Are the Ratios Equivalent? Math worksheet for students to determine if pairs of ratios are equivalent.

To determine whether the given ratios are equivalent, we need to simplify each ratio and compare them. Two ratios are equivalent if they can be reduced to the same simplest form.

Step-by-Step Solution:

#### 1. 5:20 and 1:4
- Simplify \(5:20\):
\[
\frac{5}{20} = \frac{5 \div 5}{20 \div 5} = \frac{1}{4}
\]
- The simplified form of \(5:20\) is \(1:4\).
- Compare: \(1:4\) and \(1:4\).
- Answer: Yes

#### 2. 30:10 and 10:30
- Simplify \(30:10\):
\[
\frac{30}{10} = \frac{30 \div 10}{10 \div 10} = \frac{3}{1} = 3:1
\]
- Simplify \(10:30\):
\[
\frac{10}{30} = \frac{10 \div 10}{30 \div 10} = \frac{1}{3} = 1:3
\]
- Compare: \(3:1\) and \(1:3\).
- Answer: No

#### 3. 2:12 and 12:60
- Simplify \(2:12\):
\[
\frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6} = 1:6
\]
- Simplify \(12:60\):
\[
\frac{12}{60} = \frac{12 \div 12}{60 \div 12} = \frac{1}{5} = 1:5
\]
- Compare: \(1:6\) and \(1:5\).
- Answer: No

#### 4. 50:100 and 25:50
- Simplify \(50:100\):
\[
\frac{50}{100} = \frac{50 \div 50}{100 \div 50} = \frac{1}{2} = 1:2
\]
- Simplify \(25:50\):
\[
\frac{25}{50} = \frac{25 \div 25}{50 \div 25} = \frac{1}{2} = 1:2
\]
- Compare: \(1:2\) and \(1:2\).
- Answer: Yes

#### 5. 10:20 and 100:300
- Simplify \(10:20\):
\[
\frac{10}{20} = \frac{10 \div 10}{20 \div 10} = \frac{1}{2} = 1:2
\]
- Simplify \(100:300\):
\[
\frac{100}{300} = \frac{100 \div 100}{300 \div 100} = \frac{1}{3} = 1:3
\]
- Compare: \(1:2\) and \(1:3\).
- Answer: No

#### 6. 4:16 and 15:60
- Simplify \(4:16\):
\[
\frac{4}{16} = \frac{4 \div 4}{16 \div 4} = \frac{1}{4} = 1:4
\]
- Simplify \(15:60\):
\[
\frac{15}{60} = \frac{15 \div 15}{60 \div 15} = \frac{1}{4} = 1:4
\]
- Compare: \(1:4\) and \(1:4\).
- Answer: Yes

#### 7. 5:25 and 4:18
- Simplify \(5:25\):
\[
\frac{5}{25} = \frac{5 \div 5}{25 \div 5} = \frac{1}{5} = 1:5
\]
- Simplify \(4:18\):
\[
\frac{4}{18} = \frac{4 \div 2}{18 \div 2} = \frac{2}{9} = 2:9
\]
- Compare: \(1:5\) and \(2:9\).
- Answer: No

#### 8. 10:60 and 6:36
- Simplify \(10:60\):
\[
\frac{10}{60} = \frac{10 \div 10}{60 \div 10} = \frac{1}{6} = 1:6
\]
- Simplify \(6:36\):
\[
\frac{6}{36} = \frac{6 \div 6}{36 \div 6} = \frac{1}{6} = 1:6
\]
- Compare: \(1:6\) and \(1:6\).
- Answer: Yes

#### 9. 3:24 and 8:64
- Simplify \(3:24\):
\[
\frac{3}{24} = \frac{3 \div 3}{24 \div 3} = \frac{1}{8} = 1:8
\]
- Simplify \(8:64\):
\[
\frac{8}{64} = \frac{8 \div 8}{64 \div 8} = \frac{1}{8} = 1:8
\]
- Compare: \(1:8\) and \(1:8\).
- Answer: Yes

#### 10. 10:50 and 100:450
- Simplify \(10:50\):
\[
\frac{10}{50} = \frac{10 \div 10}{50 \div 10} = \frac{1}{5} = 1:5
\]
- Simplify \(100:450\):
\[
\frac{100}{450} = \frac{100 \div 50}{450 \div 50} = \frac{2}{9} = 2:9
\]
- Compare: \(1:5\) and \(2:9\).
- Answer: No

#### 11. 3:2 and 30:22
- Simplify \(3:2\):
\[
\frac{3}{2} = 3:2 \quad (\text{already in simplest form})
\]
- Simplify \(30:22\):
\[
\frac{30}{22} = \frac{30 \div 2}{22 \div 2} = \frac{15}{11} = 15:11
\]
- Compare: \(3:2\) and \(15:11\).
- Answer: No

#### 12. 3:5 and 21:35
- Simplify \(3:5\):
\[
\frac{3}{5} = 3:5 \quad (\text{already in simplest form})
\]
- Simplify \(21:35\):
\[
\frac{21}{35} = \frac{21 \div 7}{35 \div 7} = \frac{3}{5} = 3:5
\]
- Compare: \(3:5\) and \(3:5\).
- Answer: Yes

#### 13. 10:25 and 30:75
- Simplify \(10:25\):
\[
\frac{10}{25} = \frac{10 \div 5}{25 \div 5} = \frac{2}{5} = 2:5
\]
- Simplify \(30:75\):
\[
\frac{30}{75} = \frac{30 \div 15}{75 \div 15} = \frac{2}{5} = 2:5
\]
- Compare: \(2:5\) and \(2:5\).
- Answer: Yes

#### 14. 10:45 and 15:60
- Simplify \(10:45\):
\[
\frac{10}{45} = \frac{10 \div 5}{45 \div 5} = \frac{2}{9} = 2:9
\]
- Simplify \(15:60\):
\[
\frac{15}{60} = \frac{15 \div 15}{60 \div 15} = \frac{1}{4} = 1:4
\]
- Compare: \(2:9\) and \(1:4\).
- Answer: No

Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{Yes} & \text{No} \\
\text{No} & \text{Yes} \\
\text{No} & \text{Yes} \\
\text{No} & \text{Yes} \\
\text{Yes} & \text{No} \\
\text{No} & \text{Yes} \\
\text{Yes} & \text{No} \\
\end{array}
}
\]