Find a, if `5(1)/(3)-3(2)/(3)div1 (1)/(3)diva+3(1)/(5)div1(1)/(5)=7`

Options

A. `1 1/2`
B. `2 1/3`
C. `3 1/4`
D. None of these

Answer

To solve the equation \( 5 \frac{1}{3} - 3 \frac{2}{3} \div 1 \frac{1}{3} \div a + 3 \frac{1}{5} \div 1 \frac{1}{5} = 7 \), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
1. Convert \( 5 \frac{1}{3} \):
\[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \]
2. Convert \( 3 \frac{2}{3} \):
\[ 3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]
3. Convert \( 1 \frac{1}{3} \):
\[ 1 \frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3} \]
4. Convert \( 3 \frac{1}{5} \):
\[ 3 \frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5} \]
5. Convert \( 1 \frac{1}{5} \):
\[ 1 \frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{5 + 1}{5} = \frac{6}{5} \]

### Step 2: Substitute the improper fractions into the equation
Now, substitute these values back into the equation:
\[ \frac{16}{3} - \frac{11}{3} \div \frac{4}{3} \div a + \frac{16}{5} \div \frac{6}{5} = 7 \]

### Step 3: Simplify the divisions
1. For \( \frac{11}{3} \div \frac{4}{3} \):
\[ \frac{11}{3} \div \frac{4}{3} = \frac{11}{3} \times \frac{3}{4} = \frac{11}{4} \]
2. For \( \frac{16}{5} \div \frac{6}{5} \):
\[ \frac{16}{5} \div \frac{6}{5} = \frac{16}{5} \times \frac{5}{6} = \frac{16}{6} = \frac{8}{3} \]

### Step 4: Substitute back into the equation
Now the equation looks like:
\[ \frac{16}{3} - \frac{11}{4} \div a + \frac{8}{3} = 7 \]

### Step 5: Combine like terms
Combine \( \frac{16}{3} + \frac{8}{3} \):
\[ \frac{16 + 8}{3} = \frac{24}{3} = 8 \]
So now we have:
\[ 8 - \frac{11}{4a} = 7 \]

### Step 6: Isolate the term with \( a \)
Subtract 8 from both sides:
\[ -\frac{11}{4a} = 7 - 8 \]
\[ -\frac{11}{4a} = -1 \]

### Step 7: Remove the negative sign
Multiply both sides by -1:
\[ \frac{11}{4a} = 1 \]

### Step 8: Solve for \( a \)
Cross-multiply:
\[ 11 = 4a \]
Divide both sides by 4:
\[ a = \frac{11}{4} \]

### Step 9: Convert to mixed number
Convert \( \frac{11}{4} \) to a mixed number:
\[ 11 \div 4 = 2 \quad \text{(remainder 3)} \]
So, \( a = 2 \frac{3}{4} \).

### Final Answer
\[ a = 2 \frac{3}{4} \]