NCERT Solutions

We need to calculate the total mass and the difference in gold masses using the rules of significant figures.

Concept: Significant figures

When adding or subtracting physical quantities, the final result should be rounded to the same number of decimal places as the measurement with the least number of decimal places. This ensures the precision of the result matches the least precise measurement.

Given:

• Mass of the box, $m_{\text{box}} = 2.30\,\text{kg}$
• Mass of gold piece 1, $m_1 = 20.15\,\text{g}$
• Mass of gold piece 2, $m_2 = 20.17\,\text{g}$

Solving:

1. Total mass of the box:

• First, convert the masses of the gold pieces from grams to kilograms:

$$\begin{aligned} m_1 &= \frac{20.15}{1000}\,\text{kg} = 0.02015\,\text{kg} \\ m_2 &= \frac{20.17}{1000}\,\text{kg} = 0.02017\,\text{kg} \end{aligned}$$

• Now, sum the masses to find the total mass ($m_{\text{total}}$):

$$\begin{aligned} m_{\text{total}} &= m_{\text{box}} + m_1 + m_2 \\ &= 2.30\,\text{kg} + 0.02015\,\text{kg} + 0.02017\,\text{kg} \\ &= 2.34032\,\text{kg} \end{aligned}$$

• The measurement $2.30\,\text{kg}$ has the least number of decimal places (two). Therefore, the total mass must be rounded to two decimal places:

$$m_{\text{total}} \approx 2.34\,\text{kg}$$

2. Difference in the gold masses:

• Calculate the difference ($\Delta m$) between the two gold pieces:

$$\begin{aligned} \Delta m &= m_2 - m_1 \\ &= 20.17\,\text{g} - 20.15\,\text{g} \\ &= 0.02\,\text{g} \end{aligned}$$

• Both $20.17\,\text{g}$ and $20.15\,\text{g}$ have two decimal places. Thus, the result $0.02\,\text{g}$ is already expressed to the correct number of decimal places.

Answer

(a)

$$2.34\,\text{kg}$$

(b)

$$0.02\,\text{g}$$