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.