Which of the following statements are true?
A. 1/2 < 5/12
B. 2/3 < 5/12
C. 1/4 < 5/12
D. 5/8 < 5/12
The Correct Answer and Explanation is :
The correct answer is: C is true.
To determine which of the statements are true, we will compare each fraction to ( \frac{5}{12} ).
Statement A: ( \frac{1}{2} < \frac{5}{12} )
To compare these fractions, we can convert ( \frac{1}{2} ) to twelfths:
[
\frac{1}{2} = \frac{6}{12}
]
Now we compare ( \frac{6}{12} ) and ( \frac{5}{12} ):
[
\frac{6}{12} > \frac{5}{12}
]
Conclusion: Statement A is false.
Statement B: ( \frac{2}{3} < \frac{5}{12} )
Convert ( \frac{2}{3} ) to twelfths:
[
\frac{2}{3} = \frac{8}{12}
]
Now we compare ( \frac{8}{12} ) and ( \frac{5}{12} ):
[
\frac{8}{12} > \frac{5}{12}
]
Conclusion: Statement B is false.
Statement C: ( \frac{1}{4} < \frac{5}{12} )
Convert ( \frac{1}{4} ) to twelfths:
[
\frac{1}{4} = \frac{3}{12}
]
Now we compare ( \frac{3}{12} ) and ( \frac{5}{12} ):
[
\frac{3}{12} < \frac{5}{12}
]
Conclusion: Statement C is true.
Statement D: ( \frac{5}{8} < \frac{5}{12} )
Convert ( \frac{5}{8} ) to twelfths:
[
\frac{5}{8} = \frac{15}{24} \quad \text{(or, finding a common denominator)} \quad \frac{5}{8} = \frac{15}{24} \quad \text{and} \quad \frac{5}{12} = \frac{10}{24}
]
Now we compare ( \frac{15}{24} ) and ( \frac{10}{24} ):
[
\frac{15}{24} > \frac{10}{24}
]
Conclusion: Statement D is false.
Summary
Only Statement C, ( \frac{1}{4} < \frac{5}{12} ), is true. The other statements (A, B, and D) are false because ( \frac{1}{2} ) and ( \frac{2}{3} ) are greater than ( \frac{5}{12} ), and ( \frac{5}{8} ) is also greater than ( \frac{5}{12} ).
This exercise demonstrates how to compare fractions by finding a common denominator or converting them to equivalent fractions, facilitating clear comparisons.