The value of `(4(1)/5-3(1)/(10) "of" 1(1)/7)div(6(1)/3-3(1)/5 "of"1(1)/2)` is:

Options

A. `3/5`
B. `5/9`
C. `5/7`
D. `3/7`

Answer

To solve the expression \((4(1)/5 - 3(1)/(10) \text{ of } 1(1)/7) \div (6(1)/3 - 3(1)/5 \text{ of } 1(1)/2)\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert all mixed numbers into improper fractions.

- \(4(1)/5 = \frac{4 \times 5 + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5}\)
- \(3(1)/(10) = \frac{3 \times 10 + 1}{10} = \frac{30 + 1}{10} = \frac{31}{10}\)
- \(1(1)/7 = \frac{1 \times 7 + 1}{7} = \frac{7 + 1}{7} = \frac{8}{7}\)
- \(6(1)/3 = \frac{6 \times 3 + 1}{3} = \frac{18 + 1}{3} = \frac{19}{3}\)
- \(3(1)/5 = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5}\)
- \(1(1)/2 = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}\)

### Step 2: Substitute Back into the Expression
Now we substitute these values back into the expression:

\[ \left(\frac{21}{5} - \frac{31}{10} \text{ of } \frac{8}{7}\right) \div \left(\frac{19}{3} - \frac{16}{5} \text{ of } \frac{3}{2}\right) \]

### Step 3: Calculate "of" (Multiplication)
Next, we calculate the multiplication parts:

- For \(\frac{31}{10} \text{ of } \frac{8}{7}\):
\[ \frac{31}{10} \times \frac{8}{7} = \frac{31 \times 8}{10 \times 7} = \frac{248}{70} = \frac{124}{35} \]

- For \(\frac{16}{5} \text{ of } \frac{3}{2}\):
\[ \frac{16}{5} \times \frac{3}{2} = \frac{16 \times 3}{5 \times 2} = \frac{48}{10} = \frac{24}{5} \]

### Step 4: Substitute Back into the Expression
Now substitute these results back:

\[ \left(\frac{21}{5} - \frac{124}{35}\right) \div \left(\frac{19}{3} - \frac{24}{5}\right) \]

### Step 5: Find Common Denominators and Simplify
Now we need to simplify both parts of the expression.

For \(\frac{21}{5} - \frac{124}{35}\):
- The common denominator is \(35\):
\[ \frac{21 \times 7}{5 \times 7} - \frac{124}{35} = \frac{147}{35} - \frac{124}{35} = \frac{23}{35} \]

For \(\frac{19}{3} - \frac{24}{5}\):
- The common denominator is \(15\):
\[ \frac{19 \times 5}{3 \times 5} - \frac{24 \times 3}{5 \times 3} = \frac{95}{15} - \frac{72}{15} = \frac{23}{15} \]

### Step 6: Rewrite the Expression
Now the expression becomes:
\[ \frac{23}{35} \div \frac{23}{15} \]

### Step 7: Change Division to Multiplication
Change the division into multiplication by taking the reciprocal:
\[ \frac{23}{35} \times \frac{15}{23} \]

### Step 8: Cancel Common Terms
Now we can cancel \(23\):
\[ \frac{15}{35} = \frac{3}{7} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{3}{7} \]

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