Let's define the variables and solve the problem step by step.

Given:

• The expected mass of each bolt is $5 \, \text{g}$.
• The mass variation is up to $\pm 1.9\%$. This means each bolt's mass can range between $5 \times (1 - 0.019) \, \text{g}$ and $5 \times (1 + 0.019) \, \text{g}$.
• The total mass of bolts in the box is $1 \, \text{kg} = 1000 \, \text{g}$.

We need to find the minimum and maximum number of bolts that could be in the box.

Step 1: Determine the mass range of a single bolt

The mass of a bolt can range from:

$\text{Minimum mass per bolt} = 5 \times (1 - 0.019) = 5 \times 0.981 = 4.905 \, \text{g}$

$\text{Maximum mass per bolt} = 5 \times (1 + 0.019) = 5 \times 1.019 = 5.095 \, \text{g}$

Step 2: Calculate the minimum and maximum number of bolts

Now, calculate the number of bolts for each case:

Maximum number of bolts (when each bolt is as light as possible):

$\text{Number of bolts} = \frac{\text{Total mass}}{\text{Minimum mass per bolt}} = \frac{1000}{4.905} \approx 203.86$ Since the number of bolts must be an integer, we round down to get $203$ bolts.

Minimum number of bolts (when each bolt is as heavy as possible):

$\text{Number of bolts} = \frac{\text{Total mass}}{\text{Maximum mass per bolt}} = \frac{1000}{5.095} \approx 196.27$ Rounding down gives $196$ bolts.

Conclusion:

• The minimum number of bolts is $196$.
• The maximum number of bolts is $203$.

Would you like further details on any part of the calculation or concept?

Here are some related questions to explore further:

1. How would the answer change if the variation was ±2%?
2. How would you calculate the total mass if you know the number of bolts?
3. What would happen if the total mass was 1.5 kg instead of 1 kg?
4. How can we express this problem using inequalities?
5. What would be the effect of using bolts with different masses?

Tip: When solving word problems, always start by translating the given information into mathematical equations or inequalities!