How many significant figures should be present in the answer of the following calculations? (i) (0.02856 × 298

Exercises · 1.31

Q. How many significant figures should be present in the answer of the following calculations?

(i)

\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}

(ii)

5 \times 5.364

(iii)

0.0125 + 0.7864 + 0.0215

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Concept understanding — Significant Figures Calculation

Significant Figures: The Art of Honest Measurement

Imagine you're measuring the length of a table with a ruler that has marks every millimeter. You see the table edge falls somewhere between 152.3 cm and 152.4 cm. You estimate it as 152.35 cm. But here's the truth: you're certain about 152.3, pretty sure about the 0.05, and guessing about anything beyond that. Significant figures are simply a way to communicate how much of that number you actually know.

The Core Idea

Every measurement has uncertainty. Significant figures (or "sig figs") are the digits in a number that carry meaningful information about its precision. They include all the digits you're sure of, plus one more that you estimate.

Note

A digit is "significant" if removing it would change the precision of the measurement. Zeros can be tricky — they might just be placeholders.

The Rules (Memorize These)

1. Non-zero digits are always significant

$123.45$ has 5 sig figs. Simple.

2. Zeros between non-zero digits are significant

$1002$ has 4 sig figs. The zeros are "sandwiched" — they're part of the measurement.

3. Leading zeros are never significant

$0.00123$ has 3 sig figs. Those zeros just tell you where the decimal point is.

4. Trailing zeros are significant only if there's a decimal point

$1200$ has 2 sig figs (no decimal — zeros are placeholders)
$1200.$ has 4 sig figs (decimal tells us those zeros were measured)
$1200.0$ has 5 sig figs

5. Exact numbers have infinite sig figs

If you count 5 apples, that's exactly 5 — no uncertainty. Conversion factors like $1 \text{ m} = 100 \text{ cm}$ are exact by definition.

Tip

When in doubt, write the number in scientific notation. $1.20 \times 10^3$ clearly has 3 sig figs, while $1.2 \times 10^3$ has 2.

Why This Matters: Calculations

When you multiply or add measurements, the uncertainty propagates. You can't claim more precision than your least precise measurement.

Multiplication and Division

The result should have the same number of sig figs as the measurement with the fewest sig figs.

$3.14 \times 2.5 = 7.85$ but you report $7.9$ (2 sig figs, because 2.5 has only 2)

Addition and Subtraction

The result should have the same decimal places as the measurement with the fewest decimal places.

$12.11 + 18.0 = 30.11$ but you report $30.1$ (one decimal place, because 18.0 has one) …

Why this formula?

Significant Figures: Why the Rules Work

Let’s start with the core idea: significant figures (sig figs) are a way to honestly report how precise a measurement is. The rules for addition/subtraction and multiplication/division aren’t arbitrary — they come directly from how uncertainty propagates through calculations.

1. The Fundamental Idea: Uncertainty is the Key

Every measurement has an uncertainty (error). When we say a length is 12.3 cm, we mean:

The true value lies somewhere between 12.25 cm and 12.35 cm (assuming ±0.05 cm uncertainty).
The last digit (3) is uncertain; the digits before it (1 and 2) are certain.

Why this matters: When we combine measurements, the uncertainty in the result depends on the uncertainties of the inputs. Sig fig rules are a shortcut for this uncertainty propagation.

2. Rule for Addition and Subtraction

Statement: The result should have the same number of decimal places as the measurement with the fewest decimal places.

Example:

$12.3 + 4.56 = 16.86$ → round to 16.9 (one decimal place, like 12.3)

Why this holds

Consider two measurements:

$A = 12.3 \pm 0.05$ (uncertainty in the tenths place)
$B = 4.56 \pm 0.005$ (uncertainty in the hundredths place)

When we add:

Certain digits: 12.3 has certainty up to the tenths place. 4.56 has certainty up to the hundredths place.
The weaker link: The tenths place of A is uncertain. So in the sum, the hundredths place (from B) is meaningless — because we don’t even know the tenths place of A exactly.

Mathematically, the absolute uncertainty in the sum is:

\Delta(A+B) = \sqrt{(\Delta A)^2 + (\Delta B)^2} \approx \sqrt{0.05^2 + 0.005^2} \approx 0.0502

This uncertainty is ~0.05, which affects the tenths place. So reporting the hundredths place is false precision.

Key takeaway: The result’s last significant digit is in the same decimal place as the least precise measurement’s last digit.

3. Rule for Multiplication and Division

Statement: The result should have the same number of significant figures as the measurement with the fewest significant figures.

Example:

$12.3 \times 4.56 = 56.088$ → round to 56.1 (three sig figs, like both inputs)

Why this holds

Let’s use relative uncertainty (percentage error):

$A = 12.3 \pm 0.05$ → relative uncertainty = $\frac{0.05}{12.3} \approx 0.00407$ (0.407%)
$B = 4.56 \pm 0.005$ → relative uncertainty = $\frac{0.005}{4.56} \approx 0.00110$ (0.110%)

For multiplication, relative uncertainties add (approximately):

\frac{\Delta(A \times B)}{A \times B} \approx \sqrt{\left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2}

Plugging in:

\approx \sqrt{0.00407^2 + 0.00110^2} \approx 0.00422 \text{ (0.422\%)}

Now, the absolute uncertainty in the product:

\Delta(A \times B) \approx 0.00422 \times (12.3 \times 4.56) \approx 0.00422 \times 56.088 \approx 0.237

This uncertainty (~0.2) affects the tenths place of the result. So the result 56.088 has uncertainty in the first decimal — meaning only three digits (5, 6, and the uncertain 1) are meaningful. That’s three sig figs, matching the input with fewer sig figs (both have three here).

Key takeaway: The number of sig figs in the result is limited by the least precise measurement’s number of sig figs, because relative uncertainty is dominated by the measurement with the largest relative error.

4. Why These Rules Are Different …

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The number of significant figures in a calculation result is determined by the least precise measurement used — for multiplication/division, that means the factor with the fewest significant figures; for addition/subtraction, it means the term with the fewest decimal places.

3 significant figures,
1 significant figure,
4 significant figures.

Why Significant Figures Matter

When you measure something, every digit you write carries information — but not all digits are equally trustworthy. The last digit is always an estimate. So when you combine measurements in a calculation, the result can’t be more precise than the least precise measurement you started with. That’s the core idea behind significant figures: they keep your answer honest.

For multiplication and division, the rule is: count the significant figures in each number, and the answer should have as many as the smallest count among them.

For addition and subtraction, the rule is different: look at decimal places, not total digits. The answer should have as many decimal places as the term with the fewest decimal places.

Let’s apply this to each part.

(i) $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

Step 1: Count significant figures in each factor.

$0.02856$ — the leading zeros don’t count. The digits 2, 8, 5, 6 are all significant. That’s 4 significant figures.
$298.15$ — all five digits are significant (no leading zeros, no trailing zeros without a decimal). That’s 5 significant figures.
$0.112$ — leading zero not significant. Digits 1, 1, 2 are significant. That’s 3 significant figures.
$0.5785$ — leading zero not significant. Digits 5, 7, 8, 5 are significant. That’s 4 significant figures.

Step 2: Identify the smallest count.

Among 4, 5, 3, and 4, the smallest is 3 (from $0.112$ ).

Step 3: Apply the rule.

Since this is a pure multiplication/division chain, the answer must be reported with 3 significant figures.

Tip

You don’t actually need to compute the full number to decide the significant figure count — just look at the inputs. But if you do compute it, the raw result is about $1.652\ldots$ , which rounds to $1.65$ for three significant figures.

(ii) $5 \times 5.364$

Step 1: Count significant figures.

$5$ — this is a tricky one. Is it an exact count or a measurement? In a calculation like this, if a number is written without a decimal point (just “5”), it is usually considered to have 1 significant figure. (If it were $5.0$ , that would be 2 significant figures.)
$5.364$ — all four digits are significant. That’s 4 significant figures.

Step 2: Smallest count is 1.

So the answer should have 1 significant figure.

Step 3: Compute and round. …

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Here’s a breakdown of the common mistakes students make when determining significant figures in calculations, specifically for the three problems you listed, along with how to avoid each.

General Mistake #1: Confusing the Rules for Addition vs. Multiplication

The Mistake:

Students often apply the multiplication/division rule (least number of significant figures) to addition/subtraction problems, or vice versa.

How to Avoid:

Memorise the two distinct rules:

Multiplication/Division: The answer should have the same number of significant figures as the measurement with the fewest significant figures.
Addition/Subtraction: The answer should have the same number of decimal places as the measurement with the fewest decimal places.

Problem (i): $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

Common Mistake #2: Counting Zeros Incorrectly

The Mistake:

Thinking $0.02856$ has 6 significant figures (because it has 6 digits) or thinking $0.112$ has 4 significant figures.

How to Avoid:

Remember: Leading zeros are never significant.

$0.02856$ → digits: 0.02856 → 4 significant figures (2,8,5,6)
$0.112$ → digits: 0.112 → 3 significant figures (1,1,2)
$298.15$ → 5 significant figures
$0.5785$ → 4 significant figures

Common Mistake #3: Forgetting to Compare All Terms

The Mistake:

Only comparing the first two numbers and ignoring the rest.

How to Avoid:

List the significant figures for every number in the calculation:

$0.02856$ → 4 SF
$298.15$ → 5 SF
$0.112$ → 3 SF (this is the smallest)
$0.5785$ → 4 SF

Correct Answer: The answer should have 3 significant figures.

Problem (ii): $5 \times 5.364$

Common Mistake #4: Treating Exact Numbers as Measurements

The Mistake:

Thinking $5$ has 1 significant figure, so the answer should have 1 significant figure.

How to Avoid:

Recognise that exact numbers (counted values, defined constants) have infinite significant figures.

Here, $5$ is likely an exact integer (e.g., 5 apples, 5 times). It does not limit the significant figures.
Only $5.364$ (4 SF) limits the answer.

Correct Answer: The answer should have 4 significant figures.

Problem (iii): $0.0125 + 0.7864 + 0.0215$

Common Mistake #5: Using Significant Figures Instead of Decimal Places

The Mistake:

Counting significant figures in each term (e.g., $0.0125$ has 3 SF) and giving the answer with 3 SF.

How to Avoid:

For addition/subtraction, ignore significant figures and look only at decimal places: …

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Q.How many significant figures should be present in the answer of the following calculations?

Concept understanding — Significant Figures Calculation

Significant Figures: The Art of Honest Measurement

The Core Idea

The Rules (Memorize These)

1. Non-zero digits are always significant

2. Zeros between non-zero digits are significant

3. Leading zeros are never significant

4. Trailing zeros are significant only if there's a decimal point

5. Exact numbers have infinite sig figs

Why This Matters: Calculations

Multiplication and Division

Addition and Subtraction

Why this formula?

Significant Figures: Why the Rules Work

1. The Fundamental Idea: Uncertainty is the Key

2. Rule for Addition and Subtraction

Why this holds

3. Rule for Multiplication and Division

Why this holds

4. Why These Rules Are Different …

Why Significant Figures Matter

(i) $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

(ii) $5 \times 5.364$

Method: Significant Figure Rules for Arithmetic Operations

(i) $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

(ii) $5 \times 5.364$

(iii) $0.0125 + 0.7864 + 0.0215$

General Mistake #1: Confusing the Rules for Addition vs. Multiplication

Problem (i): $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

Common Mistake #2: Counting Zeros Incorrectly

Common Mistake #3: Forgetting to Compare All Terms

Problem (ii): $5 \times 5.364$

Common Mistake #4: Treating Exact Numbers as Measurements

Problem (iii): $0.0125 + 0.7864 + 0.0215$

Common Mistake #5: Using Significant Figures Instead of Decimal Places

Q.How many significant figures should be present in the answer of the following calculations?

Why Significant Figures Matter

(i) 0.02856×298.15×0.1120.5785\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}0.57850.02856×298.15×0.112​

(ii) 5×5.3645 \times 5.3645×5.364

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(i) $\dfrac{0.02856 \times 298.15 \times 0.112}{0.5785}$

(ii) $5 \times 5.364$

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