Significant Figures Worksheet with exercises on identifying, rewriting, and rounding numbers to specific significant figures.

Let's solve each part of the Significant Figures Worksheet step by step, explaining the rules and applying them correctly.

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1) Determine the number of significant figures in each of the following:

#### Rules for Significant Figures:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros (before the first non-zero digit) are not significant.
- Trailing zeros are significant only if there is a decimal point.
- For numbers in scientific notation, only the digits in the coefficient count.

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a) 75.02 mm
- Digits: 7, 5, 0, 2 → all are significant (the zero is between non-zeros).
✔ 4 significant figures

b) 18.90 mL
- Digits: 1, 8, 9, 0 → trailing zero after decimal is significant.
✔ 4 significant figures

c) 12 test tubes
- This is a count, not a measurement. Counts are exact and have infinite significant figures.
But typically, we say such values have as many sig figs as written — here it’s 2 digits.
However, since it's a count, it's exact ⇒ infinite sig figs, but often treated as 2 sig figs unless specified.
But in most contexts like this worksheet, counts are considered to have infinite sig figs, so:
✔ Infinite significant figures (but sometimes listed as 2)

> However, in most high school/college worksheets, counts are considered exact, so they don't limit precision. But for counting purposes, 12 has 2 sig figs.

We’ll go with 2 unless otherwise stated.

✔ 2 significant figures

d) 0.0049 g
- Leading zeros are not significant.
- Non-zero digits: 4 and 9 → both significant.
✔ 2 significant figures

e) 150 cm
- No decimal point → trailing zero is ambiguous.
- Without decimal, trailing zeros are not significant.
→ So only 1 and 5 are significant.
✔ 2 significant figures

f) 150. Cm
- Note: There's a decimal point at the end → so the zero is significant.
- Also, the unit is "Cm" (likely typo for "cm"), but that doesn’t affect sig figs.
✔ 3 significant figures

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✔ Answers for #1:

| Letter | Value | Sig Figs |
|--------|------------------|----------|
| a) | 75.02 mm | 4 |
| b) | 18.90 mL | 4 |
| c) | 12 test tubes | 2 |
| d) | 0.0049 g | 2 |
| e) | 150 cm | 2 |
| f) | 150. Cm | 3 |

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2) Re-write the quantity 827,000,000,000,000 picoseconds to show:

First, write in scientific notation:

827,000,000,000,000 = 8.27 × 10¹⁴ ps

Now round to different sig figs:

a) 1 sig fig:
→ Round 8.27 to 8 → 8 × 10¹⁴ ps
✔ 8 × 10¹⁴ ps

b) 2 sig figs:
→ Round 8.27 to 8.3 → 8.3 × 10¹⁴ ps
✔ 8.3 × 10¹⁴ ps

c) 3 sig figs:
→ Already 8.27 × 10¹⁴ → no rounding needed
✔ 8.27 × 10¹⁴ ps

d) 4 sig figs:
→ Need 4 sig figs, but original has only 3 → add one more digit? But we can’t invent data.
Wait: Original number is 827,000,000,000,000 → which is 8.27 × 10¹⁴, so only 3 sig figs.
So we cannot have 4 sig figs without extra info.

But the question says "rewrite to show", so we must assume the value is exact and we can express it with more digits?

No — we should round appropriately based on the original value.

But since the original has only 3 sig figs, we can't increase them.

Wait — actually, the number 827,000,000,000,000 could be interpreted as having 3 sig figs (since no decimal), but maybe it's exact?

But in this context, likely it's 827 trillion, so 3 sig figs.

So for 4 sig figs, we would need to write it as 8.270 × 10¹⁴, implying we know the next digit is 0.

But since we don’t, we can’t assume it.

But the problem asks us to “rewrite to show” — meaning we're allowed to add zeros to indicate precision.

So for 4 sig figs, we write: 8.270 × 10¹⁴ ps (assuming the zero is known or estimated).

Similarly, for 5 sig figs: 8.2700 × 10¹⁴ ps

But that assumes more precision than exists.

But since the problem wants us to show different sig figs, we proceed accordingly.

So:

d) 4 sig figs:
→ 8.270 × 10¹⁴ ps
✔ 8.270 × 10¹⁴ ps

e) 5 sig figs:
→ 8.2700 × 10¹⁴ ps
✔ 8.2700 × 10¹⁴ ps

> Note: These imply higher precision than the original, but the problem is asking how to write it with that many sig figs.

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✔ Answers for #2:

| Part | Answer |
|------|--------------------------|
| a) | 8 × 10¹⁴ ps |
| b) | 8.3 × 10¹⁴ ps |
| c) | 8.27 × 10¹⁴ ps |
| d) | 8.270 × 10¹⁴ ps |
| e) | 8.2700 × 10¹⁴ ps |

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3) Rewrite the quantity 0.0031904 kg to show:

Original: 0.0031904 kg

Convert to scientific notation: 3.1904 × 10⁻³ kg

Now round to:

a) 1 sig fig:
→ Look at first digit: 3 → next digit is 1 < 5 → round down → 3 × 10⁻³ kg
✔ 3 × 10⁻³ kg

b) 2 sig figs:
→ 3.1904 → look at third digit: 9 ≥ 5 → round up 1 to 2 → 3.2 × 10⁻³ kg
✔ 3.2 × 10⁻³ kg

c) 3 sig figs:
→ 3.1904 → look at fourth digit: 0 < 5 → keep 3.19 → 3.19 × 10⁻³ kg
✔ 3.19 × 10⁻³ kg

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✔ Answers for #3:

| Part | Answer |
|------|----------------------|
| a) | 3 × 10⁻³ kg |
| b) | 3.2 × 10⁻³ kg |
| c) | 3.19 × 10⁻³ kg |

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4) Round each of the following to 3 significant figures:

a) 16.8477 L
→ Look at 4th digit: 4 < 5 → round down → 16.8 L
✔ 16.8 L

b) 5.6732
→ 4th digit: 2 < 5 → round down → 5.67
✔ 5.67

c) 0.14986 L
→ First non-zero is 1 → digits: 1, 4, 9 → 4th digit is 8 ≥ 5 → round up 9 → becomes 10 → carry over
→ 0.150 L (note: 150 has 3 sig figs, and trailing zero after decimal is significant)
✔ 0.150 L

d) 861.85
→ 4th digit: 8 ≥ 5 → round up → 862
✔ 862

e) 4.203 × 10⁴ km
→ Coefficient: 4.203 → 4th digit is 3 < 5 → round down → 4.20 × 10⁴ km
✔ 4.20 × 10⁴ km

f) 5.0981 × 10⁻³
→ Coefficient: 5.0981 → 4th digit: 8 ≥ 5 → round up 9 → becomes 10 → carry over
→ 5.10 × 10⁻³
✔ 5.10 × 10⁻³

g) 0.00318756 m
→ First non-zero: 3 → digits: 3,1,8 → 4th digit: 7 ≥ 5 → round up 8 → 9 → 3.19 × 10⁻³ m
✔ 3.19 × 10⁻³ m

h) 0.09025011
→ First non-zero: 9 → digits: 9,0,2 → 4th digit: 5 → round up 2 → 3 → 9.03 × 10⁻²
→ Because: 0.09025011 = 9.025011 × 10⁻² → round to 3 sig figs: 9.03 × 10⁻²
✔ 9.03 × 10⁻²

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✔ Answers for #4:

| Letter | Rounded Value |
|--------|----------------------------|
| a) | 16.8 L |
| b) | 5.67 |
| c) | 0.150 L |
| d) | 862 |
| e) | 4.20 × 10⁴ km |
| f) | 5.10 × 10⁻³ |
| g) | 3.19 × 10⁻³ m |
| h) | 9.03 × 10⁻² |

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✔ Final Answers Summary:

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#### 1) Number of Significant Figures:
a) 4
b) 4
c) 2
d) 2
e) 2
f) 3

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#### 2) Rewrite 827,000,000,000,000 ps:
a) 8 × 10¹⁴ ps
b) 8.3 × 10¹⁴ ps
c) 8.27 × 10¹⁴ ps
d) 8.270 × 10¹⁴ ps
e) 8.2700 × 10¹⁴ ps

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#### 3) Rewrite 0.0031904 kg:
a) 3 × 10⁻³ kg
b) 3.2 × 10⁻³ kg
c) 3.19 × 10⁻³ kg

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#### 4) Round to 3 sig figs:
a) 16.8 L
b) 5.67
c) 0.150 L
d) 862
e) 4.20 × 10⁴ km
f) 5.10 × 10⁻³
g) 3.19 × 10⁻³ m
h) 9.03 × 10⁻²

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