If ratio of ages of P and Q before 4 year is 9 : 10 , after 12 years sum of their ages will be 70 years, their what was P's age 2 years ago ?

Options

A. 24 years
B. 22 years
C. 20 years
D. 26 years

Answer

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

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

### Step 2: Establish the Ratio of Ages
According to the problem, the ratio of their ages 4 years ago was 9:10. Therefore, we can write the equation:
\[ \frac{X - 4}{Y - 4} = \frac{9}{10} \]
Cross-multiplying gives us:
\[ 10(X - 4) = 9(Y - 4) \]
Expanding this, we have:
\[ 10X - 40 = 9Y - 36 \]
Rearranging it leads to:
\[ 10X - 9Y = 4 \quad \text{(Equation 1)} \]

### Step 3: Establish the Sum of Ages After 12 Years
The problem states that after 12 years, the sum of their ages will be 70. Therefore, we can write the equation:
\[ (X + 12) + (Y + 12) = 70 \]
Simplifying this gives:
\[ X + Y + 24 = 70 \]
Rearranging it leads to:
\[ X + Y = 46 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( 10X - 9Y = 4 \) (Equation 1)
2. \( X + Y = 46 \) (Equation 2)

From Equation 2, we can express \( Y \) in terms of \( X \):
\[ Y = 46 - X \]

### Step 5: Substitute \( Y \) in Equation 1
Substituting \( Y \) in Equation 1:
\[ 10X - 9(46 - X) = 4 \]
Expanding this gives:
\[ 10X - 414 + 9X = 4 \]
Combining like terms results in:
\[ 19X - 414 = 4 \]
Adding 414 to both sides:
\[ 19X = 418 \]
Dividing both sides by 19 gives:
\[ X = 22 \]

### Step 6: Find \( Y \)
Now substituting \( X \) back into Equation 2 to find \( Y \):
\[ Y = 46 - 22 = 24 \]

### Step 7: Find P's Age 2 Years Ago
Since we have found \( X \) (P's present age) to be 22, we can find P's age 2 years ago:
\[ P's \, age \, 2 \, years \, ago = X - 2 = 22 - 2 = 20 \]

### Final Answer
P's age 2 years ago was **20 years**.

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