Question

Express the results of the following calculations with the correct number of significant figures: (a) \(4.884 \times 2.05\) (b) \(94.61 \div 3.7\) (c) \(3.7 \div 94.61\) (d) \(5502.3+24+0.01\) (e) \(86.3+1.42-0.09\) (f) \(5.7 \times 2.31\)

Short answer

(a) 10.0; (b) 26; (c) 0.039; (d) 5526; (e) 87.6; (f) 13.

Step-by-step solution

Multiplication for (a)

Multiply the given numbers: \(4.884 \times 2.05 = 10.0142\). The number with the least significant figures (2.05) has 3 significant figures, so the result should also have 3 significant figures: **10.0**.

Division for (b)

Divide the given numbers: \(94.61 \div 3.7 = 25.5945946\). The number with the least significant figures (3.7) has 2 significant figures, so the result should also have 2 significant figures: **26**.

Division for (c)

Divide the given numbers: \(3.7 \div 94.61 = 0.039104\). The number with the least significant figures (3.7) has 2 significant figures, so the result should also have 2 significant figures: **0.039**.

Addition for (d)

Add the given numbers: \(5502.3 + 24 + 0.01 = 5526.31\). The number with the least decimal places (24, which has no decimal place) dictates the significant decimal place, so the result should have 0 decimal places: **5526**.

Addition and Subtraction for (e)

Perform the operation: \(86.3 + 1.42 - 0.09 = 87.63\). The number 86.3 has the least decimal places (1 decimal place), so the result should also have 1 decimal place: **87.6**.

Multiplication for (f)

Multiply the given numbers: \(5.7 \times 2.31 = 13.167\). The number with the least significant figures (5.7 and 2.31) both have 2 significant figures, so the result should also have 2 significant figures: **13**.

Key concepts

multiplication with significant figures

When performing multiplication, the number of significant figures in the result should match the number with the least significant figures among the factors. Significant figures are crucial in reflecting the precision of measurements.

In example (a) from the exercise, we calculated: \(4.884 \times 2.05 = 10.0142\)

Here, 2.05, with 3 significant figures, limits our result to also have 3 significant figures. Thus, the answer is rounded to **10.0**. This ensures that we do not imply a higher precision than the calculated measurement allows.

Similarly, for example (f):
\(5.7 \times 2.31 = 13.167\)

Both numbers have 2 significant figures, so the result should be rounded to 2 significant figures as well: **13**.Ensure that when you multiply numbers, you first identify the factor with the least significant figures, and then round your final answer accordingly. This practice maintains the integrity of significant figures in calculations.

division with significant figures

Just like in multiplication, division with significant figures requires the result to reflect the precision determined by the number with fewer significant figures.

For example (b), the division is:\(94.61 \div 3.7 = 25.5945946\)

The divisor, 3.7, holds only 2 significant figures, which means the quotient should be rounded to 2 significant figures, giving us **26**.

In example (c), the operation is:\(3.7 \div 94.61 = 0.039104\)

Here again, 3.7 limits the calculation to 2 significant figures, so the result must be rounded to **0.039**.Always verify which of your numbers is limiting the calculation's precision by having the least significant figures, and adjust your final answer to match it. This rule ensures that rounding only reduces calculated precision in line with the given inputs.

addition and subtraction with significant figures

Addition and subtraction rules differ slightly from multiplication and division when it comes to significant figures. Instead of counting significant figures, we focus on decimal places.

Consider the result of example (d):\(5502.3 + 24 + 0.01 = 5526.31\)

Here, the number with the least decimal places is 24, which has zero decimal places. Therefore, the result should also have zero decimal places, making it **5526**.

In example (e), the calculation:\(86.3 + 1.42 - 0.09 = 87.63\)

The number 86.3 dictates the precision as it has only 1 decimal place, so the result must be rounded to 1 decimal place: **87.6**.It is important to identify which number has the least precise decimal placement and adjust your solution to match it, ensuring accurate results without overstating the precision of your measurements.