Let's compute each statement one by one:
A. \( \frac{1}{10} \cdot \frac{5}{2} < \frac{1}{10} \)
Calculating the left side:
\[ \frac{1}{10} \cdot \frac{5}{2} = \frac{1 \cdot 5}{10 \cdot 2} = \frac{5}{20} = \frac{1}{4} \]
Now we compare \( \frac{1}{4} < \frac{1}{10} \).
Since \( \frac{1}{4} = 0.25 \) and \( \frac{1}{10} = 0.1 \), \( 0.25 \) is not less than \( 0.1 \).
So A is false.
B. \( \frac{13}{6} \cdot \frac{4}{7} > \frac{13}{6} \)
Calculating the left side:
\[ \frac{13}{6} \cdot \frac{4}{7} = \frac{13 \cdot 4}{6 \cdot 7} = \frac{52}{42} = \frac{26}{21} \]
Now we compare \( \frac{26}{21} > \frac{13}{6} \):
Convert \( \frac{13}{6} \) to a fraction over 21:
\[ \frac{13}{6} = \frac{13 \cdot 21/13}{6 \cdot 21/13} = \frac{13 \cdot 21}{6 \cdot 21} = \frac{65}{42} \]
Now we compare:
Since \( 26/21 = 52/42 \) and \( \frac{65}{42} > \frac{52}{42} \), it shows \( \frac{26}{21} < \frac{13}{6} \).
So B is false.
C. \( \frac{11}{13} \cdot \frac{1}{8} < \frac{11}{13} \)
Calculating the left side:
\[ \frac{11}{13} \cdot \frac{1}{8} = \frac{11 \cdot 1}{13 \cdot 8} = \frac{11}{104} \]
Now we compare \( \frac{11}{104} < \frac{11}{13} \).
Convert \( \frac{11}{13} \) to a fraction over 104:
\[ \frac{11}{13} = \frac{11 \cdot 8}{13 \cdot 8} = \frac{88}{104} \]
Now we compare:
Since \( \frac{11}{104} < \frac{88}{104} \), this is indeed true.
So C is true.
D. \( \frac{4}{7} \cdot \frac{10}{3} < \frac{4}{7} \)
Calculating the left side:
\[ \frac{4}{7} \cdot \frac{10}{3} = \frac{4 \cdot 10}{7 \cdot 3} = \frac{40}{21} \]
Now we compare \( \frac{40}{21} < \frac{4}{7} \).
Convert \( \frac{4}{7} \) to a fraction over 21:
\[ \frac{4}{7} = \frac{4 \cdot 3}{7 \cdot 3} = \frac{12}{21} \]
Now we compare:
Since \( \frac{40}{21} > \frac{12}{21} \), this inequality is false.
So D is false.
In summary:
- A is false
- B is false
- C is true
- D is false
The only true statement is C: \( \frac{11}{13} \cdot \frac{1}{8} < \frac{11}{13} \).
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