Mahesh has two sons named Karan and Arjun. The ratio of present age of Mahesh and Karan is 5:2 and that of Karan and Arjun is 4:3. Also, Karan is 5 years elder than Arjun, Find the ratio of their ages 10 years ago.

Options

A. A)`10:4:3`
B. B)`7:2:1`
C. C)`8:2:1`
D. D)`8:3:1`

Answer

To solve the problem, we need to determine the ages of Mahesh, Karan, and Arjun based on the given ratios and then find the ratio of their ages 10 years ago. Let's break it down step by step.

### Step 1: Establish the Ratios
We know:
1. The ratio of the present ages of Mahesh and Karan is 5:2.
2. The ratio of the present ages of Karan and Arjun is 4:3.
3. Karan is 5 years older than Arjun.

Let’s denote:
- Mahesh's age as \( M \)
- Karan's age as \( K \)
- Arjun's age as \( A \)

From the first ratio:
\[ \frac{M}{K} = \frac{5}{2} \implies M = \frac{5}{2}K \]

From the second ratio:
\[ \frac{K}{A} = \frac{4}{3} \implies K = \frac{4}{3}A \]

### Step 2: Relate Karan and Arjun's Ages
We also know that Karan is 5 years older than Arjun:
\[ K = A + 5 \]

### Step 3: Substitute Karan's Age
Now we can substitute \( K \) from the second equation into the third equation:
\[ \frac{4}{3}A = A + 5 \]

### Step 4: Solve for Arjun's Age
To eliminate the fraction, multiply the entire equation by 3:
\[ 4A = 3A + 15 \]
\[ 4A - 3A = 15 \implies A = 15 \]

### Step 5: Find Karan's Age
Now that we have Arjun's age, we can find Karan's age:
\[ K = A + 5 = 15 + 5 = 20 \]

### Step 6: Find Mahesh's Age
Now we can find Mahesh's age using the first equation:
\[ M = \frac{5}{2}K = \frac{5}{2} \times 20 = 50 \]

### Step 7: Present Ages
Now we have:
- Mahesh's age \( M = 50 \)
- Karan's age \( K = 20 \)
- Arjun's age \( A = 15 \)

### Step 8: Ages 10 Years Ago
To find their ages 10 years ago:
- Mahesh's age 10 years ago: \( 50 - 10 = 40 \)
- Karan's age 10 years ago: \( 20 - 10 = 10 \)
- Arjun's age 10 years ago: \( 15 - 10 = 5 \)

### Step 9: Ratio of Ages 10 Years Ago
Now we can find the ratio of their ages 10 years ago:
\[ \text{Ratio} = M:K:A = 40:10:5 \]

### Step 10: Simplify the Ratio
To simplify the ratio, we divide each term by 5:
\[ \frac{40}{5} : \frac{10}{5} : \frac{5}{5} = 8 : 2 : 1 \]

### Final Answer
The ratio of their ages 10 years ago is \( 8 : 2 : 1 \).

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