To reduce each of the given fractions into its simplest form, we will follow a systematic approach. The simplest form of a fraction is achieved when the numerator and denominator have no common factors other than 1.
### Step-by-Step Solution:
#### (i) Reduce \( \frac{9}{15} \)
1. **Find the GCD (Greatest Common Divisor)** of 9 and 15.
- The factors of 9 are: 1, 3, 9
- The factors of 15 are: 1, 3, 5, 15
- The GCD is 3.
2. **Divide both the numerator and the denominator by the GCD:**
\[
\frac{9 \div 3}{15 \div 3} = \frac{3}{5}
\]
**Simplified form:** \( \frac{3}{5} \)
---
#### (ii) Reduce \( \frac{48}{60} \)
1. **Find the GCD of 48 and 60.**
- The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- The GCD is 12.
2. **Divide both the numerator and the denominator by the GCD:**
\[
\frac{48 \div 12}{60 \div 12} = \frac{4}{5}
\]
**Simplified form:** \( \frac{4}{5} \)
---
#### (iii) Reduce \( \frac{84}{98} \)
1. **Find the GCD of 84 and 98.**
- The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- The factors of 98 are: 1, 2, 7, 14, 49, 98
- The GCD is 14.
2. **Divide both the numerator and the denominator by the GCD:**
\[
\frac{84 \div 14}{98 \div 14} = \frac{6}{7}
\]
**Simplified form:** \( \frac{6}{7} \)
---
#### (iv) Reduce \( \frac{150}{60} \)
1. **Find the GCD of 150 and 60.**
- The factors of 150 are: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
- The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- The GCD is 30.
2. **Divide both the numerator and the denominator by the GCD:**
\[
\frac{150 \div 30}{60 \div 30} = \frac{5}{2}
\]
**Simplified form:** \( \frac{5}{2} \)
---
#### (v) Reduce \( \frac{72}{90} \)
1. **Find the GCD of 72 and 90.**
- The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
- The GCD is 18.
2. **Divide both the numerator and the denominator by the GCD:**
\[
\frac{72 \div 18}{90 \div 18} = \frac{4}{5}
\]
**Simplified form:** \( \frac{4}{5} \)
---
### Summary of Simplified Fractions:
1. \( \frac{9}{15} = \frac{3}{5} \)
2. \( \frac{48}{60} = \frac{4}{5} \)
3. \( \frac{84}{98} = \frac{6}{7} \)
4. \( \frac{150}{60} = \frac{5}{2} \)
5. \( \frac{72}{90} = \frac{4}{5} \)
