If P : Q = 2 : 3 and P : R = 4 : 5, then what is (P + Q) : (Q + R)?

Options

A. `20 : 27`
B. `20 : 17`
C. `10 : 11`
D. `20 : 29`

Answer

To solve the problem, we need to find the ratio \((P + Q) : (Q + R)\) given the ratios \(P : Q = 2 : 3\) and \(P : R = 4 : 5\).

### Step-by-Step Solution:

1. **Express P and Q in terms of a common variable:**
From the ratio \(P : Q = 2 : 3\), we can express \(P\) and \(Q\) as:
\[ P = 2x \quad \text{and} \quad Q = 3x \]
where \(x\) is a common multiplier.

**Hint:** Use a variable to express the quantities in ratios for easier calculations.

2. **Express R in terms of P:**
From the ratio \(P : R = 4 : 5\), we can express \(R\) in terms of \(P\):
\[ P = 4y \quad \text{and} \quad R = 5y \]
where \(y\) is another common multiplier.

**Hint:** When dealing with two ratios involving the same variable, express them in terms of a common variable.

3. **Equate the two expressions for P:**
Since both expressions represent \(P\), we can set them equal to each other:
\[ 2x = 4y \implies x = 2y \]

**Hint:** Solve for one variable in terms of the other to find a common relationship.

4. **Substitute x back into the expressions for Q and R:**
Now we substitute \(x = 2y\) into the expressions for \(Q\) and \(R\):
\[ Q = 3x = 3(2y) = 6y \]
\[ R = 5y \]

**Hint:** Substitute the value of the variable to find the other quantities.

5. **Calculate \(P + Q\) and \(Q + R\):**
Now we can calculate \(P + Q\) and \(Q + R\):
\[ P + Q = 2x + 3x = 2(2y) + 6y = 4y + 6y = 10y \]
\[ Q + R = 6y + 5y = 11y \]

**Hint:** Combine the expressions to find the total for each ratio.

6. **Form the ratio \((P + Q) : (Q + R)\):**
Now we can form the ratio:
\[ (P + Q) : (Q + R) = 10y : 11y = 10 : 11 \]

**Hint:** When forming ratios, the common variable cancels out.

### Final Answer:
The ratio \((P + Q) : (Q + R)\) is \(10 : 11\).