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Sagot :
To solve the problem of finding an equivalent form of the given rational numbers having a common denominator, we'll follow these steps for each group of fractions:
1. Identify the denominators of the given fractions.
2. Find the Least Common Multiple (LCM) of these denominators to establish a common denominator.
3. Convert each fraction so that it has the common denominator by adjusting their numerators accordingly.
### Question (i): (3/4) and (5/12)
1. Identify the denominators: The denominators are 4 and 12.
2. Find the LCM of 4 and 12: The LCM of 4 and 12 is 12.
3. Convert the fractions:
- For [tex]\( \frac{3}{4} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)[/tex].
- For [tex]\( \frac{5}{12} \)[/tex], it already has a denominator of 12, so it remains [tex]\( \frac{5}{12} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{9}{12}, \frac{5}{12} \right) \][/tex]
### Question (ii): (2/3), (7/6), and (11/12)
1. Identify the denominators: The denominators are 3, 6, and 12.
2. Find the LCM of 3, 6, and 12: The LCM of 3, 6, and 12 is 12.
3. Convert the fractions:
- For [tex]\( \frac{2}{3} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)[/tex].
- For [tex]\( \frac{7}{6} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{7 \times 2}{6 \times 2} = \frac{14}{12} \)[/tex].
- For [tex]\( \frac{11}{12} \)[/tex], it already has a denominator of 12, so it remains [tex]\( \frac{11}{12} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{7}{6}\)[/tex], and [tex]\(\frac{11}{12}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{8}{12}, \frac{14}{12}, \frac{11}{12} \right) \][/tex]
### Question (iii): (5/7), (3/8), (9/14), and (20/21)
1. Identify the denominators: The denominators are 7, 8, 14, and 21.
2. Find the LCM of 7, 8, 14, and 21: The LCM of 7, 8, 14, and 21 is 168.
3. Convert the fractions:
- For [tex]\( \frac{5}{7} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{5 \times 24}{7 \times 24} = \frac{120}{168} \)[/tex].
- For [tex]\( \frac{3}{8} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{3 \times 21}{8 \times 21} = \frac{63}{168} \)[/tex].
- For [tex]\( \frac{9}{14} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{9 \times 12}{14 \times 12} = \frac{108}{168} \)[/tex].
- For [tex]\( \frac{20}{21} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{20 \times 8}{21 \times 8} = \frac{160}{168} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{5}{7}\)[/tex], [tex]\(\frac{3}{8}\)[/tex], [tex]\(\frac{9}{14}\)[/tex], and [tex]\(\frac{20}{21}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{120}{168}, \frac{63}{168}, \frac{108}{168}, \frac{160}{168} \right) \][/tex]
1. Identify the denominators of the given fractions.
2. Find the Least Common Multiple (LCM) of these denominators to establish a common denominator.
3. Convert each fraction so that it has the common denominator by adjusting their numerators accordingly.
### Question (i): (3/4) and (5/12)
1. Identify the denominators: The denominators are 4 and 12.
2. Find the LCM of 4 and 12: The LCM of 4 and 12 is 12.
3. Convert the fractions:
- For [tex]\( \frac{3}{4} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)[/tex].
- For [tex]\( \frac{5}{12} \)[/tex], it already has a denominator of 12, so it remains [tex]\( \frac{5}{12} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{9}{12}, \frac{5}{12} \right) \][/tex]
### Question (ii): (2/3), (7/6), and (11/12)
1. Identify the denominators: The denominators are 3, 6, and 12.
2. Find the LCM of 3, 6, and 12: The LCM of 3, 6, and 12 is 12.
3. Convert the fractions:
- For [tex]\( \frac{2}{3} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)[/tex].
- For [tex]\( \frac{7}{6} \)[/tex], the equivalent fraction with a denominator of 12 is [tex]\( \frac{7 \times 2}{6 \times 2} = \frac{14}{12} \)[/tex].
- For [tex]\( \frac{11}{12} \)[/tex], it already has a denominator of 12, so it remains [tex]\( \frac{11}{12} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{7}{6}\)[/tex], and [tex]\(\frac{11}{12}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{8}{12}, \frac{14}{12}, \frac{11}{12} \right) \][/tex]
### Question (iii): (5/7), (3/8), (9/14), and (20/21)
1. Identify the denominators: The denominators are 7, 8, 14, and 21.
2. Find the LCM of 7, 8, 14, and 21: The LCM of 7, 8, 14, and 21 is 168.
3. Convert the fractions:
- For [tex]\( \frac{5}{7} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{5 \times 24}{7 \times 24} = \frac{120}{168} \)[/tex].
- For [tex]\( \frac{3}{8} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{3 \times 21}{8 \times 21} = \frac{63}{168} \)[/tex].
- For [tex]\( \frac{9}{14} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{9 \times 12}{14 \times 12} = \frac{108}{168} \)[/tex].
- For [tex]\( \frac{20}{21} \)[/tex], the equivalent fraction with a denominator of 168 is [tex]\( \frac{20 \times 8}{21 \times 8} = \frac{160}{168} \)[/tex].
Thus, the equivalent form of the fractions [tex]\(\frac{5}{7}\)[/tex], [tex]\(\frac{3}{8}\)[/tex], [tex]\(\frac{9}{14}\)[/tex], and [tex]\(\frac{20}{21}\)[/tex] with a common denominator is:
[tex]\[ \left( \frac{120}{168}, \frac{63}{168}, \frac{108}{168}, \frac{160}{168} \right) \][/tex]
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