Question

Carry out the following operations and express the answers with the appropriate number of significant numbers. (a) \((6.234+8.72) \times 0.6746\) (b) \(732.1-(892.5 / 8.2)\) (c) \(\left[\left(3.696 \times 10^{5}\right)-\left(6.234 \times 10^{3}\right)\right] \times 0.0742\) (d) \(0.006438 \times 108-(8.639+8.52)\)

Short answer

The solutions to the problems are: (a) $10.09$ (b) $623.3$ (c) $2.70 \times 10^{4}$ (d) $-16.47$

Step-by-step solution

Perform the addition

Add the numbers within the parenthesis: \(6.234 + 8.72 = 14.954\). Since the least number of decimal places is 2, we should round off the answer to 2 decimal places: \(14.95\).

Perform the multiplication

Multiply the result from Step 1 by \(0.6746\): \(14.95 \times 0.6746 = 10.08517\). With the least number of significant figures being 4, we should round off the answer to 4 significant figures: \(10.09\). The final answer for part (a) is \(10.09\). For part (b), we are given the operation \(732.1-(892.5 / 8.2)\).

Perform the division

Divide the numbers within the parenthesis: \(892.5 \div 8.2 = 108.841463\). Use the least number of significant figures, which is 3: \(108.8\).

Perform the subtraction

Subtract the result from Step 1 from \(732.1\): \(732.1 - 108.8 = 623.3\). Since the least number of decimal places is 1, we should keep the answer to 1 decimal place. The final answer for part (b) is \(623.3\). For part (c), we are given the operation $\left[\left(3.696 \times 10^{5}\right)-\left(6.234 \times 10^{3}\right)\right] \times 0.0742$.

Perform the subtraction

Subtract the numbers within the brackets: \(3.696 \times 10^{5} - 6.234 \times10^{3} = 363762\). The least number of significant figures is 4, so the answer should have 4 significant figures: \(3.638 \times 10^5\).

Perform the multiplication

Multiply the result from Step 1 by \(0.0742\): \((3.638 \times 10^5) \times 0.0742 = 27009.476\). With the least number of significant figures being 3, it should be rounded off to 3 significant figures: \(2.70 \times 10^4\). The final answer for part (c) is \(2.70 \times 10^4\). For part (d), we are given the operation \(0.006438 \times 108-(8.639+8.52)\).

Perform the multiplication

Multiply the numbers: \(0.006438 \times 108 = 0.694896\). Use the least number of significant figures, which is 4: \(0.6949\).

Perform the addition

Add the numbers within the parenthesis: \(8.639 + 8.52 = 17.159\). Use the least number of decimal places, which is 2: \(17.16\).

Perform the subtraction

Subtract the result from Step 2 from the result of Step 1: \(0.6949 - 17.16 = -16.4651\). Since the least number of decimal places is 2, the answer should have 2 decimal places: \(-16.47\). The final answer for part (d) is \(-16.47\).

Key concepts

addition and subtraction

When performing addition and subtraction with numbers, it's important to consider the number of decimal places each number has. The rule is to keep the result to the same number of decimal places as the number with the fewest decimal places in the operation. This ensures that the accuracy of the least precise number is maintained throughout.
For instance, if you are adding 6.234 and 8.72, you first add the numbers to get 14.954. Since 8.72 has only two decimal places, the result should be rounded to two decimal places, resulting in 14.95.
Similarly, in subtraction, as seen in the operation 732.1 - 108.841463, you would take into account the decimal places. The final result should match the number of decimal places of the number with the least precision, in this case, 1 decimal place, which gives 623.3.

multiplication and division

In multiplication and division, unlike addition and subtraction, the number of significant figures is more important than the number of decimal places. The rule is to have your final answer contain the same number of significant figures as the number with the least significant figures in the operation.
For example, if you multiply 14.95 by 0.6746, your answer, 10.08517, needs to be rounded to four significant figures, since 0.6746 has four. Thus, the result will be 10.09.
Likewise, for division such as 892.5 divided by 8.2, which results in 108.841463, the answer should be rounded to three significant figures because 8.2 has the least, resulting in 108.8.

scientific notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. This method helps in efficiently handling significant figures and very large or small quantities.
When you encounter operations such as \(3.696 \times 10^{5} - 6.234 \times 10^3\), you first perform the operation and then adjust the results to the appropriate significant figures. For example, \(3.696 \times 10^{5} - 6.234 \times 10^3 = 363762\). In scientific notation, this becomes \(3.638 \times 10^5\) based on significant figures.
This approach is especially useful in expressions with additional multiplication, as seen where the result is multiplied by 0.0742 to eventually form 2.70 \times 10^4, ensuring precision and clarity.

rounding rules

Rounding is a crucial process used to adjust numbers to a desired level of precision, which is critical for accuracy in scientific calculations. There are some key rules when rounding numbers:

• If the digit immediately after the place you are rounding to is less than 5, you round down.
• If it's 5 or more, you round up.

For example, in rounding 10.08517 to four significant figures, you look at the fifth digit (1). Since 1 is less than 5, the number rounds down to 10.09.
Similarly, in scenarios like 17.159 rounding to two decimal places, the digit after the second decimal place is 9, so the second place (5) rounds up, resulting in 17.16.
Using these rounding rules helps you preserve the significant figures and ensure consistent accuracy in your calculations.