Six years ago, the ratio of the ages of the two persons P and Q was `3:2` Four years hence the ratio of their ages will be `8:7`. What is P's age at present ?
A. 10 years
B. 12 years
C. 14 years
D. 8 years

Options

A. C
B. B
C. D
D. A

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define Variables
Let the present age of person P be \( P \) years and the present age of person Q be \( Q \) years.

### Step 2: Set Up the Equations
According to the problem, six years ago, the ratio of their ages was \( 3:2 \). This can be expressed as:
\[ \frac{P - 6}{Q - 6} = \frac{3}{2} \]
Cross-multiplying gives:
\[ 2(P - 6) = 3(Q - 6) \]
Expanding this, we have:
\[ 2P - 12 = 3Q - 18 \]
Rearranging gives us our first equation:
\[ 2P - 3Q = -6 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
Four years hence, the ratio of their ages will be \( 8:7 \). This can be expressed as:
\[ \frac{P + 4}{Q + 4} = \frac{8}{7} \]
Cross-multiplying gives:
\[ 7(P + 4) = 8(Q + 4) \]
Expanding this, we have:
\[ 7P + 28 = 8Q + 32 \]
Rearranging gives us our second equation:
\[ 7P - 8Q = 4 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( 2P - 3Q = -6 \)
2. \( 7P - 8Q = 4 \)

We can solve these equations using substitution or elimination. Let's use the elimination method.

First, we can multiply Equation 1 by 7 and Equation 2 by 2 to align the coefficients of \( P \):
\[ 14P - 21Q = -42 \quad \text{(Equation 3)} \]
\[ 14P - 16Q = 8 \quad \text{(Equation 4)} \]

Now, subtract Equation 4 from Equation 3:
\[ (14P - 21Q) - (14P - 16Q) = -42 - 8 \]
This simplifies to:
\[ -5Q = -50 \]
Thus, we find:
\[ Q = 10 \]

### Step 5: Substitute Back to Find P
Now substitute \( Q = 10 \) back into Equation 1:
\[ 2P - 3(10) = -6 \]
This simplifies to:
\[ 2P - 30 = -6 \]
Adding 30 to both sides gives:
\[ 2P = 24 \]
Dividing by 2 gives:
\[ P = 12 \]

### Conclusion
The present age of person P is \( 12 \) years.

### Final Answer
**P's age at present is 12 years (Option B).**

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