The sum of four numbers is 64. If you add 3 to first number , 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results become equal. What is the difference between the largest and the smallest of the original numbers ?

Options

A. 32
B. 27
C. 21
D. None of these

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the Variables
Let the four numbers be \( a, b, c, d \).

### Step 2: Set Up the Equation for the Sum
According to the problem, the sum of the four numbers is:
\[ a + b + c + d = 64 \]

### Step 3: Set Up the Equations Based on the Conditions
We are given conditions that transform the numbers:
1. \( a + 3 = k \)
2. \( b - 3 = k \)
3. \( 3c = k \)
4. \( \frac{d}{3} = k \)

From these equations, we can express \( a, b, c, d \) in terms of \( k \):
- From \( a + 3 = k \), we get \( a = k - 3 \).
- From \( b - 3 = k \), we get \( b = k + 3 \).
- From \( 3c = k \), we get \( c = \frac{k}{3} \).
- From \( \frac{d}{3} = k \), we get \( d = 3k \).

### Step 4: Substitute into the Sum Equation
Now, substitute \( a, b, c, d \) into the sum equation:
\[ (k - 3) + (k + 3) + \left(\frac{k}{3}\right) + (3k) = 64 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ k - 3 + k + 3 + \frac{k}{3} + 3k = 64 \]
This simplifies to:
\[ 5k + \frac{k}{3} = 64 \]

### Step 6: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by 3:
\[ 15k + k = 192 \]
This simplifies to:
\[ 16k = 192 \]

### Step 7: Solve for \( k \)
Now, divide both sides by 16:
\[ k = \frac{192}{16} = 12 \]

### Step 8: Find the Original Numbers
Now we can find the values of \( a, b, c, d \):
- \( a = k - 3 = 12 - 3 = 9 \)
- \( b = k + 3 = 12 + 3 = 15 \)
- \( c = \frac{k}{3} = \frac{12}{3} = 4 \)
- \( d = 3k = 3 \times 12 = 36 \)

### Step 9: Identify the Largest and Smallest Numbers
The original numbers are \( 9, 15, 4, 36 \).
- The largest number is \( 36 \).
- The smallest number is \( 4 \).

### Step 10: Calculate the Difference
Now, calculate the difference between the largest and smallest numbers:
\[ \text{Difference} = 36 - 4 = 32 \]

### Final Answer
The difference between the largest and smallest of the original numbers is \( 32 \).

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