Question

Add the following measurements and round off the answer: \( \begin{array}{llc} \text { (a) } & 0.4 \mathrm{~g} & \text { (b) } & 15.5 \mathrm{~g} \\ & 0.44 \mathrm{~g} & & 7.50 \mathrm{~g} \\ + & 0.444 \mathrm{~g} & & +0.050 \mathrm{~g} \\ \hline \end{array} \)

Short answer

The results are: (a) 1.3 g and (b) 23.0 g.

Step-by-step solution

Align the Numbers

To add the measurements, first align the decimal points of all numbers. This will ensure proper addition of the digits according to their place value.

Sum the Numbers in Each Column

Add the measurements in each column separately. For (a):\[\begin{aligned}&0.4 \&+0.44 \&+0.444 \\hline&0.40 + 0.44 + 0.444 = 1.284 \, \text{g}\end{aligned}\]For (b):\[\begin{aligned}&15.5 \&+7.50 \&+0.050 \\hline&15.50 + 7.50 + 0.050 = 23.050 \, \text{g}\end{aligned}\]

Determine the Least Precise Measurement

Check the precision (number of decimal places) in each set of measurements. - For set (a): Least precise is 0.4 (1 decimal place). - For set (b): Least precise is 15.5 (1 decimal place).

Round Off the Answers

Round the sums of each set to the same precision as the least precise measurement. - For (a): Round 1.284 to 1.3 g (1 decimal place). - For (b): Round 23.050 to 23.0 g (1 decimal place).

Record the Final Results

After rounding, the results are: - (a): 1.3 g - (b): 23.0 g

Key concepts

Rounding Off

Rounding off is a crucial skill in mathematics and sciences. It's the process of reducing the digits in a number while trying to keep its value similar to the original number. In our problem, we had sums of 1.284 g and 23.050 g. We rounded these sums to 1.3 g and 23.0 g, respectively.
\(
\)
When rounding off, especially in measurements, you always aim to match the precision of the least precise quantity. Here’s why that matters:

• It prevents the creation of a false sense of precision. You can’t measure to a tenth of a gram if your least accurate component is measured only to a half gram.
• It follows the rules of significant figures. They tell us which numbers are meaningful.

To round off to a certain number of decimal places, look at the digit in the desired place. If it's followed by a number 5 or greater, round up. If it’s less than 5, round down.

Decimal Places

Decimal places represent how precisely you measure a quantity. In the initial step, we added numbers with different decimal places, such as 0.4, 0.44, and 0.444.
\(
\)
Every measurement comes with a set decimal place which indicates its precision:

• More decimal places mean higher precision.
• Fewer decimal places mean lower precision.

In any calculation involving measurements, particularly addition and subtraction, the result cannot be more precise than the least precise measurement. This rule ensures that any conclusions drawn are based on the least amount of uncertainty, which is fair given the original data's lack of precision. Consider it a guardrail against over-interpretation of your results.

Measurement Precision

Measurement precision is about how fine a measurement is, reflecting the smallest unit that can be accurately recorded by any measuring device. In the exercise, our measurements ranged in precision from 1 to 3 decimal places, as seen in 0.4 and 0.444.
\(
\)
Here’s why measurement precision is fundamental:

• It determines the reliability of measurements. More precise measurements offer better detail, akin to zooming in with a camera.
• The least precise measurement limits the precision of calculation results. This approach avoids implying greater accuracy than is supported by the measurements.

During computations like the problem at hand, always align measurements correctly. Align numbers by their decimal point before performing any operations, and always consider the measurement with the least precision to reference the rounding of any final answer.