If Ruperts has 4 more coins than Laxmi has 1 more coin than Bill and Bill has 1 more coin than Hawkins. Finally Hawkins has 4 more coins than Ajim. Then minimum number of coins that must be transfered if all of them wish to have an equal number of coins :

Options

A. 5
B. 6
C. 7
D. none of these

Answer

To solve the problem, let's denote the number of coins each person has using variables based on the relationships provided in the question.

1. **Define Variables:**
- Let Ajim have \( A \) coins.
- Hawkins has \( A + 4 \) coins (since Hawkins has 4 more coins than Ajim).
- Bill has \( (A + 4) + 1 = A + 5 \) coins (since Bill has 1 more coin than Hawkins).
- Lakshmi has \( (A + 5) + 1 = A + 6 \) coins (since Lakshmi has 1 more coin than Bill).
- Rupert has \( (A + 6) + 4 = A + 10 \) coins (since Rupert has 4 more coins than Lakshmi).

2. **Total Coins Calculation:**
- Now, we can express the total number of coins:
\[ \text{Total coins} = A + (A + 4) + (A + 5) + (A + 6) + (A + 10) \]
- Simplifying this gives:
\[ \text{Total coins} = 5A + 25 \]

3. **Equal Distribution:**
- To find how many coins each person should have if they are to be equal, we divide the total number of coins by the number of people (5):
\[ \text{Coins per person} = \frac{5A + 25}{5} = A + 5 \]

4. **Determine Excess or Deficit:**
- Now we can find out how many coins each person has compared to \( A + 5 \):
- Ajim: \( A - (A + 5) = -5 \) (Ajim needs 5 coins)
- Hawkins: \( (A + 4) - (A + 5) = -1 \) (Hawkins needs 1 coin)
- Bill: \( (A + 5) - (A + 5) = 0 \) (Bill has enough coins)
- Lakshmi: \( (A + 6) - (A + 5) = 1 \) (Lakshmi has 1 extra coin)
- Rupert: \( (A + 10) - (A + 5) = 5 \) (Rupert has 5 extra coins)

5. **Total Coins to Transfer:**
- To equalize the coins, we need to transfer coins from those who have extra to those who are in deficit:
- Ajim needs 5 coins.
- Hawkins needs 1 coin.
- Lakshmi has 1 extra coin.
- Rupert has 5 extra coins.
- Total coins needed = 5 (Ajim) + 1 (Hawkins) = 6 coins.
- Total coins available = 1 (Lakshmi) + 5 (Rupert) = 6 coins.

6. **Minimum Transfers:**
- Since the total coins needed (6) matches the total coins available (6), the minimum number of coins that must be transferred is 6.

Thus, the minimum number of coins that must be transferred for all to have an equal number of coins is **6**.