2.1 · Measurement and CalculationPractice · 测量与计算

The calculator shows 25.67 ÷ 3.2 = 8.021875. In multiplication and division, the answer has as many significant figures as the measurement with the fewest: 25.67 has 4, 3.2 only 2, so keep 2: 8.0 g/mL. Write the trailing zero to show the answer is known to the tenths place.

The calculator shows 115.6546. In addition and subtraction, count decimal places, not significant figures. Of 12.52, 0.0346, 103.1, 103.1 is known only to the nearest 0.1 g (1 decimal place), so the total is reported to 0.1 g: 115.7 g (4 significant figures).

Do the subtraction first: 5.43 − 5.40 = 0.03. Both numbers are known to 0.01, so the difference is too, but it has only 1 significant figure: subtraction can wipe out significant figures. Then multiply by 2.6 (2 significant figures): 0.03 × 2.6 = 0.078, which is limited to 1 significant figure: 0.08.

First the volume: (2.35 cm)³ = 12.977875 cm³. That’s an intermediate result, so don’t round it yet; keep at least one extra digit (12.98 cm³). Then the density: 115.2 ÷ 12.977875 = 8.8766. 2.35 has 3 significant figures and 115.2 has 4, so report 3: 8.88 g/cm³. Rounding the volume to 13.0 first gives 8.86, with the last digit wrong.

Convert the top and bottom separately: 1 kg = 1000 g, and 1 m³ = 10⁶ cm³ (1 m = 100 cm, cubed).

The conversion factors are exact, so the answer keeps the 3 significant figures of 7.87. Multiplying by 100 instead of 100³ is the most common mistake.

1 mL = 1 cm³.

The density is a measurement, so it counts toward significant figures: 25.0 and 13.6 both have 3, so the answer has 3, including the trailing zero.

Percent error = |measured − accepted| ÷ accepted × 100% = |6.95 − 7.14| ÷ 7.14 × 100% = 2.7%. The difference, 0.19, has only 2 significant figures, so the answer has 2. Chinese textbooks keep the sign (relative error −2.7%), which shows the measurement was low.

The air pocket contains no metal, so the mass is just the metal’s, but the water displaced includes the pocket’s volume, so V is too large. In $\rho = \dfrac{m}{V}$ the denominator is too big, so the density comes out too low. Every trial with this block errs the same way: a systematic error that averaging can’t remove.

325 mg × 100 = 32500 mg = 32.5 g (1 g = 1000 mg). The 100 tablets are counted and 1000 is a definition, so neither limits the significant figures; the answer keeps the 3 figures of 325: 32.5 g.

Multiply first: 1.25 × 3.6 = 4.5, which is good to 2 significant figures, that is, to the tenths place (note it, but don’t round yet). Then add: 4.5 + 0.587 = 5.087. Addition goes by decimal places, and the product has the fewest (tenths), so report 5.1 (2 significant figures). In multi-step work, carry an extra digit or two and round only at the end.