Let's work through each of the problems step-by-step and add extra examples for practice.

Order of Operations

1. Problem: $6 + 2 \times 3$

   • Solution:

     • Follow the order of operations (PEMDAS/BODMAS).
     • Step 1: Multiply before adding: $2 \times 3 = 6$.
     • Step 2: Add: $6 + 6 = 12$.
     • Answer: $12$.

   • Practice Examples:

     • $5 + 3 \times 4$
     • $8 + 2 \times 5$
     • $10 + 4 \times 2$

2. Problem: $(4 + 5) \times 2 - 3^2$

   • Solution:

     • Step 1: Calculate inside the parentheses: $4 + 5 = 9$.
     • Step 2: Exponent first: $3^2 = 9$.
     • Step 3: Multiply: $9 \times 2 = 18$.
     • Step 4: Subtract: $18 - 9 = 9$.
     • Answer: $9$.

   • Practice Examples:

     • $(3 + 2) \times 4 - 5^2$
     • $(6 + 1) \times 3 - 2^2$
     • $(5 + 3) \times 2 - 4^2$

Fractions and Decimals

3. Adding Fractions:

   • Problem: $\frac{1}{4} + \frac{1}{3}$

     • Solution:
       • Find a common denominator (12 in this case).
       • Convert fractions: $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{3} = \frac{4}{12}$.
       • Add: $\frac{3}{12} + \frac{4}{12} = \frac{7}{12}$.
       • Answer: $\frac{7}{12}$.

   • Practice Examples:

     • $\frac{2}{5} + \frac{1}{4}$
     • $\frac{3}{8} + \frac{1}{6}$
     • $\frac{1}{2} + \frac{1}{5}$

   • Problem: $\frac{5}{6} - \frac{1}{2}$

     • Solution:
       • Common denominator: 6.
       • Convert fractions: $\frac{1}{2} = \frac{3}{6}$.
       • Subtract: $\frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$.
       • Answer: $\frac{1}{3}$.

   • Practice Examples:

     • $\frac{7}{8} - \frac{1}{4}$
     • $\frac{3}{5} - \frac{1}{3}$
     • $\frac{4}{7} - \frac{2}{5}$

4. Multiplying Decimals:

   • Problem: $0.6 \times 0.7$

     • Solution:
       • Multiply normally: $6 \times 7 = 42$.
       • Count decimal places (two in total): $0.6 \times 0.7 = 0.42$.
       • Answer: $0.42$.

   • Practice Examples:

     • $0.4 \times 0.5$
     • $0.8 \times 0.3$
     • $0.9 \times 0.6$

   • Problem: $1.5 \times 2.4$

     • Solution:
       • Multiply normally: $15 \times 24 = 360$.
       • Count decimal places (two in total): $1.5 \times 2.4 = 3.60$.
       • Answer: $3.6$.

   • Practice Examples:

     • $1.3 \times 1.6$
     • $2.5 \times 1.8$
     • $0.7 \times 2.3$

Would you like more details on any of the steps?

Additional Questions:

1. How can we find the least common denominator for adding fractions?
2. Why do we prioritize multiplication and division before addition and subtraction?
3. How can we convert mixed numbers to improper fractions for addition or subtraction?
4. What happens to the decimal places when we multiply two decimal numbers?
5. How can we simplify fractions after addition or subtraction?

Tip:

When dealing with fractions, always simplify your answer by dividing the numerator and the denominator by their greatest common factor.