Sum of the present ages of P,Q,R and S is 76 years. After 7 years ratio of their ages is `7:6:5:8`. What is R's present age ?

Options

A. 14
B. 12
C. 13
D. 8

Explanation

To find R's present age, we can follow these steps:

### Step 1: Define the variables
Let the present ages of P, Q, R, and S be represented as:
- P's age = P
- Q's age = Q
- R's age = R
- S's age = S

According to the problem, we know:
\[ P + Q + R + S = 76 \]

### Step 2: Set up the future ages
After 7 years, their ages will be:
- P's age = P + 7
- Q's age = Q + 7
- R's age = R + 7
- S's age = S + 7

### Step 3: Write the ratio of their ages after 7 years
The ratio of their ages after 7 years is given as 7:6:5:8. We can express this in terms of a variable \( x \):
- P's age after 7 years = \( 7x \)
- Q's age after 7 years = \( 6x \)
- R's age after 7 years = \( 5x \)
- S's age after 7 years = \( 8x \)

### Step 4: Set up the equation for the sum of ages after 7 years
The sum of their ages after 7 years can also be expressed as:
\[ (P + 7) + (Q + 7) + (R + 7) + (S + 7) = 76 + 28 \]
This simplifies to:
\[ P + Q + R + S + 28 = 104 \]

### Step 5: Substitute the sum of present ages
We know from Step 1 that \( P + Q + R + S = 76 \), so we can substitute this into the equation:
\[ 76 + 28 = 104 \]

### Step 6: Set up the equation with the ratios
From the ratios, we can also write:
\[ 7x + 6x + 5x + 8x = 26x \]
Thus, we have:
\[ 26x = 104 \]

### Step 7: Solve for \( x \)
Now we can solve for \( x \):
\[ x = \frac{104}{26} = 4 \]

### Step 8: Find R's age after 7 years
R's age after 7 years is given by:
\[ R + 7 = 5x \]
Substituting \( x = 4 \):
\[ R + 7 = 5 \times 4 = 20 \]

### Step 9: Calculate R's present age
To find R's present age, we subtract 7 from R's age after 7 years:
\[ R = 20 - 7 = 13 \]

### Conclusion
R's present age is **13 years**.

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