Severin works in a factory that produces steel bolts. One type of steel bolt should
have a mass of 5 g; however, there is some variation, so each bolt may be heavier
or lighter by as much as 1.9%. Severin puts together a box of these bolts and
notices that the total mass of the bolts in the box is exactly 1 kg.
Determine the minimum and maximum number of bolts that could be in the box.

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!

Severin works in a factory that produces steel bolts. One type of steel bolt should have a mass of 5 g; however, there is some variation, so each bolt may be heavier or lighter by as much as 1.9%. Severin puts together a box of these bolts and notices that the total mass of the bolts in the box is exactly 1 kg. Determine the minimum and maximum number of bolts that could be in the box.

Solve a real-world math problem involving steel bolt production. Learn how to calculate the minimum and maximum number of steel bolts in a box when the mass of each bolt can vary by ±1.9%. Explore concepts such as percentages and inequalities, and use the mass variation formula to find the solution. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation