If three numbers are in the ratio of 1:2:3 and half of their sum is 18, then what is the ratio of the squares of the numbers?

Steps to Solve

1. Identify the Numbers Based on the Ratio

The three numbers are in the ratio of 1:2:3. Let the numbers be $x$, $2x$, and $3x$. Given the sum of these numbers is 36 (since half is 18), we can set up the equation:

$$ x + 2x + 3x = 36 $$

2. Simplify and Solve for x

Combine the terms on the left side:

$$ 6x = 36 $$

Now, divide both sides by 6 to solve for $x$:

$$ x = \frac{36}{6} = 6 $$

3. Calculate the Actual Numbers

Now substitute $x$ back to find the three numbers:

• First number: $x = 6$
• Second number: $2x = 12$
• Third number: $3x = 18$

4. Find the Squares of Each Number

Now calculate the squares of each number:

• First number: $6^2 = 36$
• Second number: $12^2 = 144$
• Third number: $18^2 = 324$

5. Determine the Ratio of the Squares

The ratio of the squares can be written as:

$$ \text{Ratio} = 36 : 144 : 324 $$

To simplify, divide each term by 36:

$$ \frac{36}{36} : \frac{144}{36} : \frac{324}{36} = 1 : 4 : 9 $$

6. Express the Ratio Cleanly

Finally, the ratio of the squares, reduced identifiable, is:

$$ 1:4:9 $$