Answer :
Sure! Let's find the equivalent ratios step-by-step.
A ratio of [tex]\(a:b\)[/tex] is equivalent to another ratio [tex]\(c:d\)[/tex] if [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex]. This can also be checked using cross-multiplication: [tex]\(a \times d = b \times c\)[/tex].
Let's examine each pair:
1. Ratios [tex]\(5:6\)[/tex] and [tex]\(6:7\)[/tex]:
- Cross-multiplying gives: [tex]\(5 \times 7 = 35\)[/tex] and [tex]\(6 \times 6 = 36\)[/tex]
- Since [tex]\(35 \neq 36\)[/tex], these ratios are not equivalent.
2. Ratios [tex]\(36:35\)[/tex] and [tex]\(4:3\)[/tex]:
- Cross-multiplying gives: [tex]\(36 \times 3 = 108\)[/tex] and [tex]\(35 \times 4 = 140\)[/tex]
- Since [tex]\(108 \neq 140\)[/tex], these ratios are not equivalent.
3. Ratios [tex]\(2:3\)[/tex] and [tex]\(12:18\)[/tex]:
- Cross-multiplying gives: [tex]\(2 \times 18 = 36\)[/tex] and [tex]\(3 \times 12 = 36\)[/tex]
- Since [tex]\(36 = 36\)[/tex], these ratios are equivalent.
4. Ratios [tex]\(4:8\)[/tex] and [tex]\(20000:40000\)[/tex]:
- Cross-multiplying gives: [tex]\(4 \times 40000 = 160000\)[/tex] and [tex]\(8 \times 20000 = 160000\)[/tex]
- Since [tex]\(160000 = 160000\)[/tex], these ratios are equivalent.
5. Ratios [tex]\(45:18\)[/tex] and [tex]\(5:2\)[/tex]:
- Cross-multiplying gives: [tex]\(45 \times 2 = 90\)[/tex] and [tex]\(18 \times 5 = 90\)[/tex]
- Since [tex]\(90 = 90\)[/tex], these ratios are equivalent.
Therefore, the pairs that represent equivalent ratios are:
- [tex]\(2:3\)[/tex] and [tex]\(12:18\)[/tex]
- [tex]\(4:8\)[/tex] and [tex]\(20000:40000\)[/tex]
- [tex]\(45:18\)[/tex] and [tex]\(5:2\)[/tex]
So, the selected pairs are:
- [tex]\(3^\text{rd}~ pair\)[/tex]
- [tex]\(4^\text{th}~ pair\)[/tex]
- [tex]\(5^\text{th}~ pair\)[/tex]
These are the equivalent ratios.
A ratio of [tex]\(a:b\)[/tex] is equivalent to another ratio [tex]\(c:d\)[/tex] if [tex]\(\frac{a}{b} = \frac{c}{d}\)[/tex]. This can also be checked using cross-multiplication: [tex]\(a \times d = b \times c\)[/tex].
Let's examine each pair:
1. Ratios [tex]\(5:6\)[/tex] and [tex]\(6:7\)[/tex]:
- Cross-multiplying gives: [tex]\(5 \times 7 = 35\)[/tex] and [tex]\(6 \times 6 = 36\)[/tex]
- Since [tex]\(35 \neq 36\)[/tex], these ratios are not equivalent.
2. Ratios [tex]\(36:35\)[/tex] and [tex]\(4:3\)[/tex]:
- Cross-multiplying gives: [tex]\(36 \times 3 = 108\)[/tex] and [tex]\(35 \times 4 = 140\)[/tex]
- Since [tex]\(108 \neq 140\)[/tex], these ratios are not equivalent.
3. Ratios [tex]\(2:3\)[/tex] and [tex]\(12:18\)[/tex]:
- Cross-multiplying gives: [tex]\(2 \times 18 = 36\)[/tex] and [tex]\(3 \times 12 = 36\)[/tex]
- Since [tex]\(36 = 36\)[/tex], these ratios are equivalent.
4. Ratios [tex]\(4:8\)[/tex] and [tex]\(20000:40000\)[/tex]:
- Cross-multiplying gives: [tex]\(4 \times 40000 = 160000\)[/tex] and [tex]\(8 \times 20000 = 160000\)[/tex]
- Since [tex]\(160000 = 160000\)[/tex], these ratios are equivalent.
5. Ratios [tex]\(45:18\)[/tex] and [tex]\(5:2\)[/tex]:
- Cross-multiplying gives: [tex]\(45 \times 2 = 90\)[/tex] and [tex]\(18 \times 5 = 90\)[/tex]
- Since [tex]\(90 = 90\)[/tex], these ratios are equivalent.
Therefore, the pairs that represent equivalent ratios are:
- [tex]\(2:3\)[/tex] and [tex]\(12:18\)[/tex]
- [tex]\(4:8\)[/tex] and [tex]\(20000:40000\)[/tex]
- [tex]\(45:18\)[/tex] and [tex]\(5:2\)[/tex]
So, the selected pairs are:
- [tex]\(3^\text{rd}~ pair\)[/tex]
- [tex]\(4^\text{th}~ pair\)[/tex]
- [tex]\(5^\text{th}~ pair\)[/tex]
These are the equivalent ratios.
