Question

show the calculator answer, significant digits answer and identify the number of significant digits. show all work.
65 000(0.08 x 200 x 0.004) / (800 x 300)

make it more clear.

Answer 1

To solve the expression \( \frac{65,000 \times (0.08 \times 200 \times 0.004)}{800 \times 300} \) and determine the significant figures, we will go through each step carefully.

### Step 1: Calculate the expression

1. Calculate the numerator: \( 0.08 \times 200 \times 0.004 \).
- \( 0.08 \) has **1 significant digit**.
- \( 200 \) has **1 significant digit** (as it can be considered as \( 2 \times 10^{2} \)).
- \( 0.004 \) has **1 significant digit**.

Now, let's calculate it:
\[
0.08 \times 200 = 16 \quad (\text{1 significant digit carries over to } 0.004)
\]
\[
16 \times 0.004 = 0.064
\]

The significant figures for \( 0.064 \): **2 significant digits** (the digits 6 and 4).

2. Now calculate \( 65,000 \times 0.064 \):
- \( 65,000 \) has **2 significant digits** (the digits 6 and 5; the trailing zeros do not count unless specified).
- Using \( 0.064 \) with **2 significant digits**.

Now let's calculate:
\[
65,000 \times 0.064 = 4,160 \quad (\text{The answer ends up being 4160})
\]

The result here has **3 significant digits** (the digits 4, 1, and 6).

3. Calculate the denominator: \( 800 \times 300 \).
- \( 800 \) has **1 significant digit** (considered as \( 8 \times 10^{2} \)).
- \( 300 \) has **1 significant digit**.

So:
\[
800 \times 300 = 240,000
\]

The result here has **2 significant digits** (the digits 2 and 4; the trailing zeros do not count unless specified).

### Step 2: Now divide the numerator by the denominator
\[
\frac{4,160}{240,000}
\]

Now performing the division:
\[
4,160 \div 240,000 = 0.0173333\ldots
\]

### Step 3: Apply significant figures to the answer
The result must be rounded based on the significant figures from the numerator and denominator:
- **Numerator** (4,160): **3 significant digits**
- **Denominator** (240,000): **2 significant digits**

Since the final result will be based on the lower number of significant figures (which is 2 in this case), we should round \( 0.0173333 \) to **2 significant digits**.

### Final Result

\[
\text{Rounded: } 0.017 \quad \text{(2 significant figures)}
\]

### Summary
- **Calculator answer:** 0.0173333
- **Rounded answer with significant figures:** 0.017
- **Number of significant digits in the final answer:** 2 significant digits.