How many significant figures are in the answer to 41.9 m ÷ 6410000000 m - 6.800 m × 10 m + 7.6 m? How did it become -56?

Let's solve the problem step by step to determine how the answer comes out to be \(-56\) and how many significant figures are in the final answer.

### Step 1: Interpret the expression
We are given the expression:
\[
41.9 \, \text{m} \div 6410000000 \, \text{m} - 6.800 \, \text{m} \times 10 \, \text{m} + 7.6 \, \text{m}
\]
This can be written as:
\[
\frac{41.9}{6410000000} - (6.800 \times 10) + 7.6
\]

### Step 2: Perform the division and multiplication

1. **Division**: 
   \[
   \frac{41.9}{6410000000} = 6.5343 \times 10^{-9} \, \text{m}
   \]
   Since 41.9 has 3 significant figures, and 6410000000 has 3 significant figures, the result of the division should be expressed with 3 significant figures:
   \[
   \frac{41.9}{6410000000} = 6.53 \times 10^{-9} \, \text{m}
   \]

2. **Multiplication**:
   \[
   6.800 \times 10 = 68.00 \, \text{m}
   \]
   Since 6.800 has 4 significant figures, and 10 is exact (so it doesn't affect significant figures), the product has 4 significant figures:
   \[
   6.800 \times 10 = 68.00 \, \text{m}
   \]

### Step 3: Combine the terms

Now substitute these values into the original expression:
\[
6.53 \times 10^{-9} - 68.00 + 7.6
\]

1. **Subtract** \( 68.00 \) from the division result:
   \[
   6.53 \times 10^{-9} - 68.00 \approx -68.00
   \]
   (since \( 6.53 \times 10^{-9} \) is extremely small compared to 68.00)

2. **Add \( 7.6 \)** to the result:
   \[
   -68.00 + 7.6 = -60.4
   \]

### Step 4: Rounding to appropriate significant figures

Now, the addition involves two terms: \( -68.00 \) (which has 4 significant figures) and \( 7.6 \) (which has 2 significant figures). When adding, the result should be rounded to the least number of decimal places. Since 7.6 has only 1 decimal place, we round \( -60.4 \) to one decimal place.

The final result becomes:
\[
-60.4
\]

### Step 5: Verify the -56 value

It seems there might be a slight error or a rounding difference in how the -56 was obtained. If rounding more aggressively, the value might have been rounded to the nearest whole number from -60.4, which would give approximately -60, not -56. 

Therefore, if you are seeing -56, it may come from different intermediate rounding or input values, but the correct step-by-step result should be around -60.4, rounded according to significant figure rules.

### Significant Figures in the Final Answer

- The first operation (\(41.9 \div 6410000000\)) gives 3 significant figures.
- The second operation (\(6.800 \times 10\)) gives 4 significant figures.
- The addition step rounds to 1 decimal place due to \(7.6\).

The result should be rounded to **2 significant figures**, corresponding to the number of significant figures in \(7.6 \, \text{m}\), as it's the limiting factor in this case.

### Summary:

1. The final result is approximately \( -60.4 \).
2. The answer has 2 significant figures.
3. The difference in the -56 answer could arise from varying rounding or approximations in prior steps.

---

Do you want further clarification on any step? Here are some additional questions to consider:

1. How do significant figures affect the precision of scientific measurements?
2. Why do we round to the least number of decimal places when adding or subtracting?
3. How do you handle exact numbers in multiplication/division problems?
4. Can you apply these steps to a similar problem with different values?
5. Why is it important to track significant figures throughout a calculation?

**Tip:** When performing multiple operations, always follow the order of operations and track significant figures at every step to ensure accuracy.

Explore how to solve a problem involving significant figures in the expression 41.9 ÷ 6410000000 - 6.800 × 10 + 7.6. This detailed explanation covers division, multiplication, and addition, with a focus on significant figures and rounding rules. Suitable for Grades 9-12.

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