Understanding the Ratio of Three Numbers Problem

This question asks us to find the sum of three numbers given their ratio and the sum of their squares. We are told the ratio of the three numbers is 3 ∶ 5 ∶ 4, and the sum of the squares of these numbers is 11250.

To solve this type of ratio problem, we can represent the numbers using a common multiple, let's call it $x$. Since the ratio is 3:5:4, the three numbers can be written as $3x$, $5x$, and $4x$.

Setting Up the Equation Based on the Sum of Squares

The problem states that the sum of the squares of these numbers is 11250. We can write this as an equation:

The square of the first number is $(3x)^2$.

The square of the second number is $(5x)^2$.

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

The sum of these squares is $(3x)^2 + (5x)^2 + (4x)^2$.

So, the equation is:

$$ (3x)^2 + (5x)^2 + (4x)^2 = 11250 $$

Solving for the Value of x

Now, let's simplify and solve the equation to find the value of $x$.

Squaring each term:

$$ 9x^2 + 25x^2 + 16x^2 = 11250 $$

Combine the terms with $x^2$:

$$ (9 + 25 + 16)x^2 = 11250 $$

$$ 50x^2 = 11250 $$

Now, isolate $x^2$ by dividing both sides by 50:

$$ x^2 = \frac{11250}{50} $$

$$ x^2 = \frac{1125}{5} $$

$$ x^2 = 225 $$

To find $x$, take the square root of both sides:

$$ x = \sqrt{225} $$

The square root of 225 is 15. Since we are dealing with the magnitudes of numbers in a ratio, we usually consider the positive root.

$$ x = 15 $$

Finding the Three Numbers

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

• First number = $3x = 3 \times 15 = 45$
• Second number = $5x = 5 \times 15 = 75$
• Third number = $4x = 4 \times 15 = 60$

We can check if the sum of the squares of these numbers is indeed 11250:

$$ 45^2 + 75^2 + 60^2 = 2025 + 5625 + 3600 = 11250 $$

The numbers are correct.

Calculating the Sum of the Numbers

The question asks for the sum of the three numbers. We add the numbers we found:

Sum = First number + Second number + Third number

Sum = $45 + 75 + 60$

Sum = $120 + 60$

Sum = $180$

The sum of the three numbers is 180.

Revision Table: Ratio and Squares

Additional Information: Solving Ratio Problems

Problems involving ratios often require setting up algebraic equations. When the sum of squares is given, remember to square the terms involving the variable $x$ correctly, i.e., $(ax)^2 = a^2 x^2$. If the sum of the numbers was given instead, the equation would be $3x + 5x + 4x = \text{given sum}$, which simplifies to $12x = \text{given sum}$, making it easier to find $x$. The type of sum (sum of numbers, sum of squares, sum of cubes, etc.) dictates the form of the algebraic equation you need to solve.