Simplify the following :
(i) `1/2+25/27" (ii) "15""13/20-12""3/4" (iii) "2""3/14-1""4/35`

Answer

Let's simplify the given expressions step by step.

### (i) Simplify \( \frac{1}{2} + \frac{25}{27} \)

**Step 1: Find the LCM of the denominators.**
- The denominators are 2 and 27. The LCM of 2 and 27 is 54.

**Step 2: Convert each fraction to have the same denominator.**
- For \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{1 \times 27}{2 \times 27} = \frac{27}{54} \]
- For \( \frac{25}{27} \):
\[ \frac{25}{27} = \frac{25 \times 2}{27 \times 2} = \frac{50}{54} \]

**Step 3: Add the fractions.**
\[ \frac{27}{54} + \frac{50}{54} = \frac{27 + 50}{54} = \frac{77}{54} \]

**Step 4: Convert to a mixed fraction if necessary.**
- \( \frac{77}{54} \) can be expressed as:
\[ 1 \frac{23}{54} \]

### Final Answer for (i):
\[ \frac{1}{2} + \frac{25}{27} = \frac{77}{54} \text{ or } 1 \frac{23}{54} \]

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### (ii) Simplify \( 15 \frac{13}{20} - 12 \frac{3}{4} \)

**Step 1: Convert mixed fractions to improper fractions.**
- For \( 15 \frac{13}{20} \):
\[ 15 \frac{13}{20} = \frac{15 \times 20 + 13}{20} = \frac{300 + 13}{20} = \frac{313}{20} \]
- For \( 12 \frac{3}{4} \):
\[ 12 \frac{3}{4} = \frac{12 \times 4 + 3}{4} = \frac{48 + 3}{4} = \frac{51}{4} \]

**Step 2: Find the LCM of the denominators.**
- The denominators are 20 and 4. The LCM is 20.

**Step 3: Convert \( \frac{51}{4} \) to have the same denominator.**
\[ \frac{51}{4} = \frac{51 \times 5}{4 \times 5} = \frac{255}{20} \]

**Step 4: Subtract the fractions.**
\[ \frac{313}{20} - \frac{255}{20} = \frac{313 - 255}{20} = \frac{58}{20} \]

**Step 5: Simplify the fraction.**
\[ \frac{58}{20} = \frac{29}{10} = 2 \frac{9}{10} \]

### Final Answer for (ii):
\[ 15 \frac{13}{20} - 12 \frac{3}{4} = \frac{29}{10} \text{ or } 2 \frac{9}{10} \]

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### (iii) Simplify \( 2 \frac{3}{14} - 1 \frac{4}{35} \)

**Step 1: Convert mixed fractions to improper fractions.**
- For \( 2 \frac{3}{14} \):
\[ 2 \frac{3}{14} = \frac{2 \times 14 + 3}{14} = \frac{28 + 3}{14} = \frac{31}{14} \]
- For \( 1 \frac{4}{35} \):
\[ 1 \frac{4}{35} = \frac{1 \times 35 + 4}{35} = \frac{35 + 4}{35} = \frac{39}{35} \]

**Step 2: Find the LCM of the denominators.**
- The denominators are 14 and 35. The LCM is 70.

**Step 3: Convert each fraction to have the same denominator.**
- For \( \frac{31}{14} \):
\[ \frac{31}{14} = \frac{31 \times 5}{14 \times 5} = \frac{155}{70} \]
- For \( \frac{39}{35} \):
\[ \frac{39}{35} = \frac{39 \times 2}{35 \times 2} = \frac{78}{70} \]

**Step 4: Subtract the fractions.**
\[ \frac{155}{70} - \frac{78}{70} = \frac{155 - 78}{70} = \frac{77}{70} \]

**Step 5: Simplify the fraction.**
\[ \frac{77}{70} = \frac{11}{10} = 1 \frac{1}{10} \]

### Final Answer for (iii):
\[ 2 \frac{3}{14} - 1 \frac{4}{35} = \frac{11}{10} \text{ or } 1 \frac{1}{10} \]

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