Understanding the Problem: Positive Numbers in Ratio and Sum of Squares

The question asks us to find the average of the first two numbers from a set of three positive numbers. These numbers are in a specific ratio, and we are given the sum of their squares.

Let the three positive numbers be represented by $2x$, $3x$, and $4x$, where $x$ is a positive constant. This representation maintains the given ratio of 2 ∶ 3 ∶ 4.

Setting up the Equation with Sum of Squares

We are given that the sum of the squares of these three numbers is 2349. We can write this as an algebraic equation:

The square of the first number is $(2x)^2 = 4x^2$.

The square of the second number is $(3x)^2 = 9x^2$.

The square of the third number is $(4x)^2 = 16x^2$.

The sum of these squares is:

$(2x)^2 + (3x)^2 + (4x)^2 = 2349$

$4x^2 + 9x^2 + 16x^2 = 2349$

Solving for the Common Factor ($x$)

Now, we need to solve this equation for $x$. Combine the terms on the left side:

$(4 + 9 + 16)x^2 = 2349$

$29x^2 = 2349$

Divide both sides by 29 to isolate $x^2$:

$x^2 = \frac{2349}{29}$

Performing the division:

$x^2 = 81$

Since the numbers are positive, $x$ must be a positive value. Take the positive square root of 81:

$x = \sqrt{81}$

$x = 9$

Finding the Positive Numbers

Now that we have the value of $x$, we can find the three positive numbers:

• First number = $2x = 2 \times 9 = 18$
• Second number = $3x = 3 \times 9 = 27$
• Third number = $4x = 4 \times 9 = 36$

We can quickly check if the sum of their squares is 2349:

$18^2 + 27^2 + 36^2 = 324 + 729 + 1296$

$324 + 729 + 1296 = 1053 + 1296 = 2349$

The numbers are correct.

Calculating the Average of the First Two Numbers

The question asks for the average of the first two numbers, which are 18 and 27.

The average of two numbers is the sum of the numbers divided by 2.

Average = $\frac{\text{First number} + \text{Second number}}{2}$

Average = $\frac{18 + 27}{2}$

Average = $\frac{45}{2}$

Average = $22.5$

So, the average of the first two numbers is 22.5.

Revision Table: Key Concepts

Additional Information: Ratios and Proportions

Understanding ratios is fundamental in many mathematical problems. A ratio like 2 ∶ 3 ∶ 4 means that for every 2 units of the first number, there are 3 units of the second, and 4 units of the third. Using a common multiplier ($x$) helps translate the ratio into actual number values ($2x, 3x, 4x$) that can be used in equations.

When dealing with squares or cubes of numbers in a ratio, remember to apply the power to both the ratio part and the multiplier:

• Square of $2x$ is $(2x)^2 = 2^2 \times x^2 = 4x^2$.
• Cube of $2x$ is $(2x)^3 = 2^3 \times x^3 = 8x^3$.

This problem combined concepts of ratio, algebra (solving equations), and averages, which is common in quantitative aptitude tests.