Average of five two digit positive numbers is 38. The second number is 10 more than the first number and 20 more than the third number. The value of the fourth number is 34. If the sum of the first number and the last number is 76, what is the value of the first number?

Options

A. 45
B. 40
C. 60
D. 64

Answer

To solve the problem step by step, we will denote the five numbers as follows:

Let:
- First number = \( A \)
- Second number = \( B \)
- Third number = \( C \)
- Fourth number = \( D \)
- Fifth number = \( E \)

### Step 1: Calculate the sum of the five numbers
Given that the average of the five numbers is 38, we can find the sum of these numbers:
\[ \text{Sum} = \text{Average} \times \text{Number of items} = 38 \times 5 = 190 \]

### Step 2: Set up the relationships between the numbers
From the problem, we have the following relationships:
1. \( B = A + 10 \) (The second number is 10 more than the first number)
2. \( B = C + 20 \) (The second number is 20 more than the third number)
3. \( D = 34 \) (The fourth number is given as 34)
4. \( A + E = 76 \) (The sum of the first number and the last number is 76)

### Step 3: Express \( C \) in terms of \( A \)
From the second relationship, we can express \( C \) in terms of \( A \):
\[ B = A + 10 \implies C = B - 20 = (A + 10) - 20 = A - 10 \]

### Step 4: Substitute the known values into the sum equation
Now we substitute \( B \), \( C \), and \( D \) into the sum equation:
\[ A + B + C + D + E = 190 \]
Substituting the values we have:
\[ A + (A + 10) + (A - 10) + 34 + E = 190 \]
This simplifies to:
\[ A + A + 10 + A - 10 + 34 + E = 190 \]
\[ 3A + 34 + E = 190 \]

### Step 5: Solve for \( E \)
Now we can isolate \( E \):
\[ E = 190 - 3A - 34 = 156 - 3A \]

### Step 6: Use the relationship \( A + E = 76 \)
Now we can use the relationship \( A + E = 76 \):
\[ A + (156 - 3A) = 76 \]
This simplifies to:
\[ A + 156 - 3A = 76 \]
\[ -2A + 156 = 76 \]
\[ -2A = 76 - 156 \]
\[ -2A = -80 \implies A = 40 \]

### Step 7: Conclusion
The value of the first number \( A \) is 40.