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8/9x-3/4divided by 1/9 x -4/5dividedby 3/5

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To solve the expression \(\frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{5} \div \frac{3}{5}}\), we will first simplify the expression in the denominator.

Step 1: Simplify the denominator

The denominator contains a division:

\[ \frac{4}{5} \div \frac{3}{5} \]

Dividing by a fraction is the same as multiplying by its reciprocal:

\[ \frac{4}{5} \div \frac{3}{5} = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3} \]

Step 2: Rewrite the entire expression

Now we can rewrite the entire expression with the simplified denominator:

\[ \frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{3}} \]

Step 3: Get rid of fractions

To eliminate fractions, we can multiply both the numerator and the denominator by the least common multiple (LCM) of the denominators. The denominators in the numerator and denominator of our expression are \(9\), \(4\), and \(3\). The LCM of these numbers is \(36\).

Multiply the numerator and the denominator by \(36\):

\[ \frac{36\left(\frac{8}{9}x - \frac{3}{4}\right)}{36\left(\frac{1}{9}x - \frac{4}{3}\right)} \]

Step 4: Distribute \(36\)

Now, we can simplify each part:

  1. For the numerator: \[ 36 \left(\frac{8}{9}x - \frac{3}{4}\right) = 36 \cdot \frac{8}{9}x - 36 \cdot \frac{3}{4} = 4 \cdot 8x - 27 = 32x - 27 \]

  2. For the denominator: \[ 36 \left(\frac{1}{9}x - \frac{4}{3}\right) = 36 \cdot \frac{1}{9}x - 36 \cdot \frac{4}{3} = 4x - 48 \]

Step 5: Write the final simplified expression

Now, the expression is simplified to:

\[ \frac{32x - 27}{4x - 48} \]

This is the final simplified form of the given expression.

huh

I apologize for any confusion. Let's clarify and go through the arithmetic step-by-step in a clearer manner.

We want to simplify the expression:

\[ \frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{5} \div \frac{3}{5}} \]

Step 1: Simplify the division in the denominator

First, we simplify the division in the denominator:

\[ \frac{4}{5} \div \frac{3}{5} \]

This can be rewritten as:

\[ \frac{4}{5} \times \frac{5}{3} = \frac{4 \cdot 5}{5 \cdot 3} = \frac{4}{3} \]

Now the expression looks like this:

\[ \frac{\frac{8}{9}x - \frac{3}{4}}{\frac{1}{9}x - \frac{4}{3}} \]

Step 2: Clean up the fraction

Next, we can rewrite this as:

\[ \frac{8x - \frac{3 \cdot 9}{4}}{\frac{1 \cdot 4}{9}x - 4} \]

To eliminate fractions, we will find a common denominator. In this case, it would be easiest to deal with the fractions directly.

Find a common denominator for the numerator and the denominator:

  • In the numerator \(\frac{8}{9}x - \frac{3}{4}\), the common denominator is \(36\).
  • In the denominator \(\frac{1}{9}x - \frac{4}{3}\), the common denominator is also \(9\).

Step 3: Multiply both numerator and denominator to eliminate fractions

Multiply the numerator by \(36\):

\[ 36 \left( \frac{8}{9}x - \frac{3}{4} \right) = 36 \times \frac{8}{9}x - 36 \times \frac{3}{4} \] \[ = 4 \times 8x - 27 = 32x - 27 \]

Multiply the denominator by \(9\):

\[ 9\left(\frac{1}{9}x - \frac{4}{3}\right) = 9 \times \frac{1}{9}x - 9 \times \frac{4}{3} = x - 12 \]

Final Expression

Now the simplified expression is:

\[ \frac{32x - 27}{x - 12} \]

This is the final simplified form. If you have any specific questions or need further clarification, feel free to ask!