Find: e. `(21)/(32) div 1 (14)/(35)`

Answer

To solve the problem \( \frac{21}{32} \div 1 \frac{14}{35} \), we will follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \( 1 \frac{14}{35} \) can be converted to an improper fraction.

To do this, we multiply the whole number (1) by the denominator (35) and then add the numerator (14):
\[ 1 \frac{14}{35} = \frac{1 \times 35 + 14}{35} = \frac{35 + 14}{35} = \frac{49}{35} \]

### Step 2: Rewrite the division as multiplication
Now we rewrite the division of fractions as multiplication by the reciprocal of the second fraction:
\[ \frac{21}{32} \div \frac{49}{35} = \frac{21}{32} \times \frac{35}{49} \]

### Step 3: Simplify the fractions
Next, we simplify the fractions before multiplying. We can cancel out common factors:

1. The numerator \( 21 \) can be factored as \( 3 \times 7 \).
2. The denominator \( 49 \) can be factored as \( 7 \times 7 \).
3. The numerator \( 35 \) can be factored as \( 5 \times 7 \).

Now we can cancel:
- \( 21 \) and \( 49 \) share a common factor of \( 7 \):
\[ \frac{21}{49} = \frac{3}{7} \]
- \( 35 \) and \( 32 \) do not share any common factors, so we keep them as they are.

Now we have:
\[ \frac{3}{32} \times \frac{5}{7} \]

### Step 4: Multiply the fractions
Now we multiply the numerators and the denominators:
\[ \frac{3 \times 5}{32 \times 7} = \frac{15}{224} \]

### Final Answer
Thus, the final answer is:
\[ \frac{15}{224} \]

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