`(4)/(7) " of " (8)/(9) div (8)/(7) xx 180 " of " (1)/(9)=`?

Options

A. `(20)/(9)`
B. `(9)/(20)`
C. `(80)/(9)`
D. `(9)/(80)`

Answer

To solve the expression \(\frac{4}{7} \text{ of } \frac{8}{9} \div \frac{8}{7} \times 180 \text{ of } \frac{1}{9}\), we will follow the order of operations (BODMAS/BIDMAS rules). Let's break it down step by step.

### Step 1: Understand the terms
The term "of" means multiplication. Therefore, we can rewrite the expression as:
\[ \frac{4}{7} \times \frac{8}{9} \div \frac{8}{7} \times 180 \times \frac{1}{9} \]

### Step 2: Solve the division first
We will handle the division part first:
\[ \frac{4}{7} \times \frac{8}{9} \div \frac{8}{7} = \frac{4}{7} \times \frac{8}{9} \times \frac{7}{8} \]
Here, we flipped \(\frac{8}{7}\) to \(\frac{7}{8}\) since division by a fraction is the same as multiplying by its reciprocal.

### Step 3: Simplify the expression
Now we can simplify:
\[ = \frac{4 \times 8 \times 7}{7 \times 9 \times 8} \]
Notice that \(8\) in the numerator and denominator cancels out, and \(7\) also cancels out:
\[ = \frac{4}{9} \]

### Step 4: Multiply by \(180\) and \(\frac{1}{9}\)
Next, we multiply this result by \(180\) and \(\frac{1}{9}\):
\[ \frac{4}{9} \times 180 \times \frac{1}{9} \]

### Step 5: Calculate \(180 \times \frac{1}{9}\)
First, calculate \(180 \times \frac{1}{9}\):
\[ 180 \div 9 = 20 \]

### Step 6: Multiply by \(\frac{4}{9}\)
Now we multiply:
\[ \frac{4}{9} \times 20 = \frac{80}{9} \]

### Final Result
Thus, the final result of the expression is:
\[ \frac{80}{9} \]

### Conclusion
The answer is \(\frac{80}{9}\), which corresponds to option 3.

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