Math worksheet on simplifying ratios from Asia Academic Integrated School, Inc.

Problem: Simplify the following ratios.

The task requires simplifying each given ratio to its simplest form. To do this, we need to find the greatest common divisor (GCD) of the two numbers in each ratio and then divide both numbers by their GCD.

Let's solve each ratio step by step:

---

#### 1) \( 65 : 15 \)

- Find the GCD of 65 and 15:
- Factors of 65: \( 1, 5, 13, 65 \)
- Factors of 15: \( 1, 3, 5, 15 \)
- Common factors: \( 1, 5 \)
- GCD: \( 5 \)

- Divide both numbers by 5:
\[
65 \div 5 = 13, \quad 15 \div 5 = 3
\]

- Simplified ratio: \( 13 : 3 \)

---

#### 2) \( 56 : 45 \)

- Find the GCD of 56 and 45:
- Factors of 56: \( 1, 2, 4, 7, 8, 14, 28, 56 \)
- Factors of 45: \( 1, 3, 5, 9, 15, 45 \)
- Common factors: \( 1 \)
- GCD: \( 1 \)

- Since the GCD is 1, the ratio is already in its simplest form.

- Simplified ratio: \( 56 : 45 \)

---

#### 3) \( 98 : 24 \)

- Find the GCD of 98 and 24:
- Factors of 98: \( 1, 2, 7, 14, 49, 98 \)
- Factors of 24: \( 1, 2, 3, 4, 6, 8, 12, 24 \)
- Common factors: \( 1, 2 \)
- GCD: \( 2 \)

- Divide both numbers by 2:
\[
98 \div 2 = 49, \quad 24 \div 2 = 12
\]

- Simplified ratio: \( 49 : 12 \)

---

#### 4) \( 380 : 56 \)

- Find the GCD of 380 and 56:
- Factors of 380: \( 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380 \)
- Factors of 56: \( 1, 2, 4, 7, 8, 14, 28, 56 \)
- Common factors: \( 1, 2, 4 \)
- GCD: \( 4 \)

- Divide both numbers by 4:
\[
380 \div 4 = 95, \quad 56 \div 4 = 14
\]

- Simplified ratio: \( 95 : 14 \)

---

#### 5) \( 246 : 39 \)

- Find the GCD of 246 and 39:
- Factors of 246: \( 1, 2, 3, 6, 41, 82, 123, 246 \)
- Factors of 39: \( 1, 3, 13, 39 \)
- Common factors: \( 1, 3 \)
- GCD: \( 3 \)

- Divide both numbers by 3:
\[
246 \div 3 = 82, \quad 39 \div 3 = 13
\]

- Simplified ratio: \( 82 : 13 \)

---

#### 6) \( 98 : 112 \)

- Find the GCD of 98 and 112:
- Factors of 98: \( 1, 2, 7, 14, 49, 98 \)
- Factors of 112: \( 1, 2, 4, 7, 8, 14, 16, 28, 56, 112 \)
- Common factors: \( 1, 2, 7, 14 \)
- GCD: \( 14 \)

- Divide both numbers by 14:
\[
98 \div 14 = 7, \quad 112 \div 14 = 8
\]

- Simplified ratio: \( 7 : 8 \)

---

#### 7) \( 54 : 12 \)

- Find the GCD of 54 and 12:
- Factors of 54: \( 1, 2, 3, 6, 9, 18, 27, 54 \)
- Factors of 12: \( 1, 2, 3, 4, 6, 12 \)
- Common factors: \( 1, 2, 3, 6 \)
- GCD: \( 6 \)

- Divide both numbers by 6:
\[
54 \div 6 = 9, \quad 12 \div 6 = 2
\]

- Simplified ratio: \( 9 : 2 \)

---

#### 8) \( 87 : 15 \)

- Find the GCD of 87 and 15:
- Factors of 87: \( 1, 3, 29, 87 \)
- Factors of 15: \( 1, 3, 5, 15 \)
- Common factors: \( 1, 3 \)
- GCD: \( 3 \)

- Divide both numbers by 3:
\[
87 \div 3 = 29, \quad 15 \div 3 = 5
\]

- Simplified ratio: \( 29 : 5 \)

---

#### 9) \( 47 : 17 \)

- Find the GCD of 47 and 17:
- Factors of 47: \( 1, 47 \) (47 is a prime number)
- Factors of 17: \( 1, 17 \) (17 is a prime number)
- Common factors: \( 1 \)
- GCD: \( 1 \)

- Since the GCD is 1, the ratio is already in its simplest form.

- Simplified ratio: \( 47 : 17 \)

---

#### 10) \( 90 : 117 \)

- Find the GCD of 90 and 117:
- Factors of 90: \( 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 \)
- Factors of 117: \( 1, 3, 9, 13, 39, 117 \)
- Common factors: \( 1, 3, 9 \)
- GCD: \( 9 \)

- Divide both numbers by 9:
\[
90 \div 9 = 10, \quad 117 \div 9 = 13
\]

- Simplified ratio: \( 10 : 13 \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ 13 : 3 \\
2) & \ 56 : 45 \\
3) & \ 49 : 12 \\
4) & \ 95 : 14 \\
5) & \ 82 : 13 \\
6) & \ 7 : 8 \\
7) & \ 9 : 2 \\
8) & \ 29 : 5 \\
9) & \ 47 : 17 \\
10) & \ 10 : 13
\end{aligned}
}
\]