Assertion (A): Significant figures for 0.200 is 3 where as for 200 it is 1. Reason (R): Zero at the end or rig

NCERT Exemplar · Q40

Q. Assertion (A): Significant figures for 0.200 is 3 where as for 200 it is 1. Reason (R): Zero at the end or right of a number are significant provided they are not on the right side of the decimal point. Choose the correct option out of the choices given below.

(i) Both A and R are true and R is correct explanation of A.

(ii) Both A and R are true but R is not a correct explanation of A.

(iii) A is true but R is false.

(iv) Both A and R are false.

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Concept understanding — Unit Conversion

Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing

Imagine you're measuring the length of your desk. You pull out a ruler marked in centimetres and find it's 120 cm long. Your friend, using a metre stick, says it's 1.2 m. You're both right — you've just used different units to describe the same physical length.

That's the core idea: unit conversion is the process of changing how you express a quantity without changing the quantity itself.

The Intuition: Same Quantity, Different Labels

Think of a pizza. Whether you call it "one pizza" or "8 slices," the amount of pizza hasn't changed. You've just used a different unit (pizza vs. slice) to describe it.

Similarly, 1 metre and 100 centimetres are the same length — just like 1 pizza and 8 slices are the same amount. The number changes (1 becomes 100, or 1 becomes 8), but the actual thing being measured stays identical.

Note

This is the most important idea to hold onto: conversion changes the number, not the quantity. If you ever feel like the quantity has changed, you've made a mistake.

The Precise Statement

Unit conversion is the multiplication of a quantity by a conversion factor — a fraction equal to 1 — that cancels the old unit and introduces the new one.

A conversion factor looks like this:

\frac{\text{new unit}}{\text{old unit}} = 1

For length: $\frac{1 \text{ m}}{100 \text{ cm}} = 1$ and $\frac{100 \text{ cm}}{1 \text{ m}} = 1$ .

Why are these fractions equal to 1? Because 1 metre is 100 centimetres. The numerator and denominator describe the same physical length, so their ratio is exactly 1.

How to Convert: The Only Rule You Need

Multiply by a conversion factor that cancels the unit you have and leaves the unit you want.

Let's convert 120 cm to metres:

Start with what you have: $120 \text{ cm}$
Choose the conversion factor that has "cm" in the denominator (to cancel it) and "m" in the numerator: $\frac{1 \text{ m}}{100 \text{ cm}}$
Multiply:

120 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = \frac{120}{100} \text{ m} = 1.2 \text{ m}

The "cm" units cancel just like numbers do: $\frac{\text{cm}}{\text{cm}} = 1$ .

Tip

Always write the units explicitly. If the units don't cancel correctly, you've used the wrong conversion factor. This catches 90% of conversion mistakes.

The Reverse: Metres to Centimetres

Now convert 1.2 m to cm. This time, you want "cm" to remain and "m" to cancel. Use $\frac{100 \text{ cm}}{1 \text{ m}}$ :

1.2 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 1.2 \times 100 \text{ cm} = 120 \text{ cm}

Notice: when going from a larger unit (m) to a smaller unit (cm), the number gets larger (1.2 → 120). When going from smaller to larger, the number gets smaller (120 → 1.2). This is a useful sanity check.

Common Conversion Factors You'll Use

Quantity	Relationship	Conversion Factors
Length	1 m = 100 cm	$\frac{1 \text{ m}}{100 \text{ cm}}$ , $\frac{100 \text{ cm}}{1 \text{ m}}$
Mass	1 kg = 1000 g	$\frac{1 \text{ kg}}{1000 \text{ g}}$ , $\frac{1000 \text{ g}}{1 \text{ kg}}$
Time	1 h = 60 min	$\frac{1 \text{ h}}{60 \text{ min}}$ , $\frac{60 \text{ min}}{1 \text{ h}}$
Speed	1 km/h = $\frac{1000}{3600}$ m/s	$\frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}}$

Why this formula?

Dimensional Analysis: Why the Key Principles Hold

Dimensional Analysis is a powerful tool in physics and engineering that lets us check the consistency of equations, derive relationships, and convert units. But why does it work? Let's build the reasoning from the ground up.

1. The Core Idea: Physical Quantities Have Dimensions

Every physical quantity (like length, time, mass) can be expressed in terms of fundamental dimensions. The most common set in mechanics is:

L = Length
M = Mass
T = Time

For example:

Speed has dimensions $[L T^{-1}]$
Force has dimensions $[M L T^{-2}]$
Energy has dimensions $[M L^2 T^{-2}]$

Why this matters: Two quantities can only be meaningfully compared or equated if they have the same dimensions. You cannot add apples to oranges — and you cannot add length to time.

2. The Principle of Dimensional Homogeneity

The key formula that underpins everything is:

Every valid physical equation must be dimensionally homogeneous.

This means: the dimensions on the left-hand side must equal the dimensions on the right-hand side.

Why must this hold?

Consider an equation like:

v = u + at

Left side: $[v] = L T^{-1}$
Right side: $[u] = L T^{-1}$ , $[at] = (L T^{-2})(T) = L T^{-1}$

Both sides have dimensions $L T^{-1}$ . If they didn't match, the equation would be physically meaningless — you'd be comparing quantities that cannot be equal in any real experiment.

Reasoning: Physical laws describe relationships between measurable quantities. If the dimensions don't match, the equation cannot represent a real physical relationship, because the numerical value would depend on the arbitrary choice of units.

3. The Buckingham Pi Theorem: Why We Can Derive Relationships

This is the deeper mathematical reason. The Buckingham Pi Theorem states:

If a physical problem involves $n$ variables and $k$ fundamental dimensions, then it can be reduced to $n-k$ independent dimensionless groups (called $\pi$ groups).

Why does this work?

Imagine you have a relationship:

f(Q_1, Q_2, \dots, Q_n) = 0

where each $Q_i$ has dimensions. Because the equation must be dimensionally homogeneous, we can rearrange it into a function of dimensionless products only:

F(\pi_1, \pi_2, \dots, \pi_{n-k}) = 0

The reasoning: Dimensions act as constraints. Each fundamental dimension (M, L, T) gives one constraint. So if you have $n$ variables and $k$ constraints, you only have $n-k$ independent dimensionless combinations.

Example: For a simple pendulum, the period $T$ depends on length $L$ , mass $m$ , and gravity $g$ . That's 4 variables with 3 dimensions (M, L, T). So $4-3 = 1$ dimensionless group: $\pi = \frac{T^2 g}{L}$ . This tells us $T \propto \sqrt{L/g}$ without solving any differential equation.

4. Why We Can Convert Units Using Dimensional Analysis

The conversion factor formula:

\text{Value in new unit} = \text{Value in old unit} \times \left(\frac{\text{old unit}}{\text{new unit}}\right)^{\text{dimension exponent}}

Why this works: …

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The Assertion correctly applies significant figure rules: 0.200 has 3 significant figures, and 200 has 1. However, the Reason incorrectly states the rule for trailing zeros, making it false.

Understanding significant figures is crucial in science and engineering because they convey the precision of a measurement. When we write a number, the significant figures tell us which digits are known with certainty and which one is estimated. Zeros often cause confusion, so let's clarify their role.

The general rules for determining significant figures are:

Non-zero digits are always significant. (e.g., 123 has 3 sig figs).
Zeros between non-zero digits (confined zeros) are significant. (e.g., 101 has 3 sig figs, 1.002 has 4 sig figs).
Leading zeros (zeros before non-zero digits) are not significant. They only indicate the position of the decimal point. (e.g., 0.0025 has 2 sig figs).
Trailing zeros (zeros at the end of a number):
- If the number contains a decimal point, trailing zeros are significant. (e.g., 2.00 has 3 sig figs, 0.200 has 3 sig figs, 20.0 has 3 sig figs). This indicates that these zeros were measured or are known to be zero.
- If the number does not contain a decimal point, trailing zeros are generally not significant. They are placeholders. (e.g., 200 has 1 sig fig, 2000 has 1 sig fig). To make them significant, one would typically write the number in scientific notation (e.g., $2.00 \times 10^2$ for 3 sig figs) or add a decimal point (e.g., 200. for 3 sig figs).

Let's apply these rules to evaluate the given Assertion and Reason.

Analyze Assertion (A): Significant figures for $0.200$ is $3$ where as for $200$ it is $1$ .
- For $0.200$ :
  - The leading zero before the decimal point (0.) is not significant.
  - The digit 2 is a non-zero digit, so it is significant.
  - The two trailing zeros (00) are to the right of the decimal point. According to rule 4a, trailing zeros with a decimal point are significant.
  - Therefore, $0.200$ has $1$ (from 2) $+ 2$ (from 00) $= \textbf{3 significant figures}$ .
- For $200$ :
  - The digit 2 is a non-zero digit, so it is significant.
  - The two trailing zeros (00) are at the end of the number, but there is no decimal point explicitly shown. According to rule 4b, these trailing zeros are generally not significant. They are placeholders.
  - Therefore, $200$ has $\textbf{1 significant figure}$ .
- Since both parts of the statement are correct, Assertion (A) is True.
Analyze Reason (R): Zero at the end or right of a number are significant provided they are not on the right side of the decimal point. …

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Concept: Significant Figures (Significant Digits)

Significant figures are the digits in a number that carry meaning contributing to its precision. Rules for counting them depend on the position of zeros.

Method: Rule-Based Counting of Significant Figures

Steps:

Identify all non-zero digits — they are always significant.
Identify zeros:
- Zeros between non-zero digits are significant.
- Zeros to the left of the first non-zero digit are not significant (they are placeholders).
- Zeros to the right of the last non-zero digit:
  - Are significant if they are after a decimal point.
  - Are not significant if there is no decimal point (they are ambiguous placeholders).

Applying to the Assertion (A):

0.200
- The leading zero is not significant.
- The digits 2, 0, 0 after the decimal are all significant (trailing zeros after decimal count).
- Significant figures = 3 ✓
200
- No decimal point is shown.
- The trailing zeros are ambiguous — they are not after a decimal, so they are not significant.
- Only the digit 2 is significant.
- Significant figures = 1 ✓ …

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Q.Assertion (A): Significant figures for 0.200 is 3 where as for 200 it is 1.
Reason (R): Zero at the end or right of a number are significant provided they are not on the right side of the decimal point.
Choose the correct option out of the choices given below.

Concept understanding — Unit Conversion

Unit Conversion: Why 1 Metre and 100 Centimetres Are the Same Thing

The Intuition: Same Quantity, Different Labels

The Precise Statement

How to Convert: The Only Rule You Need

The Reverse: Metres to Centimetres

Common Conversion Factors You'll Use

Why this formula?

Dimensional Analysis: Why the Key Principles Hold

1. The Core Idea: Physical Quantities Have Dimensions

2. The Principle of Dimensional Homogeneity

Why must this hold?

3. The Buckingham Pi Theorem: Why We Can Derive Relationships

Why does this work?

4. Why We Can Convert Units Using Dimensional Analysis

Why this works: …

Concept: Significant Figures (Significant Digits)

Method: Rule-Based Counting of Significant Figures

Steps:

Applying to the Assertion (A):

The Correct Answer First

Common Mistakes Students Make

1. Misreading the Reason statement

2. Confusing trailing zeros in whole numbers vs decimals

3. Forgetting that leading zeros never count

Q.Assertion (A): Significant figures for 0.200 is 3 where as for 200 it is 1. Reason (R): Zero at the end or right of a number are significant provided they are not on the right side of the decimal point. Choose the correct option out of the choices given below.

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Q.Assertion (A): Significant figures for 0.200 is 3 where as for 200 it is 1.
Reason (R): Zero at the end or right of a number are significant provided they are not on the right side of the decimal point.
Choose the correct option out of the choices given below.

Unlock everything free for 14 days