The mass of a box measured by a grocer's balance is 2.300kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures?

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert all masses to the same unit
We have the mass of the box as 2.300 kg and the masses of the gold pieces as 20.15 g and 20.17 g. We need to convert the masses of the gold pieces from grams to kilograms for consistency.

- **Conversion**:
- 20.15 g = 20.15 / 1000 = 0.02015 kg
- 20.17 g = 20.17 / 1000 = 0.02017 kg

### Step 2: Calculate the total mass of the box
Now we can add the mass of the box and the masses of the gold pieces together.

- **Total mass calculation**:
\[ \text{Total mass} = \text{mass of the box} + \text{mass of gold piece 1} + \text{mass of gold piece 2} \]
\[ \text{Total mass} = 2.300 \, \text{kg} + 0.02015 \, \text{kg} + 0.02017 \, \text{kg} \]

### Step 3: Perform the addition
Now, we will perform the addition:

\[ \text{Total mass} = 2.300 + 0.02015 + 0.02017 = 2.34032 \, \text{kg} \]

### Step 4: Determine significant figures for the total mass
According to the rules of significant figures in addition, the final answer should have the same number of decimal places as the measurement with the least number of decimal places.

- The mass of the box (2.300 kg) has 3 decimal places.
- The masses of the gold pieces (0.02015 kg and 0.02017 kg) have 5 decimal places.

Since the mass of the box has the least number of decimal places (3), we round the total mass to 3 decimal places:

\[ \text{Total mass} = 2.340 \, \text{kg} \]

### Step 5: Calculate the difference in the masses of the gold pieces
Now we need to find the difference between the two gold pieces:

\[ \text{Difference} = 20.17 \, \text{g} - 20.15 \, \text{g} \]

### Step 6: Perform the subtraction
Calculating the difference:

\[ \text{Difference} = 20.17 - 20.15 = 0.02 \, \text{g} \]

### Step 7: Determine significant figures for the difference
The difference should also be reported with the correct number of significant figures. Both values (20.17 g and 20.15 g) have 2 decimal places, so the difference will also have 2 decimal places:

\[ \text{Difference} = 0.02 \, \text{g} \]

### Final Answers
(a) The total mass of the box is **2.340 kg**.
(b) The difference in the masses of the pieces is **0.02 g**.

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