Significant Figures Worksheet Key showing exercises for identifying significant figures and performing calculations with appropriate significant figures.

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🔢 Understanding Significant Figures (Sig Figs)

Rules for counting significant figures:

1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant.
3. Leading zeros (before the first non-zero digit) are NOT significant.
4. Trailing zeros (after the last non-zero digit):
- Are significant only if there's a decimal point.
- If no decimal point, they may be placeholders (not significant).
5. For scientific notation: Only the digits in the coefficient count.

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## ✔ Problem 1: Count the number of significant figures

Let’s evaluate each value:

| Value | Sig Figs | Explanation |
|-------------|----------|-----------|
| 246.32 | 5 | All digits are non-zero → 5 sig figs |
| 1.008 | 4 | Zeros between non-zeros are significant |
| 700000 | 1 | No decimal → trailing zeros not significant; only '7' counts |
| 107.854 | 6 | All digits are significant (zeros between non-zeros) |
| 0.00340 | 3 | Leading zeros not significant; '340' → 3 sig figs (trailing zero after decimal is significant) |
| 350.670 | 6 | Trailing zero after decimal is significant; all digits matter |
| 100.3 | 4 | Zero between non-zeros → significant |
| 14.000 | 5 | Trailing zeros after decimal → significant |
| 1.0000 | 5 | Trailing zeros after decimal → significant |
| 0.070 | 2 | Leading zeros not significant; '70' → two sig figs |
| 0.0004 | 1 | Only '4' is significant |
| 520001 | 6 | All digits are non-zero or trapped → 6 sig figs |

✔ All answers in the key are correct.

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## ✔ Problem 2: Addition/Subtraction – Round to least precise decimal place

We add/subtract numbers and round to the least number of decimal places.

Let’s compute each:

a)

```
32.567
+ 135.0
+ 1.4507
--------
169.0177 → rounded to 1 decimal place (from 135.0)
Answer: 169.0
```
✔ Correct: 169.0

b)

```
246.24
+ 236.278
+ 39.1
-------
521.618 → round to tenths (39.1 has 1 decimal)
Answer: 521.6
```
✔ Correct: 521.6

c)

```
678.0
+ 23.5478
+ 1345.25
--------
2046.7978 → round to tenths (678.0 has 1 decimal)
Answer: 2046.8
```
✔ Correct: 2046.8

---

## ✔ Problem 3: Multiplication/Division – Round to fewest sig figs

Use the least number of significant figures in any number.

a) 23.7 × 3.8

- 23.7 → 3 sig figs
- 3.8 → 2 sig figs
→ Answer should have 2 sig figs

23.7 × 3.8 = 89.06 → round to 89
✔ Correct: 89

b) 45.76 × 0.25

- 45.76 → 4 sig figs
- 0.25 → 2 sig figs
→ 2 sig figs in answer

45.76 × 0.25 = 11.44 → round to 11
✔ Correct: 11

c) 81.04 g ÷ 0.010

- 81.04 → 4 sig figs
- 0.010 → 2 sig figs (only '1' and '0' after decimal)
→ 2 sig figs

81.04 ÷ 0.010 = 8104 → round to 8.1 × 10³ or 8100? Wait — but we need 2 sig figs

8104 → rounded to 2 sig figs = 8.1 × 10³ → 8100 (but written as 8.1 × 10³)
But the key says 8.10 → that’s 3 sig figs → ✘ Incorrect

Wait — let’s check:

0.010 has 2 sig figs → so answer must have 2 sig figs.

81.04 / 0.010 = 8104 → rounded to 8.1 × 10³ → 8100 with 2 sig figs → 8.1 × 10³

But the key says 8.10 → which implies 3 sig figs → ✘ Wrong

✔ Correct answer: 8.1 × 10³ (or 8100 with underline on first two digits)

So this one is incorrect in the key.

---

d) 6.47 × 64.5

- 6.47 → 3 sig figs
- 64.5 → 3 sig figs
→ 3 sig figs

6.47 × 64.5 = 417.265 → round to 417
✔ Correct: 417

---

e) 43.678 ÷ 64.1

- 43.678 → 5 sig figs
- 64.1 → 3 sig figs
→ 3 sig figs

43.678 ÷ 64.1 ≈ 0.6813 → round to 0.681
But key says 2.8 × 10⁻¹ → 0.28 → ✘ Wrong

Wait — let’s recalculate:

43.678 ÷ 64.1 = ?

Do it:
64.1 × 0.68 = 43.648 → very close to 43.678 → so ~0.681

So answer should be 0.681 → 3 sig figs

But key says 2.8 × 10⁻¹ = 0.28 → that’s 2 sig figs and wrong.

✘ Incorrect

---

f) 1.678 ÷ 0.42

- 1.678 → 4 sig figs
- 0.42 → 2 sig figs
→ 2 sig figs

1.678 ÷ 0.42 ≈ 4.0 → wait:

1.678 ÷ 0.42 = 4.00 → actually 4.0 → but 4.0 has 2 sig figs

But 1.678 / 0.42 = 4.00 → round to 4.0

Key says 4.0 → ✔ Correct

---

g) 28.367 ÷ 3.74

- 28.367 → 5 sig figs
- 3.74 → 3 sig figs
→ 3 sig figs

28.367 ÷ 3.74 ≈ 7.584 → round to 7.58
✔ Correct: 7.58

---

h) 4270 ÷ 1.006

- 4270 → ambiguous! Is it 4270 or 4.27 × 10³?

If no decimal → trailing zeros not significant → 3 sig figs (4270 → 3 sig figs)

1.006 → 4 sig figs

→ answer should have 3 sig figs

4270 ÷ 1.006 ≈ 4244.4 → round to 4240 or 4.24 × 10³

But key says 4252 → that’s 4 sig figs → ✘ Incorrect

Wait — maybe 4270 is considered to have 4 sig figs? But without decimal, it's ambiguous.

But typically, 4270 → 3 sig figs unless specified.

So answer should be 4.24 × 10³ → but key says 4252 → which is 4 sig figs → ✘ Incorrect

---

i) (8.8 + 4.7) × 17.44

First do addition:

8.8 + 4.7 = 13.5 → both have 1 decimal → result has 1 decimal → 13.5

Then multiply: 13.5 × 17.44

- 13.5 → 3 sig figs
- 17.44 → 4 sig figs
→ answer has 3 sig figs

13.5 × 17.44 = 235.44 → round to 235

Key says 203 → ✘ Incorrect

Wait — did they do something else?

(8.8 + 4.7) = 13.5 → correct
13.5 × 17.44 = 235.44 → rounds to 235

But key says 203 → that’s off.

Unless typo?

No — clearly wrong.

✘ Incorrect

---

j) (730 - 22.7) × 3.8

First subtraction:

730 → no decimal → assume 730 has 2 sig figs (if 730±10), but ambiguity

22.7 → 1 decimal

730 - 22.7 = 707.3 → but 730 has uncertainty in tens place → so difference is uncertain to nearest 10 → so 710 (rounded to tens)

Then 710 × 3.8

710 → 2 sig figs (if 7.1×10²)
3.8 → 2 sig figs

→ product has 2 sig figs

710 × 3.8 = 2700 → 2.7 × 10³ → round to 2.7 × 10³

But key says 1.1 × 10⁴ → 11000 → ✘ Wrong

This is way off.

Let’s calculate:

730 - 22.7 = 707.3 → but if 730 has no decimal, we can’t say it's exact.

But even if we take it as exact, 730 - 22.7 = 707.3 → then × 3.8

707.3 × 3.8 = 2687.74 → round to 2 sig figs → 2.7 × 10³

Key says 1.1 × 10⁴ → 11000 → too big → ✘ Incorrect

---

k) Complex expression:

$$
\frac{(28.6) \times (61.25)}{(13.88 - 12.7) \times (21.000 - 21.000)}
$$

Wait — denominator: (21.000 - 21.000) = 0 → division by zero!

That’s undefined!

But the key says 1.89 → impossible.

Wait — let’s look again:

The expression is:

$$
\frac{(28.6) \times (61.25)}{(13.88 - 12.7) \times (21.000 - 21.000)}
$$

But (21.000 - 21.000) = 0 → denominator = 0 → undefined → infinite or undefined

But key says 1.89 → ✘ Incorrect

Possibly a typo in the problem?

Maybe it was meant to be:

$$
\frac{(28.6) \times (61.25)}{(13.88 - 12.7) \times (21.000 - 21.000)} \quad \text{→ no, still 0}
$$

Or perhaps it's supposed to be:

$$
\frac{(28.6) \times (61.25)}{(13.88 - 12.7) \times (21.000 - 21.000)} \quad \text{→ still zero}
$$

Wait — maybe it's a typo and it's 21.000 - 21.001 or something?

But as written, it's division by zero → undefined.

So the key answer 1.89 is wrong.

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## ✔ Summary of Errors in the Key

| Problem | Key Answer | Correct Answer | Status |
|--------|------------|----------------|--------|
| c) 81.04 / 0.010 | 8.10 | 8.1 × 10³ | ✘ Wrong |
| e) 43.678 / 64.1 | 2.8 × 10⁻¹ | 0.681 | ✘ Wrong |
| h) 4270 / 1.006 | 4252 | 4.24 × 10³ | ✘ Wrong |
| i) (8.8 + 4.7) × 17.44 | 203 | 235 | ✘ Wrong |
| j) (730 - 22.7) × 3.8 | 1.1 × 10⁴ | 2.7 × 10³ | ✘ Wrong |
| k) Complex | 1.89 | Undefined (÷0) | ✘ Wrong |

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## ✔ Final Verdict

- Problem 1: ✔ All correct
- Problem 2: ✔ All correct
- Problem 3: ✘ Multiple errors

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🛠️ Recommendations

1. Double-check sig fig rules in multiplication/division: use fewest sig figs.
2. Addition/subtraction: round to least precise decimal place.
3. Avoid division by zero — it’s undefined.
4. Ambiguous numbers like 730 or 4270 should be clarified with decimals or scientific notation.

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