \frac{378 \times 2}{25 \times 2} = \frac{756}{50} \]
- For \( \frac{24}{5} \):
\[ \frac{24}{5} = \frac{24 \times 10}{5 \times 10} = \frac{240}{50} \]
- For \( \frac{33}{10} \):
\[ \frac{33}{10} = \frac{33 \times 5}{10 \times 5} = \frac{165}{50} \]
- For \( \frac{83}{5} \):
\[ \frac{83}{5} = \frac{83 \times 10}{5 \times 10} = \frac{830}{50} \]

4. **Combine the Fractions**:
- Substitute back into the expression:
\[ \frac{756}{50} + \frac{240}{50} - \frac{165}{50} - \frac{830}{50} = \frac{756 + 240 - 165 - 830}{50} = \frac{1}{50} \]

### Final Answer for Part (ii):
\[ \frac{1}{50} \]

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