Question

When subtracting 61.45 g from 242.6 g, which value determines the number of significant figures in the answer? Explain.

Short answer

The value that determines the number of significant figures in the answer is 242.6 g, as it has the least number of decimal places (1 decimal place). In subtraction, the result should be rounded to the same number of decimal places as the value with the least number of decimal places.

Step-by-step solution

Identify the given values

We are given two values: 61.45 g and 242.6 g.

Recognize the rules for significant figures in subtraction

In subtraction, the result should be rounded to the same number of decimal places as the value with the least number of decimal places. In this case, we need to find which of the two given numbers has the least number of decimal places.

Determine the number of decimal places in each value

Now, let's determine the number of decimal places in each value: - 61.45 g has 2 decimal places - 242.6 g has 1 decimal place

Identify the value that determines the number of significant figures in the answer

Since 242.6 g has the least number of decimal places, it determines the number of significant figures in the answer.

Explain the result

The value with the least number of decimal places determines the number of significant figures in the answer when performing subtraction, which in this case is 242.6 g with 1 decimal place. Therefore, the number of significant figures in the answer will be determined by the value 242.6 g.

Key concepts

Measurement Precision

Understanding the concept of measurement precision is crucial when working with numbers in scientific and mathematical calculations. In the context of subtraction, precision refers to how detailed a measurement is, particularly when dealing with decimal places. It's about the extent to which a measurement is close to other measurements of the same quantity.

For instance, if we consider our original exercise where we subtract 61.45 g from 242.6 g, we notice that the measurement precision is different for both values. One has two decimal places while the other has one. This variation impacts the precision of our final answer, as the rule of thumb in subtraction is that the result cannot be more precise than the least precise measurement. Thus, the answer must be rounded to the least number of decimal places presented in any of the original measurements, ensuring that the precision of our result reflects the precision of our least exact measurement.

Decimal Places

Decimal places play a central role in determining the precision of a measurement and indicate the number of digits to the right of the decimal point. In our step-by-step solution, we focus on identifying the number of decimal places to know how to round our final answer.

But why is this important? For clarity, consider prices in a currency. If an item costs \(3.215, and the smallest currency denomination is a cent (\)0.01), the price would need to be rounded to $3.22. Similarly, when subtracting numbers with different decimal places, as seen with 61.45 g and 242.6 g, the principle of uniformity requires that our answer aligns with the number with fewer decimal places - in this case, one. Therefore, the final result, after following subtraction procedures, will be adjusted to have just one place after the decimal to match the least precise measurement, which ensures consistency across the computation. It's noteworthy that proper rounding is essential, and it accounts for the next digit just as in currency rounding.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It's represented as the product of a number between 1 and 10 and a power of 10. This approach is widely used in science and engineering because it simplifies numbers, making them easier to read, compare, and operate with, especially when dealing with measurements of vastly different scales.

When significant figures come into play, scientific notation also makes it clearer how many significant digits a number has. For example, the number 242.6 g can be expressed as 2.426 x 10^2 g in scientific notation, explicitly showing that it has four significant figures. In subtraction, if the numbers involved were in scientific notation, you would adjust them to have the same exponent before subtracting, ensuring that the precision is maintained. Although our original exercise does not require scientific notation, having a good grasp of it can be invaluable when numbers become incredibly large or small or when precision is critically important in representing the data.