Get the answers you've been searching for with IDNLearn.com. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.
Sagot :
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.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.
