Question

Expressions that are equivalent to 4 and 5/8
(4 7/8)(4 -1/4)
4(4 3/8)
(2 14/8)(4 -1/4)

Answer 1

To determine whether the expressions provided are equivalent to \(4 \frac{5}{8}\), we can convert them into improper fractions or decimals and compare.

First, let's convert \(4 \frac{5}{8}\) into an improper fraction:

\[
4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8}
\]

Next, let's evaluate each expression:

1. **Expression: \((4 \frac{7}{8})(4 - \frac{1}{4})\)**
- Convert \(4 \frac{7}{8}\) into an improper fraction:
\[
4 \frac{7}{8} = \frac{4 \times 8 + 7}{8} = \frac{32 + 7}{8} = \frac{39}{8}
\]
- Convert \(4 - \frac{1}{4}\):
\[
4 - \frac{1}{4} = \frac{4 \times 4}{4} - \frac{1}{4} = \frac{16 - 1}{4} = \frac{15}{4}
\]
- Now multiply the two fractions:
\[
\left(\frac{39}{8}\right)\left(\frac{15}{4}\right) = \frac{39 \times 15}{8 \times 4} = \frac{585}{32}
\]
- Convert \(\frac{585}{32}\) back to a mixed number:
\[
585 \div 32 = 18 \quad \text{remainder: } 9 \quad \Rightarrow \quad 18 \frac{9}{32}
\]
- This is not equivalent to \(4 \frac{5}{8}\).

2. **Expression: \(4(4 \frac{3}{8})\)**
- Convert \(4 \frac{3}{8}\) into an improper fraction:
\[
4 \frac{3}{8} = \frac{4 \times 8 + 3}{8} = \frac{32 + 3}{8} = \frac{35}{8}
\]
- Now calculate:
\[
4 \times \frac{35}{8} = \frac{4 \times 35}{8} = \frac{140}{8} = \frac{70}{4} = 17 \frac{2}{4} = 17 \frac{1}{2}
\]
- This is not equivalent to \(4 \frac{5}{8}\).

3. **Expression: \((2 \frac{14}{8})(4 - \frac{1}{4})\)**
- Convert \(2 \frac{14}{8}\) into an improper fraction:
- First, simplify \( \frac{14}{8} \) to \( \frac{7}{4} \)
\[
2 \frac{14}{8} = 2 + \frac{7}{4} = \frac{8}{4} + \frac{7}{4} = \frac{15}{4}
\]
- We already calculated \(4 - \frac{1}{4} = \frac{15}{4}\).
- Now multiply the two fractions:
\[
\left(\frac{15}{4}\right)\left(\frac{15}{4}\right) = \frac{225}{16}
\]
- Convert \(\frac{225}{16}\) back to a mixed number:
\[
225 \div 16 = 14 \quad \text{remainder:} 1 \quad \Rightarrow \quad 14 \frac{1}{16}
\]
- This is not equivalent to \(4 \frac{5}{8}\).

Therefore, none of the provided expressions are equivalent to \(4 \frac{5}{8}\).