If P : Q = 1 : 5 and Q : R = 3 : 2, then what is (P + Q) : (Q + R)?

Options

A. `6 : 7 `
B. `5 : 7`
C. `18 : 25`
D. `13 : 18`

Answer

To solve the problem, we need to find the ratio (P + Q) : (Q + R) given the ratios P : Q = 1 : 5 and Q : R = 3 : 2.

### Step 1: Express P, Q, and R in terms of a common variable.
From the ratio P : Q = 1 : 5, we can express P and Q as:
- Let P = 1x
- Let Q = 5x

From the ratio Q : R = 3 : 2, we can express Q and R as:
- Let Q = 3y
- Let R = 2y

### Step 2: Equate the values of Q.
Since Q is represented in two different ways, we set them equal to each other:
5x = 3y

### Step 3: Solve for one variable in terms of the other.
From the equation 5x = 3y, we can express y in terms of x:
y = (5/3)x

### Step 4: Substitute y back to find R.
Now, we can substitute y back into the expression for R:
R = 2y = 2 * (5/3)x = (10/3)x

### Step 5: Calculate (P + Q) and (Q + R).
Now we can find P + Q and Q + R:
- P + Q = 1x + 5x = 6x
- Q + R = 5x + (10/3)x = (15/3)x + (10/3)x = (25/3)x

### Step 6: Find the ratio (P + Q) : (Q + R).
Now we can find the ratio:
(P + Q) : (Q + R) = (6x) : ((25/3)x)

### Step 7: Simplify the ratio.
To simplify the ratio, we can eliminate x:
= 6 : (25/3)
= 6 * 3 : 25
= 18 : 25

### Final Answer:
Thus, the ratio (P + Q) : (Q + R) is 18 : 25.

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