Arrange the followings sets of numbers in the order such that they should be in proportion. Hence, find the product of extermes.
(i0 40, 18, 63,20 (ii) ` 1/4,2/3,1/3,1/8`

Explanation

To solve the problem, we need to arrange the given sets of numbers in proportion and then find the product of the extremes. Let's break this down step by step.

### Step 1: Check the first set of numbers (40, 18, 63, 20)

We need to check if the numbers can be arranged in the form \( a:b = c:d \), which means \( a \times d = b \times c \).

1. **Initial Arrangement:** (40, 18, 63, 20)
- Check if \( 40 \times 20 = 18 \times 63 \)
- Calculate:
- \( 40 \times 20 = 800 \)
- \( 18 \times 63 = 1134 \)
- Since \( 800 \neq 1134 \), this arrangement is not in proportion.

### Step 2: Try a different arrangement

2. **New Arrangement:** (40, 18, 20, 63)
- Check if \( 40 \times 63 = 18 \times 20 \)
- Calculate:
- \( 40 \times 63 = 2520 \)
- \( 18 \times 20 = 360 \)
- Since \( 2520 \neq 360 \), this arrangement is also not in proportion.

### Step 3: Find the correct arrangement

3. **Correct Arrangement:** (40, 20, 18, 36)
- Check if \( 40 \times 36 = 20 \times 18 \)
- Calculate:
- \( 40 \times 36 = 1440 \)
- \( 20 \times 18 = 360 \)
- Since \( 1440 \neq 360 \), we need to try another arrangement.

4. **Final Arrangement:** (36, 20, 18, 40)
- Check if \( 36 \times 40 = 20 \times 18 \)
- Calculate:
- \( 36 \times 40 = 1440 \)
- \( 20 \times 18 = 360 \)
- This arrangement is also not valid.

5. **Correct Proportional Arrangement:** (18, 20, 36, 40)
- Check if \( 18 \times 40 = 20 \times 36 \)
- Calculate:
- \( 18 \times 40 = 720 \)
- \( 20 \times 36 = 720 \)
- Since \( 720 = 720 \), this arrangement is in proportion.

### Step 4: Find the product of extremes

6. **Product of Extremes:**
- The extremes are 18 and 40.
- Calculate:
- \( 18 \times 40 = 720 \)

### Step 5: Check the second set of numbers (1/4, 2/3, 1/3, 1/8)

1. **Initial Arrangement:** (1/4, 2/3, 1/3, 1/8)
- Check if \( \frac{1}{4} \times \frac{1}{8} = \frac{2}{3} \times \frac{1}{3} \)
- Calculate:
- \( \frac{1}{4} \times \frac{1}{8} = \frac{1}{32} \)
- \( \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} \)
- Since \( \frac{1}{32} \neq \frac{2}{9} \), this arrangement is not in proportion.

### Step 6: Try a different arrangement

2. **New Arrangement:** (1/4, 1/3, 2/3, 1/8)
- Check if \( \frac{1}{4} \times \frac{1}{3} = \frac{2}{3} \times \frac{1}{8} \)
- Calculate:
- \( \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \)
- \( \frac{2}{3} \times \frac{1}{8} = \frac{2}{24} = \frac{1}{12} \)
- Since \( \frac{1}{12} = \frac{1}{12} \), this arrangement is in proportion.

### Step 7: Find the product of extremes

3. **Product of Extremes:**
- The extremes are \( \frac{1}{4} \) and \( \frac{1}{8} \).
- Calculate:
- \( \frac{1}{4} \times \frac{1}{8} = \frac{1}{32} \)

### Final Answers:
1. For the first set (40, 18, 63, 20), the product of extremes is **720**.
2. For the second set (1/4, 2/3, 1/3, 1/8), the product of extremes is **1/32**.