https://file.aiquickdraw.com/mathbot/file/17276751351456bu6cq48.jpg

Answer each questions with two strategies/heuristics. 

Let's break down the problem step by step, then use two different strategies to answer each question:

### Problem Breakdown:
Anna is making a design using matchsticks that looks like a row of houses. The illustration shows several houses in a row, with each house sharing matchsticks with the adjacent one. 

- Each **house** consists of 6 matchsticks: 2 vertical, 2 diagonal for the roof, and 2 horizontal for the base and the roofline.
- When adding a new house, it **shares 1 matchstick** with the previous house (the side wall), so it adds 5 more matchsticks instead of 6.

### Strategy 1: Formula Derivation (Direct Calculation)

#### 1. **Number of matchsticks for 25 houses**:
- For the first house, 6 matchsticks are needed.
- For each additional house, only 5 more matchsticks are needed due to sharing.
  
The general formula for the number of matchsticks for \( n \) houses is:
\[
\text{Total matchsticks} = 6 + 5(n - 1)
\]
This simplifies to:
\[
\text{Total matchsticks} = 5n + 1
\]
For 25 houses:
\[
\text{Total matchsticks} = 5(25) + 1 = 125 + 1 = 126
\]
Thus, 126 matchsticks are needed to make a row of 25 houses.

#### 2. **Is a box of 100 matchsticks enough?**
Since the total number of matchsticks required is 126, and the box contains 100 matchsticks:
\[
100 < 126
\]
Hence, **100 matchsticks are not enough** to complete 25 houses.

#### 3. **Number of matchsticks for \( n \) houses (Representation)**:
We already derived the formula:
\[
\text{Total matchsticks} = 5n + 1
\]
This is how you can represent the number of matchsticks for any number \( n \) of houses.

---

### Strategy 2: Visual and Incremental Approach

#### 1. **Number of matchsticks for 25 houses**:
- Let’s count the number of matchsticks visually:
  - The first house uses 6 matchsticks.
  - Each subsequent house adds 5 matchsticks (because of shared walls).

Thus, for each additional house:
\[
6 + 5 \times 24 = 6 + 120 = 126
\]
Again, we find that **126 matchsticks** are needed for 25 houses.

#### 2. **Is a box of 100 matchsticks enough?**
The total is 126 matchsticks, so:
\[
100 \, (\text{in the box}) \, < \, 126 \, (\text{needed})
\]
Clearly, **100 matchsticks are not enough**.

#### 3. **Number of matchsticks for \( n \) houses (Representation)**:
The total matchsticks can be represented as a pattern:
- Start with 6 matchsticks.
- Add 5 matchsticks for each new house.
The number of matchsticks after \( n \) houses is given by:
\[
6 + 5 \times (n - 1)
\]
Or simplified:
\[
\text{Total matchsticks} = 5n + 1
\]

---

### Summary:
- **Number of matchsticks for 25 houses**: 126 matchsticks.
- **Is 100 matchsticks enough?**: No, 100 matchsticks are not enough.
- **Representation for \( n \) houses**: \( 5n + 1 \).

Would you like more details on any part of the solution? 

### Related Questions:
1. How many matchsticks are needed for 10 houses?
2. If each house used 7 matchsticks, how would the formula change?
3. How many matchsticks are saved by sharing walls between houses?
4. Can you generalize the formula for a different house design with different matchstick patterns?
5. What is the maximum number of houses you can build with a box of 100 matchsticks?

### Tip:
When solving problems involving patterns or repetition, identifying how each new element builds on the previous ones is a key strategy for finding a formula.

Anna is making a design that looks like a row of houses for her project in Home Economics. The design looks like this: [House illustration]. If she will use matchsticks for the houses, how many matchsticks does she need to make a row of 25 houses? Is a box of matches containing 100 pieces enough for the houses? Assume we’re asked to determine the number of matchsticks Anna needs to make a row of n houses. How will you represent it?

Learn how to calculate the number of matchsticks needed to create a row of houses. The formula for n houses is 5n + 1, based on pattern recognition. This problem is suitable for students in grades 6-8 and covers key concepts in algebra and pattern recognition.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation