Question

Use the properties of operations and rules for multiplying and dividing to solve:
(8/9) x (−3/4)÷(1/9) x (−4/5)÷(3/5)

Answer 1

To solve the expression \((8/9) \times (-3/4) \div (1/9) \times (-4/5) \div (3/5)\), we can follow the rules for multiplication and division. First, let's convert the division to multiplication by using the reciprocal. The division \( a \div b \) can be rewritten as \( a \times (1/b) \).

Let's rewrite the expression step-by-step:

1. Rewrite division as multiplication by the reciprocal:
\[
(8/9) \times (-3/4) \times (9/1) \times (-4/5) \times (5/3)
\]

2. Now let's organize the expression:
\[
= (8/9) \times (-3/4) \times (9/1) \times (-4/5) \times (5/3)
\]

3. Combine the fractions together:
\[
= \frac{8 \times (-3) \times 9 \times (-4) \times 5}{9 \times 4 \times 1 \times 5 \times 3}
\]

4. Now let's calculate the numerator and the denominator separately.

- **Numerator**:
\[
8 \times (-3) \times 9 \times (-4) \times 5 = 8 \times 3 \times 9 \times 4 \times 5
\]
(Since we are multiplying two negative numbers, the product will be positive).

- Performing the multiplication in parts:
\[
= 8 \times 3 = 24
\]
\[
= 24 \times 9 = 216
\]
\[
= 216 \times 4 = 864
\]
\[
= 864 \times 5 = 4320
\]

- **Denominator**:
\[
9 \times 4 \times 1 \times 5 \times 3 = 9 \times 4 = 36
\]
\[
= 36 \times 1 = 36
\]
\[
= 36 \times 5 = 180
\]
\[
= 180 \times 3 = 540
\]

5. Now we combine the results:
\[
= \frac{4320}{540}
\]

6. To simplify \(\frac{4320}{540}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4320 and 540 is 180.

- Divide both by 180:
\[
\frac{4320 \div 180}{540 \div 180} = \frac{24}{3} = 8
\]

Thus, the final result of the expression is:
\[
\boxed{8}
\]