The sum of the present ages of A, B,C and D is 76 years. After 7 years the ratio of their ages becomes 7 : 6 : 5 : 8. What is C’s present age?

Options

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

Answer

To solve the problem step by step, we will follow these instructions:

1. **Define the variables based on the information given:**
Let the present ages of A, B, C, and D be represented as:
- A's present age = \( a \)
- B's present age = \( b \)
- C's present age = \( c \)
- D's present age = \( d \)

According to the problem, we know:
\[ a + b + c + d = 76 \]

2. **Set up the equations for their ages after 7 years:**
After 7 years, their ages will be:
- A's age = \( a + 7 \)
- B's age = \( b + 7 \)
- C's age = \( c + 7 \)
- D's age = \( d + 7 \)

The problem states that the ratio of their ages after 7 years is 7:6:5:8. We can express this as:
\[ a + 7 = 7k, \quad b + 7 = 6k, \quad c + 7 = 5k, \quad d + 7 = 8k \]
where \( k \) is a common factor.

3. **Express the present ages in terms of \( k \):**
Rearranging these equations gives us the present ages:
- \( a = 7k - 7 \)
- \( b = 6k - 7 \)
- \( c = 5k - 7 \)
- \( d = 8k - 7 \)

4. **Substitute these expressions into the sum of ages equation:**
Now we substitute \( a, b, c, d \) into the equation \( a + b + c + d = 76 \):
\[ (7k - 7) + (6k - 7) + (5k - 7) + (8k - 7) = 76 \]

5. **Combine like terms:**
Simplifying the left side:
\[ 7k + 6k + 5k + 8k - 28 = 76 \]
\[ 26k - 28 = 76 \]

6. **Solve for \( k \):**
Adding 28 to both sides:
\[ 26k = 76 + 28 \]
\[ 26k = 104 \]
Dividing both sides by 26:
\[ k = \frac{104}{26} = 4 \]

7. **Calculate C's present age:**
Now that we have \( k \), we can find C's present age:
\[ c = 5k - 7 \]
Substituting \( k = 4 \):
\[ c = 5(4) - 7 = 20 - 7 = 13 \]

Thus, C's present age is **13 years**.