Let's evaluate each of the statements one by one.
-
\( \frac{19}{20} \cdot \frac{19}{20} > \frac{19}{20} \):
- First, calculate \( \frac{19}{20} \cdot \frac{19}{20} = \frac{361}{400} \).
- Now, compare \( \frac{361}{400} \) with \( \frac{19}{20} = \frac{380}{400} \).
- Since \( \frac{361}{400} < \frac{380}{400} \), the statement is false.
-
\( 1 \frac{2}{3} \cdot 2 \frac{1}{3} < 1 \frac{2}{3} \):
- Convert mixed numbers to improper fractions:
- \( 1 \frac{2}{3} = \frac{5}{3} \)
- \( 2 \frac{1}{3} = \frac{7}{3} \)
- Now compute the product:
- \( \frac{5}{3} \cdot \frac{7}{3} = \frac{35}{9} \).
- Convert \( \frac{35}{9} \) and \( 1 \frac{2}{3} = \frac{5}{3} = \frac{15}{9} \).
- Since \( \frac{35}{9} > \frac{15}{9} \), the statement is false.
- Convert mixed numbers to improper fractions:
-
\( \frac{11}{10} \cdot \frac{10}{11} > \frac{11}{10} \):
- Calculate \( \frac{11}{10} \cdot \frac{10}{11} = 1 \).
- Compare \( 1 \) with \( \frac{11}{10} \).
- Since \( 1 < \frac{11}{10} \), the statement is false.
-
\( \frac{15}{6} \cdot 3 \frac{1}{4} > \frac{15}{6} \):
- Convert \( 3 \frac{1}{4} \) to an improper fraction:
- \( 3 \frac{1}{4} = \frac{13}{4} \).
- Now compute the product:
- \( \frac{15}{6} \cdot \frac{13}{4} = \frac{195}{24} \).
- Convert \( \frac{195}{24} \) and \( \frac{15}{6} = \frac{60}{24} \).
- Since \( \frac{195}{24} > \frac{60}{24} \), the statement is true.
- Convert \( 3 \frac{1}{4} \) to an improper fraction:
Conclusion: The only true statement is: 4. \( \frac{15}{6} \cdot 3 \frac{1}{4} > \frac{15}{6} \).
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