Question

For each of the following, give an answer with the correct number of decimal places: a. \(45.48 \mathrm{~cm}+8.057 \mathrm{~cm}\) b. \(23.45 \mathrm{~g}+104.1 \mathrm{~g}+0.025 \mathrm{~g}\) c. \(145.675 \mathrm{~mL}-24.2 \mathrm{~mL}\) d. \(1.08 \mathrm{~L}-0.585 \mathrm{~L}\)

Short answer

a. 53.54 cm, b. 127.6 g, c. 121.5 mL, d. 0.495 L

Step-by-step solution

Identify the number of decimal places in the addends (a)

For the operands in part (a), identify the decimal places: 45.48 has two decimal places, and 8.057 has three decimal places. The result should match the smallest number of decimal places, which is two.

Add the numbers in (a)

Add 45.48 and 8.057: \[ 45.48 + 8.057 = 53.537 \] Since the result should have two decimal places, round 53.537 to 53.54.

Identify the number of decimal places in the addends (b)

For part (b), the operands have different decimal places: 23.45 has two decimal places, 104.1 has one decimal place, and 0.025 has three decimal places. The result should match the smallest number of decimal places, which is one.

Add the numbers in (b)

Add 23.45, 104.1, and 0.025: \[ 23.45 + 104.1 + 0.025 = 127.575 \] Since the result should have one decimal place, round 127.575 to 127.6.

Identify the number of decimal places in the subtractends (c)

For part (c), identify the decimal places: 145.675 has three decimal places, and 24.2 has one decimal place. The result should match the smallest number of decimal places, which is one.

Subtract the numbers in (c)

Subtract 24.2 from 145.675: \[ 145.675 - 24.2 = 121.475 \] Since the result should have one decimal place, round 121.475 to 121.5.

Identify the number of decimal places in the subtractends (d)

For part (d), identify the decimal places: 1.08 has two decimal places, and 0.585 has three decimal places. The result should match the smallest number of decimal places, which is two.

Subtract the numbers in (d)

Subtract 0.585 from 1.08: \[ 1.08 - 0.585 = 0.495 \] Since the result should have two decimal places, no rounding is needed.

Key concepts

Decimal Places in Addition and Subtraction

When performing addition or subtraction in mathematics, it's crucial to consider the number of decimal places in each operand.
Let's break it down:

• Identify the decimal places in each number being added or subtracted.
• The result should match the operand with the smallest number of decimal places.

For instance, when adding 45.48 (two decimal places) and 8.057 (three decimal places), the result must have two decimal places, so we round 53.537 to 53.54.
This ensures precision and consistency in measurements and results.

Rounding Rules

Rounding is essential to obtain a result that reflects the precision of the measurements.
Here are simple rules to follow when rounding:

• Identify which decimal place you need to round to.
• If the digit immediately to the right is 5 or higher, round up.
• If the digit immediately to the right is less than 5, round down.

For example, when adding 23.45, 104.1, and 0.025, the initial sum is 127.575. Since the least number of decimal places is one (from 104.1), 127.575 rounds to 127.6.

Precision in Mathematics

Precision refers to the detail in the measurement, i.e., how close the measurements are to each other.
Higher precision means more decimal places or significant figures. In calculations, match the precision of the final result with the least precise measurement.
For example, subtracting 24.2 from 145.675 yields 121.475. Since 24.2 has one decimal place, the result is rounded to 121.5.
This maintains the integrity of the measurement accuracy.

Measurement Accuracy

Accuracy in measurement is how close a measured value is to the actual (true) value.
It's crucial to maintain accuracy by appropriately managing significant figures and decimal places in calculations.

• Always match the number of decimal places to the least accurate measurement when adding or subtracting.
• This practice minimizes errors and ensures results are consistent with the original data's accuracy.

For instance, in the case of subtracting 0.585 from 1.08, the actual calculation gives 0.495, which already adheres to the two decimal places needed. This level of detail preserves the calculation's accuracy.