To determine which of the statements is true, we can compute each one separately.
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Statement: \( \frac{4}{7} \cdot \frac{10}{3} < \frac{4}{7} \)
Computation:
\[ \frac{4}{7} \cdot \frac{10}{3} = \frac{4 \times 10}{7 \times 3} = \frac{40}{21} \]
To compare \( \frac{40}{21} \) and \( \frac{4}{7} \), we can cross-multiply:
\( 40 \times 7 = 280 \) and \( 4 \times 21 = 84 \).
Since \( 280 > 84 \),
\[ \frac{40}{21} > \frac{4}{7} \]
This statement is false. -
Statement: \( \frac{1}{10} \cdot \frac{5}{2} < \frac{1}{10} \)
Computation:
\[ \frac{1}{10} \cdot \frac{5}{2} = \frac{1 \times 5}{10 \times 2} = \frac{5}{20} = \frac{1}{4} \]
Compare \( \frac{1}{4} \) and \( \frac{1}{10} \) by cross-multiplying:
\( 1 \times 10 = 10 \) and \( 1 \times 4 = 4 \).
Since \( 10 > 4 \),
\[ \frac{1}{4} > \frac{1}{10} \]
This statement is false. -
Statement: \( \frac{11}{13} \cdot \frac{1}{8} < \frac{11}{13} \)
Computation:
\[ \frac{11}{13} \cdot \frac{1}{8} = \frac{11 \times 1}{13 \times 8} = \frac{11}{104} \]
Compare \( \frac{11}{104} \) and \( \frac{11}{13} \) by cross-multiplying:
\( 11 \times 13 = 143 \) and \( 11 \times 104 = 1144 \).
Since \( 1144 > 143 \),
\[ \frac{11}{104} < \frac{11}{13} \]
This statement is true. -
Statement: \( \frac{13}{6} \cdot \frac{4}{7} > \frac{13}{6} \)
Computation:
\[ \frac{13}{6} \cdot \frac{4}{7} = \frac{13 \times 4}{6 \times 7} = \frac{52}{42} = \frac{26}{21} \]
Compare \( \frac{26}{21} \) and \( \frac{13}{6} \) by cross-multiplying:
\( 26 \times 6 = 156 \) and \( 13 \times 21 = 273 \).
Since \( 156 < 273 \),
\[ \frac{26}{21} < \frac{13}{6} \]
This statement is false.
In conclusion, the only true statement is:
3: \( \frac{11}{13} \cdot \frac{1}{8} < \frac{11}{13} \)