Significant Figures in Addition, Subtraction Multiplication and Division

Significant figures play a crucial role in scientific calculations. When performing arithmetic operations such as addition, subtraction, multiplication, and division with measured values, the number of significant figures must be maintained correctly to ensure precision.

Rules for Algebraic Operations with Significant Figures

(a) Significant Figures in Addition and Subtraction Operation

• When adding or subtracting measured values, the final result should have the same number of decimal places as the measurement with the least decimal places.

Example: Find the sum of 420.42 m, 420.4 m, and 0.402 m.

Solution:

  420.42 m
+ 420.4 m
+  0.402 m
------------
  841.222 m

• The least precise measurement (420.4 m) has only one decimal place.
• Thus, the final answer must be rounded to one decimal place.
• Final Answer: 841.2 m

What if one or all the numbers do not have decimal places?

How do we determine the correct number of significant figures in an addition or subtraction when there is no decimal place?

Remember, the whole idea of rounding off is to drop the insignificant digits. These are all the digits to the right of the last significant digit.

For example, what is the answer to 23 – 14.6? 23 does not have a decimal place, which means the 3 is the last significant figure, and all the digits after it should be dropped following the rules for rounding off.

In other words, we can say that we rounded the answer to the ones place because 23 has the certainty of ones which is less than tenths (8.4).

Let’s look at another example: 5,289 + 100?

The sum of 5,289 and 100 is 5,389 but we need to round the final answer. The first number (5,289) ends with the ones place, while the second number (100) is significant only to the hundreds place, and therefore, we round 5,389 to the hundreds place: 5,400 (no decimal at the end).

(b) Significant Figures in Multiplication and Division Operation

• When multiplying or dividing measured values, the final result should have the same number of significant figures as the measurement with the least significant figures.

For example, in the multiplication above the exact answer is 17.75845, however, we round it to 3 significant figures according to the number 8.65 which has three significant figures, while 2.053 has four significant figures. This means the 17.75845 needs to be rounded up because of the 5 next to it.

Same for the division, 138 ÷ 11.9 = 11.596638, and we round up the 5 to 6 because of the 9 on the right. Therefore, the answer is 11.6.

Example 1: Calculate 1.2×36.72.

Solution:

  1.2 × 36.72 = 44.064

• The least number of significant figures in the given values is 2.
• Hence, the result should be rounded to 2 significant figures.
• Final Answer: 44

Example 2: Calculate 1100 ÷ 10.2.

Solution:

  1100 ÷ 10.2 = 107.8431373

• The least number of significant figures is 3 (10.2 has 3 significant figures).
• The result must be rounded to 3 significant figures.
• Final Answer: 108

Significant Numbers in Multistep Calculations

If there are more than two numbers involved in the calculation, the rule is to not round off any intermediate result and only do it for the final answer.

For example:

The calculation involves only multiplication and therefore, we keep all the digits for the intermediate numbers and round off the final answer according to the number with the fewest number of significant figures.

Now, what we mean by an intermediate result is, for example, when we multiply 3.54 by 2.6458, the answer is 9.366132. Do not round off this number to 3 significant figures yet. If we do, the answer would be 9.37 and after multiplying it by 7.48764, we get 70.159186 which, after rounding off again, gives 70.2 as a final answer.

The same would be true if there was a division in the calculation. For example,

2.1 x 5.687 ÷ 1.654 = 7.2205 ≈ 7.2

The answer has 2 significant figures based on the number 2.1.

Multistep Calculations with Mixed Operations

Let’s now discuss the significant figures when the calculation involves both subtraction division and addition subtraction. Although the same principle of rounding off only at the very end is applicable, there are some nuances we need to consider.

For example, the following calculation involves multiplication and subtraction.

4.647 x 2.5149 x (6.2594 – 2.34)

Now, the first thing is to follow the correct order of operations. Remember, the one in the parenthesis must be done before the result is multiplied by the other number. So, in this case, we first subtract 2.34 from 6.2594. The result is 3.9194 and if it was the final answer, we’d round it off to 3.92 because 2.34 has only two decimal places.

Instead, we only underline the last digit (1) to keep track of the significant figure. By the last digit, we mean the last digit that is supposed to be there based on the decimal places. Underlining the 1 indicates that the answer should have 3 significant figures after the multiplication:

Let’s do one more example of a calculation involving addition/subtraction and multiplication/division.

Calculate the answer of this calculation expressed in correct number of significant figures:

The intermediate number in the denominator is limited to two decimal places which, in this case, is equivalent to three significant figures. We do not round off this number because there is still one step it is going to be used in. However, we keep in mind that it will be used as having three significant figures and because the two numbers in the numerator have more significant figures, the final answer is rounded to three significant figures.

Conceptual Questions with Answers

Q1: Why do we use significant figures in calculations?

Answer: Significant figures indicate the precision of measured values and help maintain accuracy while performing mathematical operations.

Q2: What happens if we do not follow the rules of significant figures in calculations?

Answer: Ignoring significant figures may lead to overestimation or underestimation of results, causing errors in scientific and engineering calculations.

Multiple Choice Questions (MCQs)

Q1: What is the correct sum of 23.456, 4.1, and 0.5678 using the rules of significant figures?

A) 28.1238
B) 28.12
C) 28.13
D) 28.1

Correct Answer: C) 28.13

Explanation: The least decimal places in the given numbers is 1 decimal place (4.1). Hence, the sum must be rounded to 2 decimal places, giving 28.13.

Q2: If a measurement is given as 5.42 × 3.1, how many significant figures should the result have?

A) 1
B) 2
C) 3
D) 4

Correct Answer: B) 2

Explanation: The least number of significant figures among the given values is 2 (3.1). Thus, the answer must have 2 significant figures.

Do You Know?

• Significant figures help scientists and engineers maintain precision in experiments.
• The rounding-off rules depend on the operation being performed (addition/subtraction vs multiplication/division).
• The concept of significant figures applies to both scientific and everyday calculations.

Worksheet

Solve the following questions following the rules of significant figures:

1. Add: 25.67, 13.4, and 0.235.
2. Subtract: 98.76 – 0.321.
3. Multiply: 4.56 × 2.1.
4. Divide: 500.0 ÷ 2.5.
5. Add: 10.345 + 2.1 + 0.02.

Test Paper

Total Marks: 10

Important Points for Quick Revision

✅ In addition/subtraction, round the result to the least decimal places present in the given numbers.
✅ In multiplication/division, round the result to the least significant figures present in the given numbers.
✅ Leading zeros are not significant, while trailing zeros in a decimal number are significant.
✅ Always round-off correctly to maintain the precision of scientific calculations.

Test Your Knowledge (Quiz)

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