Worksheet for practicing significant figures and scientific notation calculations.

Here are the solutions to your worksheet, broken down step by step.

Part 1: Determine the number of significant figures

Rules to remember:
* Non-zero digits are always significant.
* Zeros between non-zero digits (captive zeros) are significant.
* Leading zeros (zeros at the start) are never significant; they just hold the decimal place.
* Trailing zeros (zeros at the end) are significant ONLY if there is a decimal point present in the number.
* In scientific notation, only the coefficient (the number before the $\times 10^x$) counts.

a) 0.02
The leading zeros are not significant. Only the '2' counts.
Answer: 1

b) 0.020
The leading zeros don't count. The '2' counts. The trailing zero *does* count because there is a decimal point.
Answer: 2

c) 501
The zero is sandwiched between two numbers, so it counts. All three digits are significant.
Answer: 3

d) 6051.00
All non-zeros count. The zero between 6 and 5 counts. The two zeros at the end count because there is a decimal point.
Answer: 6

e) 501.0
The zero between 5 and 1 counts. The zero at the end counts because of the decimal point.
Answer: 4

f) $2.000 \times 10^2$
Look only at the "2.000" part. The zeros after the decimal point count.
Answer: 4

g) $0.982 \times 10^{-3}$
Look at "0.982". The leading zero doesn't count. The 9, 8, and 2 do.
Answer: 3

h) $0.0114 \times 10^4$
Look at "0.0114". The first two zeros are leading zeros (not significant). The 1, 1, and 4 are significant.
Answer: 3

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Part 2: Calculate with correct significant digits

Rules for Math:
* Multiplication/Division: Your answer must have the same number of significant figures as the number in the problem with the *fewest* significant figures.
* Addition/Subtraction: Your answer must be rounded to the same decimal place (tenths, hundredths, etc.) as the number in the problem with the *least precise* decimal place.

a) $1.35 \times 2.467$
* Raw calculation: $3.33045$
* Sig Fig check: 1.35 has 3 sig figs. 2.467 has 4 sig figs. We need the lowest (3).
* Round to 3 sig figs: 3.33

b) $1.035 + 4.2$
* Raw calculation: $5.235$
* Decimal place check: 1.035 goes to the thousandths place. 4.2 goes to the tenths place. We must round to the tenths place (the least precise).
* Round to tenths: 5.2

c) $12.01 + 35.2 + 6.9$
* Raw calculation: $54.11$
* Decimal place check: 12.01 (hundredths), 35.2 (tenths), 6.9 (tenths). The least precise is tenths.
* Round to tenths: 54.1

d) $55.46 - 28.9$
* Raw calculation: $26.56$
* Decimal place check: 55.46 (hundredths), 28.9 (tenths). Least precise is tenths.
* Round to tenths: 26.6

e) $0.21 \times 3.2 \times 100.1$
* Raw calculation: $67.2672$
* Sig Fig check: 0.21 (2 sig figs), 3.2 (2 sig figs), 100.1 (4 sig figs). Lowest is 2.
* Round to 2 sig figs: 67

f) $505.0 - 450.25$
* Raw calculation: $54.75$
* Decimal place check: 505.0 (tenths), 450.25 (hundredths). Least precise is tenths.
* Round to tenths: 54.8

g) $(1.5 \times 10^6) / (3.76 \times 10^2)$
* Raw calculation: $3989.36...$
* Sig Fig check: 1.5 (2 sig figs), 3.76 (3 sig figs). Lowest is 2.
* We need an answer with 2 sig figs. To make it clear, we use scientific notation.
* $3989...$ becomes $4.0 \times 10^3$.
* Answer: $4.0 \times 10^3$

h) $(1.28 \times 10^3) / (1.4267 \times 10^2)$
* Raw calculation: $8.97175...$
* Sig Fig check: 1.28 (3 sig figs), 1.4267 (5 sig figs). Lowest is 3.
* Round to 3 sig figs: 8.97

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Final Answer:

1. Significant Figures Count
a) 1
b) 2
c) 3
d) 6
e) 4
f) 4
g) 3
h) 3

2. Calculations
a) 3.33
b) 5.2
c) 54.1
d) 26.6
e) 67
f) 54.8
g) $4.0 \times 10^3$
h) 8.97