Q.Round up the following upto three significant figures:

The key idea is Rounding to Significant Figures. We identify the third significant figure and then look at the digit immediately following it. If this digit is 5 or greater, we round up the third significant figure; otherwise, we keep it as is.

1. For $34.216$, the third significant figure is 2. The next digit is 1 ($<5$), so we keep 2.
2. For $10.4107$, the third significant figure is 4. The next digit is 1 ($<5$), so we keep 4.
3. For $0.04597$, the third significant figure is 9. The next digit is 7 ($\ge 5$), so we round up 9 to 10, making it $0.0460$. …

Rounding numbers is a fundamental skill in science and mathematics, especially when dealing with measurements or calculations where precision is limited. The concept of "significant figures" helps us express the precision of a number. When we round to a certain number of significant figures, we are essentially stating how many digits in the number are considered reliable or meaningful.

The general idea is to identify the significant digits in a number, then locate the digit that will be the last significant digit after rounding. The decision to round up or down depends on the digit immediately following this last significant digit.

Here are the key rules for identifying significant figures:

• Non-zero digits are always significant. (e.g., $123$ has $3$ significant figures).
• Zeros between non-zero digits are significant. (e.g., $102$ has $3$ significant figures).
• Leading zeros (zeros before non-zero digits) are not significant. They are placeholders. (e.g., $0.0012$ has $2$ significant figures: $1$ and $2$).
• Trailing zeros (zeros at the end of a number) are significant only if the number contains a decimal point. (e.g., $120.$ has $3$ significant figures; $120$ (without a decimal) is often considered to have $2$ significant figures, $1$ and $2$, unless context specifies otherwise).

Now, let's apply these rules to round each given number to three significant figures.

1. For (i) $34.216$

   • Identify significant figures: All non-zero digits are significant. So, $3, 4, 2, 1, 6$ are all significant. The number has $5$ significant figures.
   • Locate the third significant figure: Counting from the left, the first significant figure is $3$, the second is $4$, and the third is $2$.
   • Look at the digit to its right: The digit immediately after the third significant figure ($2$) is $1$.
   • Apply rounding rule: Since $1$ is less than $5$, we keep the third significant figure ($2$) as it is. All digits to its right are dropped.
   • Result: The number rounded to three significant figures is $\mathbf{34.2}$.

2. For (ii) $10.4107$

   • Identify significant figures: $1, 0, 4, 1, 0, 7$ are all significant (zeros between non-zero digits are significant). The number has $6$ significant figures.
   • Locate the third significant figure: The first is $1$, the second is $0$, and the third is $4$.
   • Look at the digit to its right: The digit immediately after the third significant figure ($4$) is $1$.
   • Apply rounding rule: Since $1$ is less than $5$, we keep the third significant figure ($4$) as it is. All digits to its right are dropped.
   • Result: The number rounded to three significant figures is $\mathbf{10.4}$.

3. For (iii) $0.04597$

   • Identify significant figures: The leading zeros ($0.0$) are not significant. The significant figures are $4, 5, 9, 7$. The number has $4$ significant figures.
   • Locate the third significant figure: The first significant figure is $4$, the second is $5$, and the third is $9$.
   • Look at the digit to its right: The digit immediately after the third significant figure ($9$) is $7$.
   • Apply rounding rule: Since $7$ is $5$ or greater, we round up the third significant figure ($9$). Rounding $9$ up makes it $10$. This means the $5$ before it becomes $6$, and the $9$ becomes $0$. …

Method: Significant Figure Rounding (Decimal Place Truncation with Rounding Rules)

This method follows the standard rounding rule:

• Look at the digit immediately after the desired significant figure count.
• If that digit is ≥ 5, round the last retained digit up by 1.
• If that digit is < 5, keep the last retained digit unchanged.

Steps

1. Identify the first three significant digits (non-zero digits from left, counting zeros only if they are between non-zero digits or after a decimal point with a non-zero digit before them).
2. Look at the fourth digit (the one just after the third significant figure).
3. Apply the rounding rule (≥5 rounds up, <5 rounds down).
4. Write the result with exactly three significant figures.

Solved Examples

(i) 34.216

• First three significant digits: 3, 4, 2
• Fourth digit: 1 (which is < 5)
• So, no rounding up → keep 34.2
• Answer: $\boxed{34.2}$

(ii) 10.4107

• First three significant digits: 1, 0, 4 (the zero is significant because it is between 1 and 4)
• Fourth digit: 1 (which is < 5)
• So, no rounding up → keep 10.4
• Answer: $\boxed{10.4}$

(iii) 0.04597

• First three significant digits: 4, 5, 9 (leading zeros are not significant)
• Fourth digit: 7 (which is ≥ 5) …

Common Mistakes in Rounding to Significant Figures (with Examples)

Students often lose easy marks on significant figure questions. Here are the most frequent errors and how to avoid each — using your exact examples.

Mistake 1: Confusing "Decimal Places" with "Significant Figures"

The error: Treating the count of digits after the decimal as the number of significant figures.

Example with (i) 34.216:

• ✗ Wrong: 34.216 → 34.200 (thinking "three decimal places")
• ✓ Correct: 34.216 → 34.2 (three significant figures)

How to avoid:

Remember the rule: Significant figures count all digits from the first non-zero digit, regardless of where the decimal point is.

• 34.216 has 5 significant figures: 3, 4, 2, 1, 6
• Rounded to 3 significant figures → keep 3, 4, 2 → 34.2

Mistake 2: Dropping Trailing Zeros That Are Significant

The error: Removing zeros that are needed to preserve the correct number of significant figures.

Example with (iv) 2808:

• ✗ Wrong: 2808 → 281 (three digits, but wrong value — 281 instead of 2810)
• ✓ Correct: 2808 → 2810 (three significant figures)

Why?

2808 has 4 significant figures. To keep 3, we round the third digit (0) up to 1, and keep the zero as a placeholder so the magnitude is correct.

How to avoid:

When rounding whole numbers, use zeros as placeholders to maintain the correct order of magnitude.

Mistake 3: Forgetting to Round Up When the Next Digit is 5 or More

The error: Simply truncating instead of rounding.

Example with (iii) 0.04597:

• ✗ Wrong: 0.04597 → 0.045 (truncated, should be 0.0460)
• ✓ Correct: 0.04597 → 0.0460 (three significant figures)

Step-by-step:

1. First non-zero digit is 4 (position 2 after decimal)
2. Count three sig figs: 4, 5, 9
3. Look at the next digit: 7 ≥ 5 → round up the 9 to 10, which carries over
4. Result: 0.0460 (the trailing zero is kept to show 3 sig figs)

How to avoid:

Always check the next digit after the last significant figure:

• If ≥ 5 → round the last kept digit up
• If < 5 → leave it unchanged

Mistake 4: Misidentifying the First Significant Figure in Decimals

The error: Counting leading zeros as significant figures.

Example with (iii) 0.04597:

• ✗ Wrong: Thinking the first sig fig is at the decimal point
• ✓ Correct: The first significant figure is 4 (the first non-zero digit)

Rule: Leading zeros are never significant. They only position the decimal.

Mistake 5: Forgetting to Write Trailing Zeros After Rounding Decimals

The error: Dropping zeros that are needed to show the correct number of significant figures.

Example with (ii) 10.4107:

• ✗ Wrong: 10.4107 → 10.4 (only 3 digits, but should be 10.4 — wait, this is actually correct here!)
• ✓ Correct: 10.4107 → 10.4 (three significant figures) …