Chapter 1: Problem 39
Carry out the following operations, and express the answers with the appropriate number of significant figures. (a) \(14.3505+2.65\) (b) \(952.7-140.7389\) (c) \(\left(3.29 \times 10^{4}\right)(0.2501)\) (d) \(0.0588 / 0.677\)
Short Answer
Expert verified
(a) 16.00
(b) 811.96
(c) \(8.23 \times 10^{4}\)
(d) 0.0869
Step by step solution
01
(a) 14.3505 + 2.65
To add numbers, align the decimals and add. However, we need to remember the significant figures rule for addition and subtraction: round your answer to the least number of decimal places of any number in the problem.
14.3505
+ 2.65
---------
16.0005 (not rounded)
Now, we will round the result to the least number of decimal places, which, in this case, is 2.
Final answer: 16.00
02
(b) 952.7 - 140.7389
For subtraction, align the decimals and subtract, then use the same significant figures rule for addition mentioned in part (a).
952.7
-140.7389
-----------
811.9611 (not rounded)
The least number of decimal places in the problem is 1, so we will round our answer to 1 decimal place.
Final answer: 811.96
03
(c) (3.29 x 10^4) (0.2501)
For multiplication, multiply the numbers as normal but follow the significant figures rule for multiplication and division: round your answer to the least number of significant figures of any number in the problem.
(3.29 x 10^4) (0.2501) = 822650
We have 3 significant figures in the first number (3.29) and 4 significant figures in the second (0.2501), so the result should have 3 significant figures.
Final answer: 8.23 x 10^4
04
(d) 0.0588 / 0.677
For division, divide the numbers as normal and use the same significant figures rule for multiplication mentioned in part (c).
0.0588 / 0.677 = 0.086850809289
We have 3 significant figures in both the numerator (0.0588) and the denominator (0.677), so our final answer should have 3 significant figures as well.
Final answer: 0.0869
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Addition and Subtraction Rules
When it comes to addition and subtraction of numbers, itβs important to consider how many decimal places are in each number involved in the calculation. To apply this correctly, you need to align the decimal points of the numbers you are adding or subtracting. This way, you can see clearly which number has the fewest decimal places. Once you've done the mathematical operation, remember to round the result to match the number with the least decimal places.
For example, if you are adding 14.3505 and 2.65, you should look at the decimal places:
For example, if you are adding 14.3505 and 2.65, you should look at the decimal places:
- 14.3505 has four decimal places after the decimal point.
- 2.65 has two decimal places.
Multiplication and Division Rules
In multiplication and division, significant figures guide you on how to round off your final result. Unlike addition and subtraction, what matters here is the total number of significant figures in the numbers being multiplied or divided. Here, the answer must be rounded to match the number with the fewest significant figures in the problem.
For example, if you're multiplying 3.29 (which has 3 significant figures) by 0.2501 (which has 4 significant figures), the intermediate product is 8226.505. Despite the intricacy of the numbers and multiplication, the most critical rule is focusing on the number with the least significant figures, which in this scenario is 3. Thus, you'll round the result to 3 significant figures, providing an outcome of 8.23 x 10β΄.
Similarly, for an operation like dividing 0.0588 by 0.677, both numbers have 3 significant figures. Therefore, the answer will need to be rounded to 3 significant figures as well, following the division process.
For example, if you're multiplying 3.29 (which has 3 significant figures) by 0.2501 (which has 4 significant figures), the intermediate product is 8226.505. Despite the intricacy of the numbers and multiplication, the most critical rule is focusing on the number with the least significant figures, which in this scenario is 3. Thus, you'll round the result to 3 significant figures, providing an outcome of 8.23 x 10β΄.
Similarly, for an operation like dividing 0.0588 by 0.677, both numbers have 3 significant figures. Therefore, the answer will need to be rounded to 3 significant figures as well, following the division process.
Rounding Significant Figures
Rounding significant figures is an essential step in maintaining the precision and accuracy of scientific calculations. When rounding numbers, it's pivotal to look at the digits immediately after the last included significant figure.
Here's a concise way to round:
Here's a concise way to round:
- If the next digit is less than 5, you retain the last included figure and drop the subsequent ones.
- If the digit is 5 or greater, increase the last significant figure by one and drop what follows.