To determine which two ratios are equivalent to [tex]\(10:12\)[/tex], we can follow a systematic approach to compare the given ratios.
First, let's simplify the initial ratio [tex]\(10:12\)[/tex]:
### Step 1: Simplify the Ratio
[tex]\[ \frac{10}{12} \][/tex]
To simplify [tex]\( \frac{10}{12} \)[/tex], we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[ \frac{10 \div 2}{12 \div 2} = \frac{5}{6} \][/tex]
So [tex]\(10:12\)[/tex] simplifies to [tex]\(5:6\)[/tex].
### Step 2: Compare each option with simplified ratio [tex]\(5:6\)[/tex]
1. Option A: [tex]\(11:13\)[/tex] and [tex]\(20:24\)[/tex]
- Simplify [tex]\(20:24\)[/tex]:
[tex]\[ \frac{20}{24} \][/tex]
Divide both the numerator and the denominator by their GCD, which is 4:
[tex]\[ \frac{20 \div 4}{24 \div 4} = \frac{5}{6} \][/tex]
This matches with the simplified ratio [tex]\(5:6\)[/tex].
2. Option B: [tex]\(10:12\)[/tex] and [tex]\(11:13\)[/tex]
- [tex]\(10:12\)[/tex] simplifies to [tex]\(5:6\)[/tex], as shown earlier.
- So half of this option is correct.
3. Option C: [tex]\(5:6\)[/tex] and [tex]\(20:24\)[/tex]
- [tex]\(5:6\)[/tex] is already simplified and matches exactly.
- [tex]\(20:24\)[/tex] simplifies to [tex]\(5:6\)[/tex], as shown earlier.
- Both ratios in this option match with the simplified ratio [tex]\(5:6\)[/tex].
4. Option D: [tex]\(5:6\)[/tex] and [tex]\(11:13\)[/tex]
- [tex]\(5:6\)[/tex] is already simplified and matches exactly.
- So half of this option is correct.
### Step 3: Conclusion
Based on these comparisons, Option C [tex]\((5:6 \text{ and } 20:24)\)[/tex] contains both ratios that are equivalent to the initial ratio [tex]\(10:12\)[/tex].
Therefore, the correct answer is:
(C) [tex]\(5:6\)[/tex] and [tex]\(20:24\)[/tex]
