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Sagot :
To determine which of the given statements is true, we need to compare each pair of fractions. Let’s go through them one by one:
1. First statement: [tex]\(\frac{5}{6} > \frac{11}{12}\)[/tex]
To compare [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{11}{12}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 6 and 12 is 12.
- Convert [tex]\(\frac{5}{6}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \][/tex]
- Now compare [tex]\(\frac{10}{12}\)[/tex] with [tex]\(\frac{11}{12}\)[/tex]:
[tex]\[ \frac{10}{12} < \frac{11}{12} \][/tex]
Thus, [tex]\(\frac{5}{6} > \frac{11}{12}\)[/tex] is false.
2. Second statement: [tex]\(\frac{9}{15} < \frac{4}{5}\)[/tex]
To compare [tex]\(\frac{9}{15}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 15 and 5 is 15.
- Convert [tex]\(\frac{4}{5}\)[/tex] to an equivalent fraction with a denominator of 15:
[tex]\[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \][/tex]
- Now compare [tex]\(\frac{9}{15}\)[/tex] with [tex]\(\frac{12}{15}\)[/tex]:
[tex]\[ \frac{9}{15} < \frac{12}{15} \][/tex]
Thus, [tex]\(\frac{9}{15} < \frac{4}{5}\)[/tex] is true.
3. Third statement: [tex]\(\frac{18}{27} = \frac{1}{3}\)[/tex]
To compare [tex]\(\frac{18}{27}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], we can simplify the fraction [tex]\(\frac{18}{27}\)[/tex]:
- Simplify [tex]\(\frac{18}{27}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 9:
[tex]\[ \frac{18}{27} = \frac{18 \div 9}{27 \div 9} = \frac{2}{3} \][/tex]
- Now compare [tex]\(\frac{2}{3}\)[/tex] with [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{2}{3} \neq \frac{1}{3} \][/tex]
Thus, [tex]\(\frac{18}{27} = \frac{1}{3}\)[/tex] is false.
4. Fourth statement: [tex]\(\frac{3}{4} < \frac{2}{3}\)[/tex]
To compare [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 4 and 3 is 12.
- Convert [tex]\(\frac{3}{4}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
- Convert [tex]\(\frac{2}{3}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \][/tex]
- Now compare [tex]\(\frac{9}{12}\)[/tex] with [tex]\(\frac{8}{12}\)[/tex]:
[tex]\[ \frac{9}{12} > \frac{8}{12} \][/tex]
Thus, [tex]\(\frac{3}{4} < \frac{2}{3}\)[/tex] is false.
After evaluating all the statements, the only true statement is the second one:
[tex]\(\boxed{2}\)[/tex]
1. First statement: [tex]\(\frac{5}{6} > \frac{11}{12}\)[/tex]
To compare [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{11}{12}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 6 and 12 is 12.
- Convert [tex]\(\frac{5}{6}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \][/tex]
- Now compare [tex]\(\frac{10}{12}\)[/tex] with [tex]\(\frac{11}{12}\)[/tex]:
[tex]\[ \frac{10}{12} < \frac{11}{12} \][/tex]
Thus, [tex]\(\frac{5}{6} > \frac{11}{12}\)[/tex] is false.
2. Second statement: [tex]\(\frac{9}{15} < \frac{4}{5}\)[/tex]
To compare [tex]\(\frac{9}{15}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 15 and 5 is 15.
- Convert [tex]\(\frac{4}{5}\)[/tex] to an equivalent fraction with a denominator of 15:
[tex]\[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \][/tex]
- Now compare [tex]\(\frac{9}{15}\)[/tex] with [tex]\(\frac{12}{15}\)[/tex]:
[tex]\[ \frac{9}{15} < \frac{12}{15} \][/tex]
Thus, [tex]\(\frac{9}{15} < \frac{4}{5}\)[/tex] is true.
3. Third statement: [tex]\(\frac{18}{27} = \frac{1}{3}\)[/tex]
To compare [tex]\(\frac{18}{27}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], we can simplify the fraction [tex]\(\frac{18}{27}\)[/tex]:
- Simplify [tex]\(\frac{18}{27}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 9:
[tex]\[ \frac{18}{27} = \frac{18 \div 9}{27 \div 9} = \frac{2}{3} \][/tex]
- Now compare [tex]\(\frac{2}{3}\)[/tex] with [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{2}{3} \neq \frac{1}{3} \][/tex]
Thus, [tex]\(\frac{18}{27} = \frac{1}{3}\)[/tex] is false.
4. Fourth statement: [tex]\(\frac{3}{4} < \frac{2}{3}\)[/tex]
To compare [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex], we should convert them to the same denominator. The least common denominator (LCD) of 4 and 3 is 12.
- Convert [tex]\(\frac{3}{4}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
- Convert [tex]\(\frac{2}{3}\)[/tex] to an equivalent fraction with a denominator of 12:
[tex]\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \][/tex]
- Now compare [tex]\(\frac{9}{12}\)[/tex] with [tex]\(\frac{8}{12}\)[/tex]:
[tex]\[ \frac{9}{12} > \frac{8}{12} \][/tex]
Thus, [tex]\(\frac{3}{4} < \frac{2}{3}\)[/tex] is false.
After evaluating all the statements, the only true statement is the second one:
[tex]\(\boxed{2}\)[/tex]
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