Understanding the Problem: Average and Ratio

The question asks us to find the smaller of two numbers. We are given information about the average of a set of 11 numbers and the average of 9 of those numbers. We also know the ratio between the two remaining numbers.

Calculating Total Sum

First, let's find the sum of all 11 numbers. The formula for the average is:

Average = $\frac{\text{Sum of numbers}}{\text{Count of numbers}}$

Therefore, the sum of numbers can be calculated as:

Sum = Average $\times$ Count of numbers

For the 11 numbers:

• Count = 11
• Average = 18
• Sum = $11 \times 18 = 198$

Calculating Sum of 9 Numbers

Next, we calculate the sum of the 9 numbers whose average is given:

• Count = 9
• Average = 17
• Sum = $9 \times 17 = 153$

Finding the Sum of the Remaining Two Numbers

The sum of the remaining two numbers is the difference between the total sum of 11 numbers and the sum of the 9 numbers:

Sum of remaining two numbers = (Sum of 11 numbers) - (Sum of 9 numbers)

Sum of remaining two numbers = $198 - 153 = 45$

Using the Ratio to Find the Numbers

We are told the ratio of the remaining two numbers is 2 : 3. Let the two numbers be $2x$ and $3x$.

The sum of these two numbers is $2x + 3x = 5x$.

We know their sum is 45, so we can set up the equation:

$5x = 45$

Solving for x

To find the value of $x$, we divide the sum by 5:

$x = \frac{45}{5}$

$x = 9$

Identifying the Smaller Number

The two numbers are $2x$ and $3x$. Since $x=9$:

• The first number = $2x = 2 \times 9 = 18$
• The second number = $3x = 3 \times 9 = 27$

Comparing the two numbers, 18 and 27, the smaller number is 18.

Summary Table

The smaller of the two remaining numbers is 18.