Out of four numbers the average of the first three is 26 and that of the last three is 24.5. If the last number is 19, then find the first number ?

Options

A. 24.5
B. 23.5
C. 25
D. 25.5

Answer

To solve the problem step by step, we will follow the given information about the averages and the last number.

### Step 1: Define the Variables
Let the four numbers be A, B, C, and D. Here, A is the first number, B is the second number, C is the third number, and D is the fourth number.

**Hint:** Identify the variables clearly to avoid confusion later.

### Step 2: Use the Average of the First Three Numbers
We know that the average of the first three numbers (A, B, C) is 26. The formula for average is:
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
So, we can write:
\[ \frac{A + B + C}{3} = 26 \]
Multiplying both sides by 3 gives us:
\[ A + B + C = 26 \times 3 = 78 \quad \text{(Equation 1)} \]

**Hint:** Remember to multiply by the number of observations to find the sum.

### Step 3: Use the Average of the Last Three Numbers
The average of the last three numbers (B, C, D) is 24.5. Using the same average formula:
\[ \frac{B + C + D}{3} = 24.5 \]
Multiplying both sides by 3 gives us:
\[ B + C + D = 24.5 \times 3 = 73.5 \quad \text{(Equation 2)} \]

**Hint:** Again, multiply by the number of observations to find the sum.

### Step 4: Substitute the Value of D
We know that the last number D is 19. Substituting D into Equation 2:
\[ B + C + 19 = 73.5 \]
Now, we can solve for B + C:
\[ B + C = 73.5 - 19 = 54.5 \quad \text{(Equation 3)} \]

**Hint:** Always substitute known values into equations to simplify them.

### Step 5: Substitute B + C into Equation 1
Now, we can substitute the value of B + C from Equation 3 into Equation 1:
\[ A + (B + C) = 78 \]
Substituting B + C:
\[ A + 54.5 = 78 \]
Now, solve for A:
\[ A = 78 - 54.5 = 23.5 \]

**Hint:** Isolate the variable you want to find by performing inverse operations.

### Conclusion
The first number A is 23.5.

**Final Answer:** The first number is **23.5**.