If the density of a solution is 3.12g mL⁻¹, the mass of 1.5 mL solution in significant figures is ___. (i) 4.7

NCERT Exemplar · Q11

3.12\ \text{g mL}^{-1}

(i) 4.7 g

(ii)

4680 \times 10^{-3}

(iii) 4.680 g

(iv) 46.80 g

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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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Multiply density by volume, then round the result to match the least number of significant figures in the given data (two). The mass is 4.7 g.

Why significant figures matter in multiplication

When you multiply measured quantities, the precision of your answer cannot exceed the precision of your least-precise input. Significant figures capture this idea: they tell us how many digits in a number are meaningful based on the measurement's uncertainty.

The rule for multiplication and division is straightforward: the result should have as many significant figures as the measurement with the fewest significant figures.

Here we're using the relationship between density, mass, and volume:

\text{Mass} = \text{Density} \times \text{Volume}

Let's identify the significant figures in our given data and then calculate.

Step-by-step calculation

Count significant figures in the given values
- Density: $3.12\ \text{g mL}^{-1}$ has 3 significant figures (all non-zero digits count)
- Volume: $1.5\ \text{mL}$ has 2 significant figures (the trailing zero after the decimal would need to be written explicitly to count)
Perform the raw calculation

\text{Mass} = 3.12 \times 1.5 = 4.68\ \text{g}

Apply the significant figure rule

The limiting factor is the volume with only 2 significant figures. Therefore, our answer must be rounded to 2 significant figures.

$4.68\ \text{g}$ rounded to 2 significant figures becomes $4.7\ \text{g}$ . …

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Here are the common mistakes students make on this significant figures calculation, along with how to avoid each.

Mistake 1: Forgetting to apply the rule for multiplication with significant figures

The error:

Students multiply $3.12 \times 1.5$ and get $4.68$ , then write the answer as 4.68 g or 4.680 g without checking the least number of significant figures in the inputs.

Why it happens:

They focus only on the arithmetic and ignore the significant figure rule:

In multiplication/division, the result should have the same number of significant figures as the factor with the fewest.

How to avoid:

Identify significant figures in each given value:
- $3.12$ has 3 significant figures.
- $1.5$ has 2 significant figures.
The result must be rounded to 2 significant figures.
So $4.68$ → round to 4.7 g (2 sig figs).

✓ Correct answer: (i) 4.7 g

Mistake 2: Confusing significant figures with decimal places

The error:

A student sees $4.680$ and thinks “four decimal places means more precision” — but the rule is about significant figures, not decimal places.

Why it happens:

Decimal place rules (for addition/subtraction) are mixed up with significant figure rules (for multiplication/division).

How to avoid:

Memorise the distinction:
- Addition/Subtraction: round to least decimal places.
- Multiplication/Division: round to least significant figures.
Here it’s multiplication, so use significant figures, not decimal places.

Mistake 3: Misreading the options — especially scientific notation

The error:

Option (ii) is $4680 \times 10^{-3}$ g. A student might think this equals $4.680$ g and pick it, missing that $4680$ has 4 significant figures (not 2).

Why it happens:

They convert $4680 \times 10^{-3} = 4.680$ without checking the significant figures in the original form.

How to avoid:

Always check the number of significant figures in the given form, not just the numeric value.
$4680 \times 10^{-3}$ implies 4 sig figs (the digits 4,6,8,0 are all significant).
Since the answer must have 2 sig figs, this option is wrong.

Mistake 4: Forgetting to round correctly

The error: …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

KCET 2025Set D-41 markMCQ
$Q. Select the INCORRECT statement/s from the following:$
(a) 22 books have infinite significant figures
(b) In the answer of calculation $2.5 \times 1.25$ has four significant figures
(c) Zero's preceding to first non-zero digit are significant
(d) In the answer of calculation $12.11 + 18.0 + 1.012$ has three significant figures (A) b, c and d only (B) b and c only (C) b and d only (D) a and b only
›Reveal solutionSolution
Evaluate each of the four statements against the significant-figure rules (exact numbers, multiplication rule, leading zeros, addition rule) and collect the false ones.

Statement (a): "22 books have infinite significant figures."

"22 books" is an exact counted number, not a measurement. Counting is not subject to measurement uncertainty — there are precisely $22$ , not $22 \pm 0.5$ . Exact numbers (and defined constants such as $1\ \mathrm{km} = 1000\ \mathrm{m}$ ) are treated as having infinite significant figures, so they never limit the precision of a calculation.

$\Rightarrow$ (a) is CORRECT.

Statement (b): "In the answer of $2.5 \times 1.25$ there are four significant figures."

The multiplication/division rule: the result carries as many significant figures as the factor with the fewest.

$2.5$ has 2 significant figures.

$1.25$ has 3 significant figures.

The raw product is
$2.5 \times 1.25 = 3.125$
but it must be rounded to the smaller count, 2 significant figures:
$\Rightarrow 3.1$
The claim of four significant figures is wrong (it just reports every digit the calculator shows).

$\Rightarrow$ (b) is INCORRECT.

Statement (c): "Zeros preceding the first non-zero digit are significant."

Leading zeros are never significant — they are placeholders that merely fix the decimal point. For example $0.0025$ has only 2 significant figures ( $2$ and $5$ ); writing it as $2.5 \times 10^{-3}$ makes this obvious, since the leading zeros vanish entirely in scientific notation.

$\Rightarrow$ (c) is INCORRECT.

Statement (d): "In the answer of $12.11 + 18.0 + 1.012$ there are three significant figures." …
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COMEDK 2024Set 2024-A1 markMCQ
$1.5 \mathrm{~kg}$
›Reveal solutionSolution
$P = \dfrac{mgh}{t} = 3.465\,\text{W}$ . The least precise datum (mass $1.5\,\text{kg}$ , two significant figures) fixes the precision, so the answer to the appropriate significant figures is $3.5\,\text{W}$ — option (D).

Concept

The motor lifts the load against gravity, so the work done equals the gain in gravitational potential energy, $W = mgh$ , and the power is that work divided by the time, $P = mgh/t$ . The phrase "appropriate significant figures" is the real point of the question: a calculated result can carry no more significant figures than the least precise measurement used.

Solution

Formula: $P = \dfrac{mgh}{t}$ .

Substitute: $P = \dfrac{1.5 \times 9.81 \times 1.128}{4.79}$ .

Evaluate: $1.5 \times 9.81 = 14.715$ ; $14.715 \times 1.128 = 16.59852\,\text{J}$ ; $16.59852 / 4.79 = 3.465\,\text{W}$ . …
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KCET 2023Set D-21 markMCQ
$\sqrt{3}$
›Reveal solutionSolution

Use the BCC body-diagonal contact relation to get the edge $a$ from the radius, then cube it — and convert Å to metres carefully.

1. The BCC radius–edge relation

In a body-centred cubic cell the atoms touch along the body diagonal, whose length is $\sqrt3\,a$ and which contains 4 radii:
$4r = \sqrt{3}\,a \qquad\Longrightarrow\qquad a = \frac{4r}{\sqrt3}$
2. Substitute $r = \sqrt3$ Å
$a = \frac{4 \times \sqrt3}{\sqrt3} = 4\ \text{Å}$
The $\sqrt3$ was chosen precisely so that it cancels — a clean edge length of 4 Å.

3. Volume of the cubic cell
$V = a^3 = (4\ \text{Å})^3 = 64\ \text{Å}^3$
4. Convert to $\mathrm{m^3}$

Since $1\ \text{Å} = 10^{-10}\ \mathrm{m}$ ,
$1\ \text{Å}^3 = (10^{-10})^3 = 10^{-30}\ \mathrm{m^3}$ $V = 64 \times 10^{-30}\ \mathrm{m^3} = \mathbf{6.4 \times 10^{-29}\ \mathrm{m^3}}$
5. Reading the distractors …
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Q.If the density of a solution is $3.12\ \text{g mL}^{-1}$ , the mass of 1.5 mL solution in 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

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 in multiplication

Step-by-step calculation

Method: Multiplication with Significant Figures

Step 1: Count significant figures in each given value

Step 2: Perform the multiplication exactly

Mistake 1: Forgetting to apply the rule for multiplication with significant figures

Mistake 2: Confusing significant figures with decimal places

Mistake 3: Misreading the options — especially scientific notation

Mistake 4: Forgetting to round correctly

Concept

Solution

1. The BCC radius–edge relation

2. Substitute $r = \sqrt3$ Å

3. Volume of the cubic cell

4. Convert to $\mathrm{m^3}$

5. Reading the distractors …

Q.If the density of a solution is 3.12 g mL−13.12\ \text{g mL}^{-1}3.12 g mL−1, the mass of 1.5 mL solution in significant figures is ______.

Why significant figures matter in multiplication

Step-by-step calculation

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Q.If the density of a solution is $3.12\ \text{g mL}^{-1}$ , the mass of 1.5 mL solution in significant figures is ______.

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