Distance Measurement with Gunter's Chain and Engineer's Chain

Introduction:
In the question, we are given one complete Gunter's chain and one complete Engineer's chain. These chains are used for measuring distances in surveying and engineering applications. We need to determine the distance between two stations A and B using these chains.

Explanation:
To solve this problem, we need to understand the measurements of Gunter's chain and Engineer's chain.

1. Gunter's Chain:
- A Gunter's chain is 66 feet long.
- It consists of 100 links, with each link measuring 0.66 feet (66/100 = 0.66 ft).
- Therefore, one complete Gunter's chain measures 66 ft.

2. Engineer's Chain:
- An Engineer's chain is 100 feet long.
- It consists of 100 links, with each link measuring 1 foot (100/100 = 1 ft).
- Therefore, one complete Engineer's chain measures 100 ft.

Now, let's calculate the distance between stations A and B using the given chains.

- We have one complete Gunter's chain, which measures 66 ft.
- We have one complete Engineer's chain, which measures 100 ft.

To calculate the distance between A and B, we can use the concept of ratio and proportion.

Since the ratio of Gunter's chain to Engineer's chain is 66:100, we can set up the following proportion:

66 ft / 100 ft = x ft / d ft

Where:
- x ft is the unknown distance between A and B.
- d ft is the total distance covered by both chains.

Simplifying the proportion:

66/100 = x/d

Cross-multiplying:

66d = 100x

Dividing both sides by 66:

d = (100x)/66

Now, substituting the given options into the equation, we can find the correct answer.

a) 166 ft:
d = (100 * 166) / 66 = 150 ft (approximately)

b) 266 ft:
d = (100 * 266) / 66 = 404 ft (approximately)

c) 133 ft:
d = (100 * 133) / 66 = 201 ft (approximately)

d) 233 ft:
d = (100 * 233) / 66 = 353 ft (approximately)

Therefore, the correct answer is option 'A' - 166 ft.

Summary:
Using the concept of ratio and proportion, we can calculate the distance between two stations using Gunter's chain and Engineer's chain. In this question, the correct answer is option 'A' - 166 ft.