https://file.aiquickdraw.com/mathbot/file/1725461928496bgki88m1.jpg

Let's work through the problems in the image one by one.

### 1. Simplify
   - **(a)** \( \frac{21}{63} \)
   - **(b)** \( \frac{0.21}{0.63} \)

### Solutions:

**1a.** Simplify \( \frac{21}{63} \):
- The greatest common divisor (GCD) of 21 and 63 is 21.
- Simplifying, we have:
  \[
  \frac{21 \div 21}{63 \div 21} = \frac{1}{3}
  \]
  So, \( \frac{21}{63} = \frac{1}{3} \).

**1b.** Simplify \( \frac{0.21}{0.63} \):
- We can multiply the numerator and the denominator by 100 to eliminate the decimals:
  \[
  \frac{0.21 \times 100}{0.63 \times 100} = \frac{21}{63}
  \]
- As solved in **1a**, \( \frac{21}{63} = \frac{1}{3} \).

### 2. Calculate
   - **(a)** \( \frac{1}{4} + \frac{2}{5} \)
   - **(b)** \( \frac{5}{4} - \frac{12}{15} \)

### Solutions:

**2a.** Calculate \( \frac{1}{4} + \frac{2}{5} \):
- Find the common denominator, which is 20.
  \[
  \frac{1}{4} = \frac{5}{20}, \quad \frac{2}{5} = \frac{8}{20}
  \]
- Adding these, we get:
  \[
  \frac{5}{20} + \frac{8}{20} = \frac{13}{20}
  \]

**2b.** Calculate \( \frac{5}{4} - \frac{12}{15} \):
- Simplify \( \frac{12}{15} \) to \( \frac{4}{5} \).
- Find a common denominator (which is 20):
  \[
  \frac{5}{4} = \frac{25}{20}, \quad \frac{4}{5} = \frac{16}{20}
  \]
- Subtracting, we get:
  \[
  \frac{25}{20} - \frac{16}{20} = \frac{9}{20}
  \]

### 3. Calculate
   - **(a)** \( 2 \times \frac{4}{15} \)
   - **(b)** \( 11 - 1 \frac{1}{3} \)

### Solutions:

**3a.** Calculate \( 2 \times \frac{4}{15} \):
- Multiply the whole number by the numerator:
  \[
  2 \times \frac{4}{15} = \frac{8}{15}
  \]

**3b.** Calculate \( 11 - 1 \frac{1}{3} \):
- Convert \( 1 \frac{1}{3} \) to an improper fraction:
  \[
  1 \frac{1}{3} = \frac{4}{3}
  \]
- Find a common denominator (which is 3):
  \[
  11 = \frac{33}{3}
  \]
- Subtracting, we get:
  \[
  \frac{33}{3} - \frac{4}{3} = \frac{29}{3}
  \]
- Convert back to a mixed number:
  \[
  \frac{29}{3} = 9 \frac{2}{3}
  \]

### 4. Calculate
   - **(a)** \( 25 \div (1 + 27) \times 2 \)
   - **(b)** \( 5 + 12 \div 6 \times 2 - 18 \div 3^2 \)
   - **(c)** \( \frac{2 \times 2 \times 27 \div 3^2}{16 - 3 \times 4} \)

### Solutions:

**4a.** Calculate \( 25 \div (1 + 27) \times 2 \):
- Simplify inside the parentheses:
  \[
  25 \div 28 \times 2 = \frac{25}{28} \times 2 = \frac{50}{28} = \frac{25}{14}
  \]

**4b.** Calculate \( 5 + 12 \div 6 \times 2 - 18 \div 3^2 \):
- Apply the order of operations (BODMAS):
  \[
  5 + 2 \times 2 - 18 \div 9 = 5 + 4 - 2 = 7
  \]

**4c.** Calculate \( \frac{2 \times 2 \times 27 \div 3^2}{16 - 3 \times 4} \):
- Simplify the numerator:
  \[
  2 \times 2 \times 27 \div 9 = 4 \times 3 = 12
  \]
- Simplify the denominator:
  \[
  16 - 12 = 4
  \]
- The final fraction is:
  \[
  \frac{12}{4} = 3
  \]

### 5. Insert brackets in this expression to make it correct:
   - \( 2 \times 3 + 3 \div 3 = 3 \)

### Solution:
To make the equation correct, the brackets should be placed as follows:
\[ 2 \times (3 + 3) \div 3 = 3 \]

This simplifies to:
\[ 2 \times 6 \div 3 = 3 \]
\[ 12 \div 3 = 3 \]
\[ 3 = 3 \]

### Do you want more details or have any questions? 

### Related Questions:
1. How do you find the greatest common divisor (GCD) of two numbers?
2. What is the difference between a mixed number and an improper fraction?
3. How can we apply the order of operations when dealing with fractions?
4. How can we convert a decimal to a fraction and simplify it?
5. What are the key steps in solving equations involving fractions?

**Tip:** Always remember the order of operations, also known as BODMAS/BIDMAS, to avoid errors when calculating expressions.

Give all answers, where appropriate, as fractions in their lowest terms.

Learn how to simplify fractions, perform arithmetic operations, and apply the order of operations in mathematical expressions. This guide provides step-by-step solutions to problems suitable for students in Grades 6-8.

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