Ratio Analysis for p and q

We are given the ratios: $(p + q) : (p - q) : pq = 7 : 1 : 60$.

From the first part of the ratio, we consider $(p + q) : (p - q) = 7 : 1$.

This implies we can set $p + q = 7k$ and $p - q = 1k$ for some non-zero constant $k$.

• Adding these two equations yields: $(p + q) + (p - q) = 7k + k$, which simplifies to $2p = 8k$. Solving for $p$, we get $p = 4k$.
• Subtracting the second equation from the first yields: $(p + q) - (p - q) = 7k - k$, which simplifies to $2q = 6k$. Solving for $q$, we get $q = 3k$.

Determining the Scaling Factor k

Next, we use the ratio $(p + q) : pq = 7 : 60$.

Substitute the expressions $p = 4k$ and $q = 3k$ into this ratio:

• $p + q = 4k + 3k = 7k$.
• $pq = (4k)(3k) = 12k^2$.

The ratio $(p + q) : pq$ is therefore $(7k) : (12k^2)$.

We set this equal to the given ratio $7 : 60$: $ \frac{7k}{12k^2} = \frac{7}{60} $

Since $k \neq 0$ (otherwise $p=q=0$, which doesn't fit the ratios), we can simplify the equation:

Solving for $k$, we find $12k = 60$, which means $k = \frac{60}{12} = 5$.

Calculating r from p^2 + q^2

Now that we have found $k=5$, we can determine the specific values of $p$ and $q$:

• $p = 4k = 4(5) = 20$.
• $q = 3k = 3(5) = 15$.

The problem states that $p^2 + q^2 = r^2$.

Substitute the calculated values of $p$ and $q$ into this equation:

• $r^2 = (20)^2 + (15)^2$.
• $r^2 = 400 + 225$.
• $r^2 = 625$.

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