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
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
Answers
GPT-4o mini
To find the value that makes each equation true using the mixed number four and one half (which is \(4 \frac{1}{2} = \frac{9}{2}\)), we can solve each equation one by one.
1. **For** \(2 \frac{3}{8} + __ = 6 \frac{1}{2}\):
- Convert mixed numbers to improper fractions:
- \(2 \frac{3}{8} = \frac{16 + 3}{8} = \frac{19}{8}\)
- \(6 \frac{1}{2} = \frac{12 + 1}{2} = \frac{13}{2} = \frac{26}{4} \text{(to have a common denominator)}\)
- Rewriting the equation:
- \(\frac{19}{8} + x = \frac{26}{4}\)
- Solving for \(x\):
- \(x = \frac{26}{4} - \frac{19 \cdot 2}{8} = \frac{26}{4} - \frac{19}{8}\)
- Get a common denominator for \(x\):
- \(\frac{52}{8} - \frac{19}{8} = \frac{33}{8}\)
- Since we’re looking for \( \frac{9}{2} \) and it does not match with \( \frac{33}{8}\), continue to other equations.
2. **For** \(9 \frac{2}{3} - __ = 5 \frac{1}{6}\):
- \(9 \frac{2}{3} = \frac{27 + 2}{3} = \frac{29}{3}\)
- \(5 \frac{1}{6} = \frac{30 + 1}{6} = \frac{31}{6}\)
- Rewrite the equation:
- \(\frac{29}{3} - x = \frac{31}{6}\)
- Solving for \(x\):
- \(x = \frac{29}{3} - \frac{31}{6}\)
- Common denominator:
- \(\frac{58}{6} - \frac{31}{6} = \frac{27}{6}\)
- Not \( \frac{9}{2} \), continue.
3. **For** \(\{3 \frac{1}{2}+2 \frac{5}{8}\}- __ = 1 \frac{3}{4}\):
- \(3 \frac{1}{2} = \frac{7}{2}\) and \(2 \frac{5}{8} = \frac{21}{8}\)
- Combine:
- Rewrite:
- \(x = \{3 \frac{1}{2}+2 \frac{5}{8}\}- 1 \frac{3}{4}\)
- Solve for \(x\):
- \(\frac{7}{2} + \frac{21}{8} = \frac{28 + 21}{8} = \frac{49}{8}\)
- \(1 \frac{3}{4} = \frac{7}{4} = \frac{14}{8}\)
- Then:
- \(x = \frac{49}{8} - \frac{14}{8} = \frac{35}{8}\)
- Not \(9/2\).
4. **For** \( __ + \{7 \frac{2}{5} - 2 \frac{3}{10}\} = 9 \frac{3}{5}\):
- Find the internal value:
- \(7 \frac{2}{5} = \frac{35 + 2}{5} = \frac{37}{5}\)
- \(2 \frac{3}{10} = \frac{23}{10}\)
- Rewrite:
- \( \frac{37}{5} - \frac{23}{10} = x\)
- Rewrite \(\frac{37}{5} = \frac{74}{10}\)
- Leads to:
- \(= \frac{74}{10} - \frac{23}{10} = \frac{51}{10} = 5 \frac{1}{10}\)
- Now:
- Solving for \(x\):
- \(x + 5 \frac{1}{10} = 9 \frac{3}{5}\)
- Solve:
- \((x + \frac{51}{10} = 9 + \frac{0}{10})\)
- \(x = 9 \frac{3}{5} - 5 \frac{1}{10} = 9 \frac{6}{10} - 5 \frac{1}{10} = 4 \frac{5}{10} = 4 \frac{1}{2}\)
5. **For** \(__ - \{1 \frac{1}{8} = 2 \frac{3}{5}\} = \frac{31}{40}\), it was leading unfit.
The equation that holds true for the mixed number four and one-half \(4 \frac{1}{2}\) is the fourth equation:
\[
\text{ } __ + \{7 \frac{2}{5} - 2 \frac{3}{10}\} = 9 \frac{3}{5}
\]
So, the result should have been:
\[
\text{} = 4 \frac{1}{2}
\]
This is the only equation suitable with the query, and is true.
1. **For** \(2 \frac{3}{8} + __ = 6 \frac{1}{2}\):
- Convert mixed numbers to improper fractions:
- \(2 \frac{3}{8} = \frac{16 + 3}{8} = \frac{19}{8}\)
- \(6 \frac{1}{2} = \frac{12 + 1}{2} = \frac{13}{2} = \frac{26}{4} \text{(to have a common denominator)}\)
- Rewriting the equation:
- \(\frac{19}{8} + x = \frac{26}{4}\)
- Solving for \(x\):
- \(x = \frac{26}{4} - \frac{19 \cdot 2}{8} = \frac{26}{4} - \frac{19}{8}\)
- Get a common denominator for \(x\):
- \(\frac{52}{8} - \frac{19}{8} = \frac{33}{8}\)
- Since we’re looking for \( \frac{9}{2} \) and it does not match with \( \frac{33}{8}\), continue to other equations.
2. **For** \(9 \frac{2}{3} - __ = 5 \frac{1}{6}\):
- \(9 \frac{2}{3} = \frac{27 + 2}{3} = \frac{29}{3}\)
- \(5 \frac{1}{6} = \frac{30 + 1}{6} = \frac{31}{6}\)
- Rewrite the equation:
- \(\frac{29}{3} - x = \frac{31}{6}\)
- Solving for \(x\):
- \(x = \frac{29}{3} - \frac{31}{6}\)
- Common denominator:
- \(\frac{58}{6} - \frac{31}{6} = \frac{27}{6}\)
- Not \( \frac{9}{2} \), continue.
3. **For** \(\{3 \frac{1}{2}+2 \frac{5}{8}\}- __ = 1 \frac{3}{4}\):
- \(3 \frac{1}{2} = \frac{7}{2}\) and \(2 \frac{5}{8} = \frac{21}{8}\)
- Combine:
- Rewrite:
- \(x = \{3 \frac{1}{2}+2 \frac{5}{8}\}- 1 \frac{3}{4}\)
- Solve for \(x\):
- \(\frac{7}{2} + \frac{21}{8} = \frac{28 + 21}{8} = \frac{49}{8}\)
- \(1 \frac{3}{4} = \frac{7}{4} = \frac{14}{8}\)
- Then:
- \(x = \frac{49}{8} - \frac{14}{8} = \frac{35}{8}\)
- Not \(9/2\).
4. **For** \( __ + \{7 \frac{2}{5} - 2 \frac{3}{10}\} = 9 \frac{3}{5}\):
- Find the internal value:
- \(7 \frac{2}{5} = \frac{35 + 2}{5} = \frac{37}{5}\)
- \(2 \frac{3}{10} = \frac{23}{10}\)
- Rewrite:
- \( \frac{37}{5} - \frac{23}{10} = x\)
- Rewrite \(\frac{37}{5} = \frac{74}{10}\)
- Leads to:
- \(= \frac{74}{10} - \frac{23}{10} = \frac{51}{10} = 5 \frac{1}{10}\)
- Now:
- Solving for \(x\):
- \(x + 5 \frac{1}{10} = 9 \frac{3}{5}\)
- Solve:
- \((x + \frac{51}{10} = 9 + \frac{0}{10})\)
- \(x = 9 \frac{3}{5} - 5 \frac{1}{10} = 9 \frac{6}{10} - 5 \frac{1}{10} = 4 \frac{5}{10} = 4 \frac{1}{2}\)
5. **For** \(__ - \{1 \frac{1}{8} = 2 \frac{3}{5}\} = \frac{31}{40}\), it was leading unfit.
The equation that holds true for the mixed number four and one-half \(4 \frac{1}{2}\) is the fourth equation:
\[
\text{ } __ + \{7 \frac{2}{5} - 2 \frac{3}{10}\} = 9 \frac{3}{5}
\]
So, the result should have been:
\[
\text{} = 4 \frac{1}{2}
\]
This is the only equation suitable with the query, and is true.