The mass and volume of a body are 4.237 g and 2.5 cm$^3$, respectively. The density of the material of the body in correct significant figures is (a) $1.6048$ g cm$^{-3}$ (b) $1.69$ g cm$^{-3}$ (c) $1.7$ g cm$^{-3}$ (d) $1.695$ g cm$^{-3}$

The mass and volume of a body are 4.237 g and 2.5 cm³, respectively. The density of the material of the body i

NCERT Exemplar · Q3

^3

(a)

1.6048

g cm

^{-3}

(b)

1.69

g cm

^{-3}

(c)

1.7

g cm

^{-3}

(d)

1.695

g cm

^{-3}

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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)

Watch out

These two rules are different! For multiplication, count sig figs. For addition, count decimal places. Mixing them up is the most common mistake.

A Concrete Example

You measure a rectangular field:

Length: $152.3$ m (4 sig figs)
Width: $45.0$ m (3 sig figs)

Area = $152.3 \times 45.0 = 6853.5$ m²

But your width measurement only has 3 sig figs, so you report $6.85 \times 10^3$ m² (or 6850 m², but that's ambiguous — use scientific notation).

The Big Picture

Significant figures aren't about being pedantic. They're about honesty in science. When you write $3.0$ instead of $3$ , you're telling the reader: "I measured this to the tenths place, and it was exactly 3.0 — not 2.9, not 3.1." That's valuable information.

Important

Final rule of thumb: Your answer cannot be more precise than your least precise measurement. Sig figs enforce that.

Queries such as "significant figures rules class 11 physics" and "significant figures calculation examples" are common around exam season, reflecting how central this topic is to the Units and Measurements chapter of the NCERT/CBSE Class 11 Physics curriculum. It's also a frequent source of numerical-based questions in JEE Main and NEET.

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

Operation	Rule	Why different?
+ / −	Decimal places	Uncertainty is absolute — it depends on the position of the last digit
× / ÷	Sig figs	Uncertainty is relative — it depends on the fraction of the value

Example to see the difference:

$1000 + 0.001 = 1000$ (decimal places rule: 1000 has 0 decimal places, so result is 1000)
$1000 \times 0.001 = 1$ (sig figs rule: 1000 has 4 sig figs? Actually ambiguous — but if 1000 has 1 sig fig, result is $1 \times 10^0$ )

5. The Deeper Reason: It’s All About Honest Reporting

The rules exist because:

Measurements have inherent uncertainty — no measurement is exact.
Calculations propagate uncertainty — the result cannot be more precise than the least precise input.
Sig figs are a practical shortcut — they avoid doing full error propagation for every calculation, while still giving a reasonable estimate of precision.

Bottom line: The rules aren’t arbitrary — they’re derived from the mathematics of uncertainty. When you round to the correct number of sig figs, you’re saying: “This is how precisely I actually know the answer, given the precision of my measurements.”

Quick Exam Tip

Addition/Subtraction: Look at decimal places — the weakest link is the one with fewest decimals.
Multiplication/Division: Look at sig figs — the weakest link is the one with fewest sig figs.
Mixed operations: Follow order of operations, applying the appropriate rule at each step.

To find the density with correct significant figures, divide the mass by the volume. The result must have the same number of significant figures as the measurement with the fewest significant figures. Here, the volume has 2 significant figures, so the density is $\boxed{1.7 \text{ g cm}^{-3}}$ .

When we perform calculations with measured quantities, the precision of our final answer cannot exceed the precision of the least precise measurement used in the calculation. This is the fundamental idea behind significant figures. Significant figures tell us how many digits in a measurement are known with certainty, plus one estimated digit.

For multiplication and division, the rule is straightforward: the result should be reported with the same number of significant figures as the measurement that has the least number of significant figures. This ensures that our calculated value doesn't falsely imply a higher level of precision than our original measurements allow.

Let's apply this concept to find the density of the material.

Identify the given measurements and their significant figures.
- Mass ( $m$ $m$ ) = $4.237$ $4.237$ g.
  - All non-zero digits are significant. So, $4.237$ has 4 significant figures.
- Volume ( $V$ $V$ ) = $2.5$ $2.5$ cm $^3$ $^{3}$ .
  - All non-zero digits are significant. So, $2.5$ has 2 significant figures.
Recall the formula for density.
$\rho$
Perform the calculation without considering significant figures initially.
- $\rho = \frac{4.237 \text{ g}}{2.5 \text{ cm}^3}$
- $\rho = 1.6948 \text{ g cm}^{-3}$
Determine the limiting number of significant figures.
- The mass ( $4.237$ g) has 4 significant figures.
- The volume ( $2.5$ cm $^3$ ) has 2 significant figures.
- According to the rule for multiplication and division, the result must be rounded to the same number of significant figures as the measurement with the least significant figures. In this case, the least number of significant figures is 2 (from the volume).
Round the calculated value to the correct number of significant figures.
- Our calculated value is $1.6948$ .
- We need to round this to 2 significant figures.
- The first two significant figures are $1$ and $6$ .
- We look at the digit immediately following the second significant figure, which is $9$ . Since $9$ is greater than or equal to $5$ , we round up the preceding digit ( $6$ ).
- Rounding $1.6948$ to 2 significant figures gives $1.7$ .
Watch out

A common mistake is to round only at the very end of a multi-step calculation. While it's generally good practice to carry extra digits through intermediate steps to minimize rounding errors, for a single-step calculation like this, applying the significant figure rule at the end is correct. Always ensure the final answer reflects the precision of the least precise input.

The density of the material, expressed with the correct number of significant figures, is $1.7$ g cm $^{-3}$ .

✓Final answer

\boxed{\text{(C)} 1.7 \text{ g cm}^{-3}}

Common Mistakes in Significant Figures Calculation (Density Problem)

The Correct Answer First

The density is:

\rho = \frac{4.237\ \text{g}}{2.5\ \text{cm}^3} = 1.6948\ \text{g cm}^{-3}

Correct answer: (C) $1.7$ g cm $^{-3}$ — because the volume (2.5) has 2 significant figures, so the result must be rounded to 2 significant figures.

Mistake #1: Reporting Too Many Decimal Places

What students do:

They divide $4.237 \div 2.5 = 1.6948$ and pick option (A) $1.6048$ or (D) $1.695$ without considering significant figures.

Why it’s wrong:

The rule for multiplication/division: the result must have the same number of significant figures as the measurement with the fewest significant figures.

Mass $4.237$ → 4 significant figures
Volume $2.5$ → 2 significant figures
So density must have 2 significant figures

How to avoid:

Always identify the least precise measurement (fewest significant figures) before doing the calculation. Here, volume $2.5$ has only 2 significant figures — that’s your limit.

Mistake #2: Confusing Decimal Places with Significant Figures

What students do:

They see $2.5$ has one decimal place, so they round the answer to one decimal place → $1.7$ (which happens to be correct here, but for the wrong reason).

Why it’s dangerous:

If the volume were $2.50$ (3 significant figures), the answer would be $1.69$ — but the same decimal-place logic would still give $1.7$ , which would be wrong.

How to avoid:

Count significant figures, not decimal places.

$2.5$ → 2 sig figs
$2.50$ → 3 sig figs
$0.025$ → 2 sig figs

Mistake #3: Rounding Too Early

What students do:

They round $2.5$ to $3$ or $4.237$ to $4.2$ before dividing, getting a rough answer like $1.4$ or $1.6$ .

Why it’s wrong:

Intermediate rounding introduces error. The rule is:

Do the full calculation first, then round to the correct significant figures at the end.

How to avoid:

Keep all digits during calculation. Only round the final answer.

Mistake #4: Forgetting the Trailing Zero Rule

What students do:

They get $1.7$ and think it’s wrong because it has only 2 digits — but they forget that $1.7$ does have 2 significant figures.

Why it’s correct:

$1.7$ has 2 significant figures (the trailing zero is not needed here).

If the answer were $1.70$ , that would be 3 significant figures — which would be incorrect.

How to avoid:

Remember:

$1.7$ → 2 sig figs
$1.70$ → 3 sig figs
$1.700$ → 4 sig figs

Only add trailing zeros if the measurement actually justifies that precision.

Quick Checklist to Avoid These Mistakes

Step	Action
1	Identify fewest sig figs among given data
2	Do the full calculation (no rounding)
3	Round final answer to that many sig figs
4	Check: is the answer in the correct units?

For this problem:

Fewest sig figs = 2 (from $2.5$ )
Full calculation: $1.6948$
Rounded to 2 sig figs: $1.7$ g cm $^{-3}$ → Option (C)

←Previous questionQ2 — The sum of the numbers 436.32, 227.2 and 0.301 in appropriate signific…→Next questionQ4 — The numbers 2.745 and 2.735 on rounding off to 3 significant figures w…

Q.The mass and volume of a body are 4.237 g and 2.5 cm $^3$ , respectively. The density of the material of the body in correct significant figures is

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

A Concrete Example

The Big Picture

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

5. The Deeper Reason: It’s All About Honest Reporting

Quick Exam Tip

Method: Significant Figures in Division (Least Precise Measurement Rule)

Concept

Step-by-Step Solution

Final Answer

Why Other Options Are Wrong

Common Mistakes in Significant Figures Calculation (Density Problem)

The Correct Answer First

Mistake #1: Reporting Too Many Decimal Places

Mistake #2: Confusing Decimal Places with Significant Figures

Mistake #3: Rounding Too Early

Mistake #4: Forgetting the Trailing Zero Rule

Quick Checklist to Avoid These Mistakes

Q.The mass and volume of a body are 4.237 g and 2.5 cm3^33, respectively. The density of the material of the body in correct significant figures is

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Q.The mass and volume of a body are 4.237 g and 2.5 cm $^3$ , respectively. The density of the material of the body in correct significant figures is

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