Question

Using the appropriate number of significant digits, what is the answer to the following math problem? (Note: Assume all numbers are the results of measurements.) 3.060 × 4.10 + 200. = (A) 210 (B) 213 (C) 212.5 (D) 212.55

Short answer

The correct answer is (B) 213.

Step-by-step solution

- Perform the Multiplication

Multiply 3.060 by 4.10. Using significant figures, 3.060 has 4 significant digits and 4.10 has 3 significant digits, so the result should be rounded to 3 significant figures. Perform the calculation: 3.060 × 4.10 = 12.306 Rounded to 3 significant figures, this gives us 12.3.

- Perform the Addition

Add the rounded result from Step 1 to 200. 12.3 + 200 = 212.3

- Adjust for Significant Figures in Addition

In addition, the answer should have the same number of decimal places as the measurement with the fewest decimal places (which is 200, with no decimal places): So, 212.3 should be rounded to 212 to match the precision of 200.

Key concepts

Significant Digits

Significant digits, also known as significant figures, are the digits in a number that convey meaningful information about its precision. These are crucial when dealing with measurements. All non-zero digits are always significant. Any zeros between significant digits are also significant. Leading zeros (zeros before all non-zero digits) are not significant. Trailing zeros in a number containing a decimal point are significant.
For example, in the number 3.060, all four digits are significant because the zeros are between significant digits or follow a decimal point.
In contrast, in the number 200 (with no decimal point), only the digit '2' is considered significant, making it important to know how many significant digits your measurements have before performing calculations.

Multiplication

When multiplying numbers, the result should be reported with the same number of significant digits as the number with the fewest significant digits in the calculation.
For example, in the exercise you have 3.060 (4 significant digits) and 4.10 (3 significant digits).
When you multiply them: \(3.060 \times 4.10\ = 12.306\). This result should be rounded to three significant digits (since 4.10 has 3 significant digits): \(12.306 \approx 12.3\).
This ensures that the precision of the result matches the precision of the least precise measurement used in the calculation.

Addition

In addition, precision is determined by the least number of decimal places in the values being added.
For example, after calculating 3.060 × 4.10 and rounding it to 12.3, the next step is to add it to 200.0.
When adding 12.3 and 200, we note the number with the fewest decimal places is 200.0 (0 decimals).
Therefore, the sum should retain only up to the same decimal place: \(12.3 + 200 = 212.3\).
This ensures the addition respects the precision of the least precise measurement.

Rounding

Rounding is used to ensure numbers reflect the proper level of precision. When rounding numbers:
1. Identify the digit at the place you are rounding to.
2. Look at the next digit. If it is 5 or higher, round up. If it is less than 5, round down.
In the final step of the exercise, the sum from the previous step (212.3) is rounded to match the precision of the least precise number (200.0): \(212.3 \rightarrow 212\).
This gives us the final answer, ensuring the result is reported with the correct level of precision.
With the appropriate significant figures, 3.060 × 4.10 + 200 gives us the final answer as \boxed{212}.