Give answers to the following problems in the correct number of significant figures:
Sum of 23.15, 0.984, 124, and 12.5
Sum of 2.115, 23.04, 13.8, and 199.66
Product of 127.08 and 13.1
Quotient of 4466.83 divided by 35.61
The correct answer and explanation is :
Let’s solve each of the problems one by one, ensuring that we adhere to the rules of significant figures.
1. Sum of 23.15, 0.984, 124, and 12.5
Step 1: Identify the number of decimal places
- 23.15 has 2 decimal places
- 0.984 has 3 decimal places
- 124 has no decimal places
- 12.5 has 1 decimal place
Step 2: Apply the rule of significant figures for addition/subtraction
When adding or subtracting numbers, the result should be rounded to the least number of decimal places in the data. In this case, 124 has no decimal places, so the result must be rounded to the nearest whole number.
Step 3: Perform the sum
[
23.15 + 0.984 + 124 + 12.5 = 160.634
]
Step 4: Round the result
The least number of decimal places is 0 (from 124), so we round the result to the nearest whole number:
[
160.634 \rightarrow 161
]
Answer: 161
2. Sum of 2.115, 23.04, 13.8, and 199.66
Step 1: Identify the number of decimal places
- 2.115 has 3 decimal places
- 23.04 has 2 decimal places
- 13.8 has 1 decimal place
- 199.66 has 2 decimal places
Step 2: Apply the rule for addition/subtraction
The least number of decimal places is 1 (from 13.8), so the result should be rounded to 1 decimal place.
Step 3: Perform the sum
[
2.115 + 23.04 + 13.8 + 199.66 = 238.615
]
Step 4: Round the result
Since the least number of decimal places is 1, we round to 1 decimal place:
[
238.615 \rightarrow 238.6
]
Answer: 238.6
3. Product of 127.08 and 13.1
Step 1: Identify the number of significant figures
- 127.08 has 5 significant figures
- 13.1 has 3 significant figures
Step 2: Apply the rule for multiplication/division
When multiplying or dividing, the result should have the same number of significant figures as the number with the least significant figures. Here, 13.1 has 3 significant figures, so the result should have 3 significant figures.
Step 3: Perform the multiplication
[
127.08 \times 13.1 = 1665.248
]
Step 4: Round the result
Since the least number of significant figures is 3, we round the result to 3 significant figures:
[
1665.248 \rightarrow 1660
]
Answer: 1660
4. Quotient of 4466.83 divided by 35.61
Step 1: Identify the number of significant figures
- 4466.83 has 5 significant figures
- 35.61 has 4 significant figures
Step 2: Apply the rule for multiplication/division
The result should have the same number of significant figures as the number with the least significant figures. Here, 35.61 has 4 significant figures, so the result should have 4 significant figures.
Step 3: Perform the division
[
4466.83 \div 35.61 = 125.639
]
Step 4: Round the result
Since the least number of significant figures is 4, we round the result to 4 significant figures:
[
125.639 \rightarrow 125.6
]
Answer: 125.6
Explanation of Significant Figures
Significant figures (sig figs) are used in scientific measurements to convey the precision of a measurement. The rules for significant figures help determine how precise a calculated result can be. Here are the key rules that apply to the operations:
- Addition and Subtraction: When adding or subtracting, the result should be rounded to the least number of decimal places in the numbers involved.
- Multiplication and Division: When multiplying or dividing, the result should have the same number of significant figures as the number with the least significant figures.
For instance, in the sum of 23.15, 0.984, 124, and 12.5, the least number of decimal places is zero (from 124), so the sum must be rounded to the nearest whole number, yielding 161. Similarly, for the product of 127.08 and 13.1, since 13.1 has the fewest significant figures (3), the product is rounded to 3 significant figures (1660).
This system ensures that the results of calculations are presented in a way that reflects the precision of the original measurements.