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Let's express the given ratios in terms of their lowest common denominators (LCD) and compare them. Here’s a detailed, step-by-step solution for each pair of ratios:
### a) [tex]\(1:2\)[/tex] and [tex]\(3:4\)[/tex]
We need to express these ratios with the same denominator for comparison.
1. Find the least common multiple (LCM) of the denominators 2 and 4, which is 4.
2. Convert each ratio to have a denominator of 4.
- [tex]\(1:2\)[/tex] becomes [tex]\(2:4\)[/tex] (since [tex]\(1 \times 2 = 2\)[/tex]).
- [tex]\(3:4\)[/tex] remains [tex]\(3:4\)[/tex].
So, [tex]\(1:2\)[/tex] and [tex]\(3:4\)[/tex] expressed with the common denominator 4 are:
[tex]\[ \frac{2}{4} \text{ and } \frac{3}{4} \][/tex]
### b) [tex]\(2:5\)[/tex] and [tex]\(1:3\)[/tex]
1. Find the LCM of the denominators 5 and 3, which is 15.
2. Convert each ratio to have a denominator of 15.
- [tex]\(2:5\)[/tex] becomes [tex]\(6:15\)[/tex] (since [tex]\(2 \times 3 = 6\)[/tex]).
- [tex]\(1:3\)[/tex] becomes [tex]\(5:15\)[/tex] (since [tex]\(1 \times 5 = 5\)[/tex]).
So, [tex]\(2:5\)[/tex] and [tex]\(1:3\)[/tex] expressed with the common denominator 15 are:
[tex]\[ \frac{6}{15} \text{ and } \frac{5}{15} \][/tex]
### c) [tex]\(2:3\)[/tex] and [tex]\(5:6\)[/tex]
1. Find the LCM of the denominators 3 and 6, which is 6.
2. Convert each ratio to have a denominator of 6.
- [tex]\(2:3\)[/tex] becomes [tex]\(4:6\)[/tex] (since [tex]\(2 \times 2 = 4\)[/tex]).
- [tex]\(5:6\)[/tex] remains [tex]\(5:6\)[/tex].
So, [tex]\(2:3\)[/tex] and [tex]\(5:6\)[/tex] expressed with the common denominator 6 are:
[tex]\[ \frac{4}{6} \text{ and } \frac{5}{6} \][/tex]
### d) [tex]\(4:7\)[/tex] and [tex]\(3:8\)[/tex]
1. Find the LCM of the denominators 7 and 8, which is 56.
2. Convert each ratio to have a denominator of 56.
- [tex]\(4:7\)[/tex] becomes [tex]\(32:56\)[/tex] (since [tex]\(4 \times 8 = 32\)[/tex]).
- [tex]\(3:8\)[/tex] becomes [tex]\(21:56\)[/tex] (since [tex]\(3 \times 7 = 21\)[/tex]).
So, [tex]\(4:7\)[/tex] and [tex]\(3:8\)[/tex] expressed with the common denominator 56 are:
[tex]\[ \frac{32}{56} \text{ and } \frac{21}{56} \][/tex]
### e) [tex]\(7:9\)[/tex] and [tex]\(5:8\)[/tex]
1. Find the LCM of the denominators 9 and 8, which is 72.
2. Convert each ratio to have a denominator of 72.
- [tex]\(7:9\)[/tex] becomes [tex]\(56:72\)[/tex] (since [tex]\(7 \times 8 = 56\)[/tex]).
- [tex]\(5:8\)[/tex] becomes [tex]\(45:72\)[/tex] (since [tex]\(5 \times 9 = 45\)[/tex]).
So, [tex]\(7:9\)[/tex] and [tex]\(5:8\)[/tex] expressed with the common denominator 72 are:
[tex]\[ \frac{56}{72} \text{ and } \frac{45}{72} \][/tex]
### f) [tex]\(7:12\)[/tex] and [tex]\(4:9\)[/tex]
1. Find the LCM of the denominators 12 and 9, which is 36.
2. Convert each ratio to have a denominator of 36.
- [tex]\(7:12\)[/tex] becomes [tex]\(21:36\)[/tex] (since [tex]\(7 \times 3 = 21\)[/tex]).
- [tex]\(4:9\)[/tex] becomes [tex]\(16:36\)[/tex] (since [tex]\(4 \times 4 = 16\)[/tex]).
So, [tex]\(7:12\)[/tex] and [tex]\(4:9\)[/tex] expressed with the common denominator 36 are:
[tex]\[ \frac{21}{36} \text{ and } \frac{16}{36} \][/tex]
By expressing the ratios in the lowest common denominators, we can easily compare the numerators directly.
### a) [tex]\(1:2\)[/tex] and [tex]\(3:4\)[/tex]
We need to express these ratios with the same denominator for comparison.
1. Find the least common multiple (LCM) of the denominators 2 and 4, which is 4.
2. Convert each ratio to have a denominator of 4.
- [tex]\(1:2\)[/tex] becomes [tex]\(2:4\)[/tex] (since [tex]\(1 \times 2 = 2\)[/tex]).
- [tex]\(3:4\)[/tex] remains [tex]\(3:4\)[/tex].
So, [tex]\(1:2\)[/tex] and [tex]\(3:4\)[/tex] expressed with the common denominator 4 are:
[tex]\[ \frac{2}{4} \text{ and } \frac{3}{4} \][/tex]
### b) [tex]\(2:5\)[/tex] and [tex]\(1:3\)[/tex]
1. Find the LCM of the denominators 5 and 3, which is 15.
2. Convert each ratio to have a denominator of 15.
- [tex]\(2:5\)[/tex] becomes [tex]\(6:15\)[/tex] (since [tex]\(2 \times 3 = 6\)[/tex]).
- [tex]\(1:3\)[/tex] becomes [tex]\(5:15\)[/tex] (since [tex]\(1 \times 5 = 5\)[/tex]).
So, [tex]\(2:5\)[/tex] and [tex]\(1:3\)[/tex] expressed with the common denominator 15 are:
[tex]\[ \frac{6}{15} \text{ and } \frac{5}{15} \][/tex]
### c) [tex]\(2:3\)[/tex] and [tex]\(5:6\)[/tex]
1. Find the LCM of the denominators 3 and 6, which is 6.
2. Convert each ratio to have a denominator of 6.
- [tex]\(2:3\)[/tex] becomes [tex]\(4:6\)[/tex] (since [tex]\(2 \times 2 = 4\)[/tex]).
- [tex]\(5:6\)[/tex] remains [tex]\(5:6\)[/tex].
So, [tex]\(2:3\)[/tex] and [tex]\(5:6\)[/tex] expressed with the common denominator 6 are:
[tex]\[ \frac{4}{6} \text{ and } \frac{5}{6} \][/tex]
### d) [tex]\(4:7\)[/tex] and [tex]\(3:8\)[/tex]
1. Find the LCM of the denominators 7 and 8, which is 56.
2. Convert each ratio to have a denominator of 56.
- [tex]\(4:7\)[/tex] becomes [tex]\(32:56\)[/tex] (since [tex]\(4 \times 8 = 32\)[/tex]).
- [tex]\(3:8\)[/tex] becomes [tex]\(21:56\)[/tex] (since [tex]\(3 \times 7 = 21\)[/tex]).
So, [tex]\(4:7\)[/tex] and [tex]\(3:8\)[/tex] expressed with the common denominator 56 are:
[tex]\[ \frac{32}{56} \text{ and } \frac{21}{56} \][/tex]
### e) [tex]\(7:9\)[/tex] and [tex]\(5:8\)[/tex]
1. Find the LCM of the denominators 9 and 8, which is 72.
2. Convert each ratio to have a denominator of 72.
- [tex]\(7:9\)[/tex] becomes [tex]\(56:72\)[/tex] (since [tex]\(7 \times 8 = 56\)[/tex]).
- [tex]\(5:8\)[/tex] becomes [tex]\(45:72\)[/tex] (since [tex]\(5 \times 9 = 45\)[/tex]).
So, [tex]\(7:9\)[/tex] and [tex]\(5:8\)[/tex] expressed with the common denominator 72 are:
[tex]\[ \frac{56}{72} \text{ and } \frac{45}{72} \][/tex]
### f) [tex]\(7:12\)[/tex] and [tex]\(4:9\)[/tex]
1. Find the LCM of the denominators 12 and 9, which is 36.
2. Convert each ratio to have a denominator of 36.
- [tex]\(7:12\)[/tex] becomes [tex]\(21:36\)[/tex] (since [tex]\(7 \times 3 = 21\)[/tex]).
- [tex]\(4:9\)[/tex] becomes [tex]\(16:36\)[/tex] (since [tex]\(4 \times 4 = 16\)[/tex]).
So, [tex]\(7:12\)[/tex] and [tex]\(4:9\)[/tex] expressed with the common denominator 36 are:
[tex]\[ \frac{21}{36} \text{ and } \frac{16}{36} \][/tex]
By expressing the ratios in the lowest common denominators, we can easily compare the numerators directly.
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