Question

Which of the following statements is true Which number would make this statement 5 2/9 • 7/8 > 5 2/9 B 4 1/2 • 5/4 < 4 1/2 C 17/4 • 3/2 < 17/4 D 3 5/8 • 9/4 > 3 5/8

Answer 1

Let's evaluate each statement to determine which is true.

1. **Statement A:** \( 5 \frac{2}{9} \cdot \frac{7}{8} > 5 \frac{2}{9} \)

First, convert \( 5 \frac{2}{9} \) to an improper fraction:
\[
5 \frac{2}{9} = \frac{5 \times 9 + 2}{9} = \frac{45 + 2}{9} = \frac{47}{9}
\]

Now calculate \( 5 \frac{2}{9} \cdot \frac{7}{8} \):
\[
\frac{47}{9} \cdot \frac{7}{8} = \frac{47 \cdot 7}{9 \cdot 8} = \frac{329}{72}
\]

Now we compare \( \frac{329}{72} \) and \( \frac{47}{9} \):
To do this, convert \( \frac{47}{9} \) to a denominator of 72:
\[
\frac{47}{9} = \frac{47 \cdot 8}{9 \cdot 8} = \frac{376}{72}
\]

Now compare:
\[
\frac{329}{72} > \frac{376}{72} \text{ is false (since 329 < 376)}.
\]
So, statement A is **false**.

2. **Statement B:** \( 4 \frac{1}{2} \cdot \frac{5}{4} < 4 \frac{1}{2} \)

Convert \( 4 \frac{1}{2} \) to an improper fraction:
\[
4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}
\]

Now calculate \( 4 \frac{1}{2} \cdot \frac{5}{4} \):
\[
\frac{9}{2} \cdot \frac{5}{4} = \frac{9 \cdot 5}{2 \cdot 4} = \frac{45}{8}
\]

Now compare \( \frac{45}{8} \) and \( \frac{9}{2} \):
Convert \( \frac{9}{2} \) to a denominator of 8:
\[
\frac{9}{2} = \frac{9 \cdot 4}{2 \cdot 4} = \frac{36}{8}
\]

Now compare:
\[
\frac{45}{8} < \frac{36}{8} \text{ is false (since 45 > 36)}.
\]
So, statement B is **false**.

3. **Statement C:** \( \frac{17}{4} \cdot \frac{3}{2} < \frac{17}{4} \)

Calculate \( \frac{17}{4} \cdot \frac{3}{2} \):
\[
\frac{17}{4} \cdot \frac{3}{2} = \frac{17 \cdot 3}{4 \cdot 2} = \frac{51}{8}
\]

Now compare \( \frac{51}{8} \) and \( \frac{17}{4} \):
Convert \( \frac{17}{4} \) to a denominator of 8:
\[
\frac{17}{4} = \frac{17 \cdot 2}{4 \cdot 2} = \frac{34}{8}
\]

Now compare:
\[
\frac{51}{8} < \frac{34}{8} \text{ is false (since 51 > 34)}.
\]
So, statement C is **false**.

4. **Statement D:** \( 3 \frac{5}{8} \cdot \frac{9}{4} > 3 \frac{5}{8} \)

Convert \( 3 \frac{5}{8} \) to an improper fraction:
\[
3 \frac{5}{8} = \frac{3 \cdot 8 + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8}
\]

Calculate \( 3 \frac{5}{8} \cdot \frac{9}{4} \):
\[
\frac{29}{8} \cdot \frac{9}{4} = \frac{29 \cdot 9}{8 \cdot 4} = \frac{261}{32}
\]

Now compare \( \frac{261}{32} \) with \( \frac{29}{8} \):
Convert \( \frac{29}{8} \) to a denominator of 32:
\[
\frac{29}{8} = \frac{29 \cdot 4}{8 \cdot 4} = \frac{116}{32}
\]

Now compare:
\[
\frac{261}{32} > \frac{116}{32} \text{ is true (since 261 > 116)}.
\]
So, statement D is **true**.

**Conclusion:** The true statement is **D: \( 3 \frac{5}{8} \cdot \frac{9}{4} > 3 \frac{5}{8} \)**.