4 7/32x-(3 3/4x - 1 7/32x)

To simplify the expression \( 4 \frac{7}{32}x - \left(3 \frac{3}{4}x - 1 \frac{7}{32}x \right) \), we first need to simplify inside the parentheses and convert the mixed numbers to improper fractions.

1. Convert the mixed numbers to improper fractions:

• \( 4 \frac{7}{32} \) as an improper fraction is \( \frac{4 \times 32 + 7}{32} = \frac{128 + 7}{32} = \frac{135}{32} \)
• \( 3 \frac{3}{4} \) as an improper fraction is \( \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \)
• \( 1 \frac{7}{32} \) as an improper fraction is \( \frac{1 \times 32 + 7}{32} = \frac{32 + 7}{32} = \frac{39}{32} \)

2. Rewrite the expression using the improper fractions and distribute the negative sign inside the parentheses:

\( \frac{135}{32}x - \left(\frac{15}{4}x - \frac{39}{32}x \right) \)

3. Simplify inside the parentheses:

To subtract \( \frac{15}{4}x \) and \( -\frac{39}{32}x \), we need a common denominator. The least common denominator (LCD) for 4 and 32 is 32.

\( \frac{15}{4}x \) can be written as \( \frac{15 \times 8}{4 \times 8}x = \frac{120}{32}x \)

Now we subtract \( \frac{120}{32}x \) from \( -\frac{39}{32}x \):

\( \frac{135}{32}x - \left(\frac{120}{32}x - \frac{39}{32}x \right) \) \( \frac{135}{32}x - \left(\frac{-120 + 39}{32}x \right) \) \( \frac{135}{32}x - \left(\frac{-81}{32}x \right) \)

4. Distribute the negative sign (which is the same as multiplying by -1):

\( \frac{135}{32}x - \left(-\frac{81}{32}x\right) \) \( \frac{135}{32}x + \frac{81}{32}x \)

5. Add the x terms together:

\( \frac{135 + 81}{32}x \) \( \frac{216}{32}x \)

6. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8:

\( \frac{216 \div 8}{32 \div 8}x \) \( \frac{27}{4}x \)

So the simplified expression is \( \frac{27}{4}x \) or \( 6 \frac{3}{4}x \) in mixed number form.