Understanding the Ratio Problem

The problem provides us with ratios involving the sums of three variables, p, q, and r, and also gives us the total sum of these variables. We are asked to find the ratio p : q : r.

The given information is:

• (p + q) ∶ (q + r) ∶ (r + p) = 5 ∶ 6 ∶ 7
• p + q + r = 18

We need to find the ratio p ∶ q ∶ r.

Setting up Equations from Ratios

From the first ratio, we can introduce a constant multiplier, let's call it 'k'. This means:

• p + q = 5k (Equation 1)
• q + r = 6k   (Equation 2)
• r + p = 7k   (Equation 3)

Solving for p, q, and r in terms of k

We can add the three equations above:

\((p + q) + (q + r) + (r + p) = 5k + 6k + 7k\)

\(2p + 2q + 2r = 18k\)

\(2(p + q + r) = 18k\)

Dividing by 2, we get:

\(p + q + r = 9k\)   (Equation 4)

Now, we can find expressions for p, q, and r individually by subtracting Equations 1, 2, and 3 from Equation 4:

• To find r, subtract Equation 1 from Equation 4: \((p + q + r) - (p + q) = 9k - 5k \implies r = 4k\)
• To find p, subtract Equation 2 from Equation 4: \((p + q + r) - (q + r) = 9k - 6k \implies p = 3k\)
• To find q, subtract Equation 3 from Equation 4: \((p + q + r) - (r + p) = 9k - 7k \implies q = 2k\)

So, from the ratio of sums, we find that p = 3k, q = 2k, and r = 4k.

The ratio p ∶ q ∶ r is therefore \(3k \ratio; 2k \ratio; 4k\), which simplifies to \(3 \ratio; 2 \ratio; 4\).

Using the Total Sum to Find the Value of k

We are given that p + q + r = 18.

Substitute the expressions for p, q, and r in terms of k into this equation:

\(3k + 2k + 4k = 18\)

\(9k = 18\)

Dividing by 9:

\(k = \frac{18}{9} = 2\)

Finding the Actual Values and the Ratio

Now that we have the value of k, we can find the actual values of p, q, and r:

• \(p = 3k = 3 \times 2 = 6\)
• \(q = 2k = 2 \times 2 = 4\)
• \(r = 4k = 4 \times 2 = 8\)

The actual values that satisfy both conditions are p=6, q=4, and r=8. Let's check:

• p+q = 6+4 = 10 (This is 5k with k=2)
• q+r = 4+8 = 12 (This is 6k with k=2)
• r+p = 8+6 = 14 (This is 7k with k=2)
• The ratio (p+q):(q+r):(r+p) = 10:12:14 = 5:6:7. This matches the first condition.
• p+q+r = 6+4+8 = 18. This matches the second condition.

The ratio p ∶ q ∶ r is \(6 \ratio; 4 \ratio; 8\).

This ratio can be simplified by dividing all terms by their greatest common divisor, which is 2:

\(\frac{6}{2} \ratio; \frac{4}{2} \ratio; \frac{8}{2} = 3 \ratio; 2 \ratio; 4\)

The ratio p ∶ q ∶ r is \(3 \ratio; 2 \ratio; 4\).

Now let's look at the given options. The options are 4 ∶ 5 ∶ 6, 5 ∶ 6 ∶ 7, 9 ∶ 6 ∶ 12, and 2 ∶ 3 ∶ 4.

Our derived ratio is \(3 \ratio; 2 \ratio; 4\).

Let's examine the third option: \(9 \ratio; 6 \ratio; 12\).

We can see that \(9 \ratio; 6 \ratio; 12\) is proportional to \(3 \ratio; 2 \ratio; 4\), as \(3 \times 3 = 9\), \(2 \times 3 = 6\), and \(4 \times 3 = 12\). So, \(9 \ratio; 6 \ratio; 12\) is \(3 \times (3 \ratio; 2 \ratio; 4)\).

Therefore, the ratio p ∶ q ∶ r is \(9 \ratio; 6 \ratio; 12\).

Revision Table: Steps to Solve Ratio Problems

Additional Information: Proportional Ratios

A ratio represents a relationship between quantities. For example, the ratio 3:2:4 means that the quantities are in the proportion 3 parts, 2 parts, and 4 parts, for a total of 3+2+4 = 9 parts. The actual values could be 3, 2, 4 (if the total is 9), or 6, 4, 8 (if the total is 18, as 9 parts = 18, so 1 part = 2), or 9, 6, 12 (if the total is 27, as 9 parts = 27, so 1 part = 3), and so on.

All ratios like 3:2:4, 6:4:8, 9:6:12, etc., are considered equivalent because they represent the same relative proportion between the quantities. They are all scalar multiples of the simplest form (3:2:4).

In this problem, our calculations based on the given conditions lead directly to the ratio 3:2:4 (or actual values 6, 4, 8 which have the ratio 3:2:4). However, the option provided as the correct answer is 9:6:12. This ratio 9:6:12 is a proportional representation of the ratio 3:2:4, obtained by multiplying each term by 3. Often, ratio problems might list a scaled version of the simplest ratio as an option.