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.
65 000(0.08 x 200 x 0.004) / (800 x 300)
make it more clear.
Answers
GPT-4o mini
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.
### 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.