Answered

Which of the following is a true statement?


A$\frac{8}{16}=\frac{1}{4}$
B$\frac{11}{15}<\frac{4}{5}$
C$\frac{3}{4}<\frac{4}{6}$
D$\frac{5}{6}>\frac{10}{12}$

Verified Answer :

  • Simplify \(\frac{8}{16}\) to \(\frac{1}{2}\) and show \(\frac{1}{2} eq \frac{1}{4}\).
  • Convert \(\frac{4}{5}\) to \(\frac{12}{15}\) and confirm \(\frac{11}{15} < \frac{12}{15}\) is true.
  • Compare \(\frac{3}{4}\) and \(\frac{4}{6}\) by finding a common denominator, showing \(\frac{9}{12} > \frac{8}{12}\).
  • Show \(\frac{5}{6}\) equals \(\frac{10}{12}\), making the inequality \(\frac{5}{6} > \frac{10}{12}\) false.
  • The correct statement is \(\boxed{\frac{11}{15} < \frac{4}{5}}\).

Explanation

  1. Evaluating the Statements

To determine which statement is true, we examine each option individually by comparing the two rational numbers given.

  • Option 1: \(\frac{8}{16} = \frac{1}{4}\)

We simplify the fraction \(\frac{8}{16}\) by dividing both the numerator and denominator by their greatest common divisor, which is 8:
\[\frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}\]
Comparing \(\frac{1}{2}\) and \(\frac{1}{4}\), we see that \(\frac{1}{2} eq \frac{1}{4}\). Thus, this statement is false.

  • Option 2: \(\frac{11}{15} < \frac{4}{5}\)

To compare these fractions, we find a common denominator. The least common multiple of 15 and 5 is 15. We convert \(\frac{4}{5}\) to have a denominator of 15:
\[\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}\]
Now we compare the numerators: since \(11 < 12\), it follows that \(\frac{11}{15} < \frac{12}{15}\). Thus, this statement is true.

  • Option 3: \(\frac{3}{4} < \frac{4}{6}\)

We find a common denominator for 4 and 6, which is 12:
\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\]
\[\frac{4}{6} = \frac{4 \times 2}{6 \times 2} = \frac{8}{12}\]
Since \(9 > 8\), \(\frac{9}{12} > \frac{8}{12}\), so \(\frac{3}{4} > \frac{4}{6}\). Thus, this statement is false.

  • Option 4: \(\frac{5}{6} > \frac{10}{12}\)

We simplify \(\frac{10}{12}\) by dividing by 2:
\[\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}\]
Comparing \(\frac{5}{6}\) and \(\frac{5}{6}\), we see that \(\frac{5}{6} = \frac{5}{6}\). The statement \(\frac{5}{6} > \frac{5}{6}\) is false.

Consequently, only the second statement is true.

  1. Conclusion

Based on our analysis, we have evaluated each of the four mathematical statements and found that only the second one, \(\frac{11}{15} < \frac{4}{5}\), is correct. The final answer is \(\boxed{\frac{11}{15} < \frac{4}{5}}\).

Examples

Comparing fractions is a fundamental skill in everyday tasks, such as cooking or managing finances. For example, if a recipe calls for \(\frac{11}{15}\) of a cup of flour and you only have \(\frac{4}{5}\) of a cup, comparing these two fractions allows you to determine if you have enough flour. Since \(\frac{4}{5} = \frac{12}{15}\) and \(\frac{11}{15} < \frac{12}{15}\), you can correctly conclude that you have more than enough flour to proceed with the recipe. Mastering this simple comparison ensures accuracy in measurements and resource management.