https://file.aiquickdraw.com/mathbot/file/17374666949957d3pbrk4.jpg

I see you've uploaded an image with three mathematical problems labeled as 25, 26, and 27. Let me go through each problem step by step.

---

### **Problem 25**  
**Question:**  
The distance between Arnold's and Beno's houses on a map is 15 cm. Calculate the actual distance between their houses if the map scale is:  
a) 1:2000  
b) 1:36,000  
c) 1:240,000  
d) 1:14,765  

**Solution:**  
The actual distance can be calculated using the formula:  
\[
\text{Actual Distance} = \text{Map Distance} \times \text{Scale Factor}
\]  

For each scale:  
1. **a) 1:2000**  
   \[
   \text{Actual Distance} = 15 \, \text{cm} \times 2000 = 30,000 \, \text{cm} = 300 \, \text{m}
   \]  

2. **b) 1:36,000**  
   \[
   \text{Actual Distance} = 15 \, \text{cm} \times 36,000 = 540,000 \, \text{cm} = 5.4 \, \text{km}
   \]  

3. **c) 1:240,000**  
   \[
   \text{Actual Distance} = 15 \, \text{cm} \times 240,000 = 3,600,000 \, \text{cm} = 36 \, \text{km}
   \]  

4. **d) 1:14,765**  
   \[
   \text{Actual Distance} = 15 \, \text{cm} \times 14,765 = 221,475 \, \text{cm} = 2.21 \, \text{km}
   \]  

---

### **Problem 26**  
**Question:**  
The length of a microchip in a microscope image is 12 cm. Calculate the actual length of the microchip if the image scale is:  
a) 2:1  
b) 10:1  
c) 25:1  
d) 8.5:1  
e) \(8\frac{2}{3}:1\)  

**Solution:**  
For these problems, the actual length is calculated as:  
\[
\text{Actual Length} = \frac{\text{Image Length}}{\text{Scale Factor}}
\]  

1. **a) 2:1**  
   \[
   \text{Actual Length} = \frac{12 \, \text{cm}}{2} = 6 \, \text{cm}
   \]  

2. **b) 10:1**  
   \[
   \text{Actual Length} = \frac{12 \, \text{cm}}{10} = 1.2 \, \text{cm}
   \]  

3. **c) 25:1**  
   \[
   \text{Actual Length} = \frac{12 \, \text{cm}}{25} = 0.48 \, \text{cm}
   \]  

4. **d) 8.5:1**  
   \[
   \text{Actual Length} = \frac{12 \, \text{cm}}{8.5} \approx 1.41 \, \text{cm}
   \]  

5. **e) \(8\frac{2}{3}:1\) or 8.6667:1**  
   \[
   \text{Actual Length} = \frac{12 \, \text{cm}}{8.6667} \approx 1.38 \, \text{cm}
   \]  

---

### **Problem 27**  
**Question:**  
The distance between Vilnius Cathedral Square and the Seimas of the Republic of Lithuania is 1.7 km. The distance on the map is 10 cm. Determine the map scale.  

**Solution:**  
The scale is calculated as:  
\[
\text{Scale} = \frac{\text{Actual Distance}}{\text{Map Distance}}
\]  

1. Convert actual distance to the same units as the map:  
   \[
   1.7 \, \text{km} = 1.7 \times 100,000 = 170,000 \, \text{cm}
   \]  

2. Divide:  
   \[
   \text{Scale} = \frac{170,000}{10} = 17,000
   \]  

So, the scale is **1:17,000**.  

---

### Would you like detailed explanations for any of these solutions? 😊  

Here are **5 related questions** for practice:  
1. How would the actual distance change in Problem 25 if the map distance were 20 cm?  
2. What is the actual length in Problem 26 if the scale is 12:1?  
3. If a different map in Problem 27 had a scale of 1:20,000, what would be the map distance for 1.7 km?  
4. How do scale calculations change if the unit conversions (e.g., cm to km) are skipped?  
5. Why is it important to use consistent units when working with scales?

**Tip:** Always ensure units are consistent when working with scale calculations—convert all measurements to the same unit before solving! 🌟

Calculate the actual distances and map scales based on the given problems: (1) Map distance between Arnold's and Beno's houses is 15 cm with multiple scales, (2) Microchip length of 12 cm in a microscope image with various scales, (3) Map distance of 10 cm for a real-world distance of 1.7 km between Vilnius Cathedral Square and the Seimas of Lithuania.

Step-by-step solutions to problems involving map scales, microchip image scales, and proportional reasoning. Learn how to calculate actual distances, lengths, and scales using key formulas and unit conversions.

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