Question

Select the equation for which the mixed number four and one half makes the equation true
2 3/8+__=6 1/2
9 2/3-__=5 1/6
{3 1/2+2 5/8}- __ =1 3/4
__+{7 2/5-2 3/10=9 3/5
__-{1 1/8=2 3/5}=31/40

Answer 1

To determine the appropriate equation where the mixed number \( 4 \frac{1}{2} \) (or \( \frac{9}{2} \) in improper fraction form) will make the equation true, we can evaluate each equation one by one.

1. **Equation: \( 2 \frac{3}{8} + __ = 6 \frac{1}{2} \)**

Convert everything to improper fractions:
- \( 2 \frac{3}{8} = \frac{19}{8} \)
- \( 6 \frac{1}{2} = \frac{13}{2} = \frac{26}{4} \) (but for simplicity, let's convert to eighths)
- \( 6 \frac{1}{2} = \frac{52}{8} \)

Solve for \( x \) in \( \frac{19}{8} + x = \frac{52}{8} \):
\[
x = \frac{52}{8} - \frac{19}{8} = \frac{33}{8} = 4 \frac{1}{8}
\]
(not \( 4 \frac{1}{2} \))

2. **Equation: \( 9 \frac{2}{3} - __ = 5 \frac{1}{6} \)**

Convert everything to improper fractions:
- \( 9 \frac{2}{3} = \frac{29}{3} \)
- \( 5 \frac{1}{6} = \frac{31}{6} \)

Solve for \( x \) in \( \frac{29}{3} - x = \frac{31}{6} \):
\[
x = \frac{29}{3} - \frac{31}{6} = \frac{58}{6} - \frac{31}{6} = \frac{27}{6} = 4 \frac{1}{2}
\]
(This matches!)

3. **Equation: \( \{3 \frac{1}{2} + 2 \frac{5}{8}\} - __ = 1 \frac{3}{4} \)**

Convert to improper fractions:
- \( 3 \frac{1}{2} = \frac{7}{2} \)
- \( 2 \frac{5}{8} = \frac{21}{8} \)
- \( 1 \frac{3}{4} = \frac{7}{4} \)

Find the common denominator (8):
- \( \frac{7}{2} = \frac{28}{8} \)
- \( \frac{7}{4} = \frac{14}{8} \)

Solve for \( x \):
\[
x = \left( \frac{28}{8} + \frac{21}{8} \right) - \frac{14}{8} = \frac{49}{8} - \frac{14}{8} = \frac{35}{8} = 4 \frac{3}{8}
\]
(not \( 4 \frac{1}{2} \))

4. **Equation: \( __ + \{7 \frac{2}{5} - 2 \frac{3}{10}\} = 9 \frac{3}{5} \)**

First calculate \( 7 \frac{2}{5} - 2 \frac{3}{10} \):
- \( 7 \frac{2}{5} = \frac{36}{5} \)
- \( 2 \frac{3}{10} = \frac{23}{10} = \frac{46}{20} \) converting \( \frac{36}{5} = \frac{144}{20} \)

Thus:
\[
\frac{144}{20} - \frac{46}{20} = \frac{98}{20} = 4 \frac{18}{20} \text{ or } 4 \frac{9}{10}
\]

Solve for \( x \):
\[
x + 4 \frac{9}{10} = 9 \frac{3}{5} = \frac{48}{10}
\]
\[
x = \frac{48}{10} - \frac{49}{10} = \frac{-1}{10}
\]
(not \( 4 \frac{1}{2} \))

5. **Equation: \( __ - \{1 \frac{1}{8} = 2 \frac{3}{5}\} = \frac{31}{40} \)**

This notation appears off; let’s assume it means something like:
\( x - 1 \frac{1}{8} = 2 \frac{3}{5} + \frac{31}{40} \)

However, let's resolve this ambiguity:
- \( 1 \frac{1}{8} = \frac{9}{8} \)
- \( 2 \frac{3}{5} = \frac{13}{5} \)
- \( \frac{31}{40} = \frac{31}{40} \)

You'd solve this but initially seems far-fetched for \( 4 \frac{1}{2} \).

From our evaluation, the equation that \( 4 \frac{1}{2} \) fits best is:

**\( 9 \frac{2}{3} - __ = 5 \frac{1}{6} \)**.