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Sagot :
To determine which set of fractions is ordered from greatest to least, we'll compare the given fractions: [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{3}{8}\)[/tex], and [tex]\(\frac{5}{12}\)[/tex].
### Step-by-Step Solution:
1. Identify the fractions to be compared:
- [tex]\(\frac{2}{3}\)[/tex]
- [tex]\(\frac{3}{8}\)[/tex]
- [tex]\(\frac{5}{12}\)[/tex]
2. Order the fractions from greatest to least:
Comparing [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex]:
- Convert them to have a common denominator (let's use 12):
[tex]\[\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\][/tex]
[tex]\[\frac{5}{12} = \frac{5}{12}\][/tex]
- Compare the numerators: [tex]\(8 > 5\)[/tex]
- So, [tex]\(\frac{2}{3} > \frac{5}{12}\)[/tex]
Comparing [tex]\(\frac{5}{12}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]:
- Convert them to have a common denominator (let's use 24):
[tex]\[\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}\][/tex]
[tex]\[\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\][/tex]
- Compare the numerators: [tex]\(10 > 9\)[/tex]
- So, [tex]\(\frac{5}{12} > \frac{3}{8}\)[/tex]
Comparing [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]:
- Convert them to have a common denominator (let's use 24):
[tex]\[\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\][/tex]
[tex]\[\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\][/tex]
- Compare the numerators: [tex]\(16 > 9\)[/tex]
- So, [tex]\(\frac{2}{3} > \frac{3}{8}\)[/tex]
3. Order the fractions:
- Based on the comparisons above:
- [tex]\(\frac{2}{3}\)[/tex] is the greatest.
- [tex]\(\frac{5}{12}\)[/tex] is the next largest.
- [tex]\(\frac{3}{8}\)[/tex] is the smallest.
So, the ordered set from greatest to least is: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex]
4. Match this order with the given options:
- Option A: [tex]\(\frac{2}{3}, \frac{3}{8}, \frac{5}{12}\)[/tex]
- Option B: [tex]\(\frac{5}{12}, \frac{3}{8}, \frac{2}{3}\)[/tex]
- Option C: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex]
- Option D: [tex]\(\frac{3}{8} \cdot \frac{2}{3}, \frac{5}{12}\)[/tex] (This involves a product, isn't relevant for comparison)
Therefore, the correct answer is option C: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex].
### Step-by-Step Solution:
1. Identify the fractions to be compared:
- [tex]\(\frac{2}{3}\)[/tex]
- [tex]\(\frac{3}{8}\)[/tex]
- [tex]\(\frac{5}{12}\)[/tex]
2. Order the fractions from greatest to least:
Comparing [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex]:
- Convert them to have a common denominator (let's use 12):
[tex]\[\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\][/tex]
[tex]\[\frac{5}{12} = \frac{5}{12}\][/tex]
- Compare the numerators: [tex]\(8 > 5\)[/tex]
- So, [tex]\(\frac{2}{3} > \frac{5}{12}\)[/tex]
Comparing [tex]\(\frac{5}{12}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]:
- Convert them to have a common denominator (let's use 24):
[tex]\[\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}\][/tex]
[tex]\[\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\][/tex]
- Compare the numerators: [tex]\(10 > 9\)[/tex]
- So, [tex]\(\frac{5}{12} > \frac{3}{8}\)[/tex]
Comparing [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]:
- Convert them to have a common denominator (let's use 24):
[tex]\[\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\][/tex]
[tex]\[\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\][/tex]
- Compare the numerators: [tex]\(16 > 9\)[/tex]
- So, [tex]\(\frac{2}{3} > \frac{3}{8}\)[/tex]
3. Order the fractions:
- Based on the comparisons above:
- [tex]\(\frac{2}{3}\)[/tex] is the greatest.
- [tex]\(\frac{5}{12}\)[/tex] is the next largest.
- [tex]\(\frac{3}{8}\)[/tex] is the smallest.
So, the ordered set from greatest to least is: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex]
4. Match this order with the given options:
- Option A: [tex]\(\frac{2}{3}, \frac{3}{8}, \frac{5}{12}\)[/tex]
- Option B: [tex]\(\frac{5}{12}, \frac{3}{8}, \frac{2}{3}\)[/tex]
- Option C: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex]
- Option D: [tex]\(\frac{3}{8} \cdot \frac{2}{3}, \frac{5}{12}\)[/tex] (This involves a product, isn't relevant for comparison)
Therefore, the correct answer is option C: [tex]\(\frac{2}{3}, \frac{5}{12}, \frac{3}{8}\)[/tex].
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