The average of three numbers is 135. The largest number is 195 and the difference between the other two is 20. The smallest number is:

Options

A. 65
B. 95
C. 105
D. 115

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Given Information
We know:
- The average of three numbers is 135.
- The largest number is 195.
- The difference between the other two numbers is 20.

### Step 2: Calculate the Sum of the Three Numbers
The average of three numbers is given by the formula:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of terms}} \]
From the problem, we can rearrange this to find the sum:
\[ \text{Sum of numbers} = \text{Average} \times \text{Number of terms} = 135 \times 3 = 405 \]

### Step 3: Set Up the Equation for the Other Two Numbers
Let the three numbers be \( A \), \( B \), and \( C \), where:
- \( A = 195 \) (the largest number)
- \( B \) and \( C \) are the other two numbers.

From the sum we calculated:
\[ A + B + C = 405 \]
Substituting \( A \):
\[ 195 + B + C = 405 \]
This simplifies to:
\[ B + C = 405 - 195 = 210 \]

### Step 4: Use the Difference Between the Two Numbers
We know the difference between the two numbers \( B \) and \( C \) is 20. We can set up the following equations:
1. \( B + C = 210 \) (from the previous step)
2. \( B - C = 20 \) (given in the problem)

### Step 5: Solve the System of Equations
Now, we can solve these two equations simultaneously. We can add the two equations:
\[ (B + C) + (B - C) = 210 + 20 \]
This simplifies to:
\[ 2B = 230 \implies B = \frac{230}{2} = 115 \]

Now, substitute \( B \) back into one of the equations to find \( C \):
\[ B + C = 210 \implies 115 + C = 210 \implies C = 210 - 115 = 95 \]

### Step 6: Identify the Smallest Number
From our calculations:
- The largest number \( A = 195 \)
- The second number \( B = 115 \)
- The smallest number \( C = 95 \)

Thus, the smallest number is **95**.

### Summary of the Solution
The smallest number is **95**.