If 40 is added to a list of natural numbers , the average is increased by 4. When 30 is added to the new list , the average of the number in the new list is increased by 1. How many numbers were is the original list ?

Options

A. A)8
B. B)5
C. C)4
D. D)6

Answer

To solve the problem step by step, let's denote:

- \( n \) = number of natural numbers in the original list
- \( k \) = average of the original list

### Step 1: Calculate the sum of the original list
The sum of the original list can be expressed as:
\[ \text{Sum} = n \times k \]

### Step 2: Adding 40 to the list
When 40 is added to the list, the new sum becomes:
\[ \text{New Sum} = n \times k + 40 \]
The new number of elements in the list is \( n + 1 \), and the new average is \( k + 4 \). Therefore, we can set up the equation:
\[ \frac{n \times k + 40}{n + 1} = k + 4 \]

### Step 3: Cross multiply to eliminate the fraction
Cross multiplying gives:
\[ n \times k + 40 = (n + 1)(k + 4) \]

Expanding the right side:
\[ n \times k + 40 = n \times k + 4n + k + 4 \]

### Step 4: Simplify the equation
Subtract \( n \times k \) from both sides:
\[ 40 = 4n + k + 4 \]
Rearranging gives:
\[ 4n + k = 36 \quad \text{(Equation 1)} \]

### Step 5: Adding 30 to the new list
Now, when 30 is added to the new list, the new sum becomes:
\[ \text{New Sum} = n \times k + 40 + 30 = n \times k + 70 \]
The new number of elements is \( n + 2 \), and the new average is \( k + 5 \). Thus, we can set up the equation:
\[ \frac{n \times k + 70}{n + 2} = k + 5 \]

### Step 6: Cross multiply again
Cross multiplying gives:
\[ n \times k + 70 = (n + 2)(k + 5) \]

Expanding the right side:
\[ n \times k + 70 = n \times k + 5n + 2k + 10 \]

### Step 7: Simplify the equation
Subtract \( n \times k \) from both sides:
\[ 70 = 5n + 2k + 10 \]
Rearranging gives:
\[ 5n + 2k = 60 \quad \text{(Equation 2)} \]

### Step 8: Solve the system of equations
Now we have two equations:
1. \( 4n + k = 36 \)
2. \( 5n + 2k = 60 \)

From Equation 1, we can express \( k \) in terms of \( n \):
\[ k = 36 - 4n \]

Substituting \( k \) into Equation 2:
\[ 5n + 2(36 - 4n) = 60 \]
Expanding gives:
\[ 5n + 72 - 8n = 60 \]
Combining like terms:
\[ -3n + 72 = 60 \]
Solving for \( n \):
\[ -3n = 60 - 72 \]
\[ -3n = -12 \]
\[ n = 4 \]

### Conclusion
The number of natural numbers in the original list is \( n = 4 \).