If the ratio of the age of P to that of Q 4 years ago was `5:4` and after 12 years the sum of their ages will be 68 years, what was P’s age 2 years ago?

Options

A. 24 years
B. 22 years
C. 18. years
D. 26 years

Answer

To solve the problem step by step, we will use algebra to find the ages of P and Q.

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

### Step 2: Set Up the First Equation
According to the problem, the ratio of the ages of P to Q 4 years ago was \( 5:4 \). This can be expressed mathematically as:
\[ \frac{X - 4}{Y - 4} = \frac{5}{4} \]
Cross-multiplying gives us:
\[ 4(X - 4) = 5(Y - 4) \]
Expanding this, we get:
\[ 4X - 16 = 5Y - 20 \]
Rearranging gives us our first equation:
\[ 4X - 5Y = -4 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that after 12 years, the sum of their ages will be 68 years. Therefore, we can express this as:
\[ (X + 12) + (Y + 12) = 68 \]
Simplifying this gives:
\[ X + Y + 24 = 68 \]
Rearranging gives us our second equation:
\[ X + Y = 44 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
From Equation 2, we can express \( Y \) in terms of \( X \):
\[ Y = 44 - X \]
Now, substitute this value of \( Y \) into Equation 1:
\[ 4X - 5(44 - X) = -4 \]
Expanding this gives:
\[ 4X - 220 + 5X = -4 \]
Combining like terms results in:
\[ 9X - 220 = -4 \]
Adding 220 to both sides gives:
\[ 9X = 216 \]
Dividing by 9 gives:
\[ X = 24 \]

### Step 5: Find the Present Age of Q
Now, substitute \( X \) back into Equation 2 to find \( Y \):
\[ Y = 44 - 24 = 20 \]

### Step 6: Calculate P's Age 2 Years Ago
Finally, to find P's age 2 years ago, we subtract 2 from P's present age:
\[ P's \, age \, 2 \, years \, ago = X - 2 = 24 - 2 = 22 \]

### Final Answer
P's age 2 years ago was **22 years**.