Math worksheet with problems and answer choices for selecting correct expressions.

Problem Analysis:

The task involves solving equations from Column A and matching them with their corresponding solutions in Column B. Each equation in Column A must be solved, and the result must be matched with one of the options in Column B.

Step-by-Step Solution:

#### 1. Solve Equation 1:
Equation (Column A):
$$
\frac{1}{2} + x = \frac{3}{4}
$$

- Subtract $\frac{1}{2}$ from both sides:
$$
x = \frac{3}{4} - \frac{1}{2}
$$

- Convert $\frac{1}{2}$ to a fraction with a denominator of 4:
$$
\frac{1}{2} = \frac{2}{4}
$$

- Perform the subtraction:
$$
x = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}
$$

- Solution: $x = \frac{1}{4}$

- Match with Column B: The solution $\frac{1}{4}$ corresponds to option 1.

---

#### 2. Solve Equation 2:
Equation (Column A):
$$
\frac{3}{4} + x = \frac{5}{6}
$$

- Subtract $\frac{3}{4}$ from both sides:
$$
x = \frac{5}{6} - \frac{3}{4}
$$

- Find a common denominator for $\frac{5}{6}$ and $\frac{3}{4}$. The least common denominator is 12:
$$
\frac{5}{6} = \frac{10}{12}, \quad \frac{3}{4} = \frac{9}{12}
$$

- Perform the subtraction:
$$
x = \frac{10}{12} - \frac{9}{12} = \frac{1}{12}
$$

- Solution: $x = \frac{1}{12}$

- Match with Column B: The solution $\frac{1}{12}$ corresponds to option 2.

---

#### 3. Solve Equation 3:
Equation (Column A):
$$
\frac{3}{4} \cdot x = \frac{1}{2}
$$

- Divide both sides by $\frac{3}{4}$:
$$
x = \frac{\frac{1}{2}}{\frac{3}{4}}
$$

- Dividing by a fraction is equivalent to multiplying by its reciprocal:
$$
x = \frac{1}{2} \cdot \frac{4}{3} = \frac{4}{6} = \frac{2}{3}
$$

- Solution: $x = \frac{2}{3}$

- Match with Column B: The solution $\frac{2}{3}$ corresponds to option 3.

---

#### 4. Solve Equation 4:
Equation (Column A):
$$
-0.5x + 0.3 = 0.2x + 0.1
$$

- Combine like terms by moving all $x$-terms to one side and constants to the other:
$$
-0.5x - 0.2x = 0.1 - 0.3
$$

- Simplify:
$$
-0.7x = -0.2
$$

- Divide both sides by $-0.7$:
$$
x = \frac{-0.2}{-0.7} = \frac{2}{7}
$$

- Solution: $x = \frac{2}{7}$

- Match with Column B: The solution $\frac{2}{7}$ corresponds to option 4.

---

#### 5. Solve Equation 5:
Equation (Column A):
$$
0.2x + 1 = 0.8x - 2
$$

- Combine like terms by moving all $x$-terms to one side and constants to the other:
$$
0.2x - 0.8x = -2 - 1
$$

- Simplify:
$$
-0.6x = -3
$$

- Divide both sides by $-0.6$:
$$
x = \frac{-3}{-0.6} = 5
$$

- Solution: $x = 5$

- Match with Column B: The solution $5$ corresponds to option 5.

---

Final Answer:

Match each equation in Column A with its solution in Column B:
1. $\frac{1}{2} + x = \frac{3}{4}$ → $\frac{1}{4}$ (Option 1)
2. $\frac{3}{4} + x = \frac{5}{6}$ → $\frac{1}{12}$ (Option 2)
3. $\frac{3}{4} \cdot x = \frac{1}{2}$ → $\frac{2}{3}$ (Option 3)
4. $-0.5x + 0.3 = 0.2x + 0.1$ → $\frac{2}{7}$ (Option 4)
5. $0.2x + 1 = 0.8x - 2$ → $5$ (Option 5)

$$
\boxed{1, 2, 3, 4, 5}
$$