Six years hence, the age of A would be `(5)/(6th)` of the age of B. 4 years ago, the ages of A and B were in the ratio of `10 : 13`. What is the present age of C, who is 5 years elder to A?

Options

A. 24 years
B. 29 years
C. 30 years
D. 34 years

Answer

To solve the problem step by step, we need to determine the present ages of A and B based on the information provided, and then find the age of C.

### Step 1: Define Variables
Let the present age of A be \( a \) and the present age of B be \( b \).

### Step 2: Set Up Future Age Equation
According to the problem, six years from now, the age of A will be \( \frac{5}{6} \) of the age of B. This can be expressed mathematically as:
\[ a + 6 = \frac{5}{6}(b + 6) \]

### Step 3: Set Up Past Age Ratio
The problem also states that four years ago, the ages of A and B were in the ratio of \( 10:13 \). This can be expressed as:
\[ \frac{a - 4}{b - 4} = \frac{10}{13} \]
Cross-multiplying gives:
\[ 13(a - 4) = 10(b - 4) \]

### Step 4: Simplify the Equations
From the first equation:
\[ 6(a + 6) = 5(b + 6) \]
Expanding this gives:
\[ 6a + 36 = 5b + 30 \implies 6a - 5b = -6 \quad \text{(Equation 1)} \]

From the second equation:
\[ 13a - 52 = 10b - 40 \implies 13a - 10b = 12 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have a system of linear equations:
1. \( 6a - 5b = -6 \)
2. \( 13a - 10b = 12 \)

We can solve these equations simultaneously. We can multiply Equation 1 by 2 to align the coefficients of \( b \):
\[ 12a - 10b = -12 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (12a - 10b) - (13a - 10b) = -12 - 12 \]
This simplifies to:
\[ -1a = -24 \implies a = 24 \]

### Step 6: Find the Age of B
Substituting \( a = 24 \) back into Equation 1:
\[ 6(24) - 5b = -6 \]
\[ 144 - 5b = -6 \implies 5b = 150 \implies b = 30 \]

### Step 7: Find the Age of C
C is 5 years older than A:
\[ \text{Age of C} = a + 5 = 24 + 5 = 29 \]

### Final Answer
The present age of C is \( 29 \).