Answer :
Let's evaluate each statement step by step.
### Statement (i)
[tex]\[ \frac{-3}{8} \geq 0 \][/tex]
First, consider the fraction [tex]\(\frac{-3}{8}\)[/tex]:
- The numerator is -3, which is negative.
- The denominator is 8, which is positive.
- A negative number divided by a positive number is negative.
Since [tex]\(\frac{-3}{8}\)[/tex] is negative, it is not greater than or equal to 0. Therefore, the statement [tex]\(\frac{-3}{8} \geq 0\)[/tex] is False.
To correct the statement:
[tex]\[ \frac{-3}{8} < 0 \][/tex]
### Statement (ii)
[tex]\[ \text{If } \frac{1}{2} > \frac{1}{3} \text{ then } \frac{1}{2} - \frac{1}{3} \text{ is positive.} \][/tex]
First, verify that [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex]:
- To compare [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], convert them to a common denominator.
- The least common denominator of 2 and 3 is 6.
Convert each fraction:
[tex]\[ \frac{1}{2} = \frac{3}{6} \][/tex]
[tex]\[ \frac{1}{3} = \frac{2}{6} \][/tex]
Clearly, [tex]\(\frac{3}{6} > \frac{2}{6}\)[/tex], so [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex] is true.
Next, verify the condition:
[tex]\[ \frac{1}{2} - \frac{1}{3} \][/tex]
To subtract these fractions, again convert to a common denominator:
[tex]\[ \frac{1}{2} = \frac{3}{6} \][/tex]
[tex]\[ \frac{1}{3} = \frac{2}{6} \][/tex]
Now subtract:
[tex]\[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] is positive, the statement [tex]\(\frac{1}{2} - \frac{1}{3}\)[/tex] is positive is true.
Therefore, the statement in (ii) holds:
[tex]\[ \frac{1}{2} > \frac{1}{3} \text{ implies } \frac{1}{2} - \frac{1}{3} \text{ is positive.} \][/tex]
Thus, this statement is True.
### Summary:
- Statement (i): False. Corrected statement: [tex]\(\frac{-3}{8} < 0\)[/tex]
- Statement (ii): True
The results of the evaluation are:
[tex]\[ \text{(i) False} \][/tex]
[tex]\[ \text{(ii) True} \][/tex]
### Statement (i)
[tex]\[ \frac{-3}{8} \geq 0 \][/tex]
First, consider the fraction [tex]\(\frac{-3}{8}\)[/tex]:
- The numerator is -3, which is negative.
- The denominator is 8, which is positive.
- A negative number divided by a positive number is negative.
Since [tex]\(\frac{-3}{8}\)[/tex] is negative, it is not greater than or equal to 0. Therefore, the statement [tex]\(\frac{-3}{8} \geq 0\)[/tex] is False.
To correct the statement:
[tex]\[ \frac{-3}{8} < 0 \][/tex]
### Statement (ii)
[tex]\[ \text{If } \frac{1}{2} > \frac{1}{3} \text{ then } \frac{1}{2} - \frac{1}{3} \text{ is positive.} \][/tex]
First, verify that [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex]:
- To compare [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex], convert them to a common denominator.
- The least common denominator of 2 and 3 is 6.
Convert each fraction:
[tex]\[ \frac{1}{2} = \frac{3}{6} \][/tex]
[tex]\[ \frac{1}{3} = \frac{2}{6} \][/tex]
Clearly, [tex]\(\frac{3}{6} > \frac{2}{6}\)[/tex], so [tex]\(\frac{1}{2} > \frac{1}{3}\)[/tex] is true.
Next, verify the condition:
[tex]\[ \frac{1}{2} - \frac{1}{3} \][/tex]
To subtract these fractions, again convert to a common denominator:
[tex]\[ \frac{1}{2} = \frac{3}{6} \][/tex]
[tex]\[ \frac{1}{3} = \frac{2}{6} \][/tex]
Now subtract:
[tex]\[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \][/tex]
Since [tex]\(\frac{1}{6}\)[/tex] is positive, the statement [tex]\(\frac{1}{2} - \frac{1}{3}\)[/tex] is positive is true.
Therefore, the statement in (ii) holds:
[tex]\[ \frac{1}{2} > \frac{1}{3} \text{ implies } \frac{1}{2} - \frac{1}{3} \text{ is positive.} \][/tex]
Thus, this statement is True.
### Summary:
- Statement (i): False. Corrected statement: [tex]\(\frac{-3}{8} < 0\)[/tex]
- Statement (ii): True
The results of the evaluation are:
[tex]\[ \text{(i) False} \][/tex]
[tex]\[ \text{(ii) True} \][/tex]
