Five years ago, Somi's age was `1/ 3` of Amit's age at that time. The ratio of Amit's age six years hence to Somi's age twelve years hence will be `7:4`. What was Somi's age three years ago (in years)

Options

A. 13
B. 29
C. 17
D. 25

Answer

To solve the problem step by step, let's define the present ages of Somi and Amit.

Let:
- Somi's present age = X years
- Amit's present age = Y years

### Step 1: Establish the first equation based on the information given.
Five years ago, Somi's age was \( \frac{1}{3} \) of Amit's age at that time.

This can be expressed as:
\[ X - 5 = \frac{1}{3}(Y - 5) \]

### Step 2: Clear the fraction by multiplying through by 3.
Multiplying both sides by 3 gives:
\[ 3(X - 5) = Y - 5 \]
Expanding this results in:
\[ 3X - 15 = Y - 5 \]
Rearranging gives us:
\[ 3X - Y = 10 \quad \text{(Equation 1)} \]

### Step 3: Establish the second equation based on the future ages.
The ratio of Amit's age six years hence to Somi's age twelve years hence is \( 7:4 \).

This can be expressed as:
\[ \frac{Y + 6}{X + 12} = \frac{7}{4} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:
\[ 4(Y + 6) = 7(X + 12) \]
Expanding both sides results in:
\[ 4Y + 24 = 7X + 84 \]
Rearranging gives us:
\[ 4Y - 7X = 60 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations.
We have two equations now:
1. \( 3X - Y = 10 \) (Equation 1)
2. \( 4Y - 7X = 60 \) (Equation 2)

From Equation 1, we can express \( Y \) in terms of \( X \):
\[ Y = 3X - 10 \]

### Step 6: Substitute \( Y \) in Equation 2.
Substituting \( Y \) in Equation 2 gives:
\[ 4(3X - 10) - 7X = 60 \]
Expanding this results in:
\[ 12X - 40 - 7X = 60 \]
Combining like terms gives:
\[ 5X - 40 = 60 \]
Adding 40 to both sides results in:
\[ 5X = 100 \]
Dividing by 5 gives:
\[ X = 20 \]

### Step 7: Find Somi's age three years ago.
Somi's present age is 20 years, so three years ago, Somi's age was:
\[ 20 - 3 = 17 \text{ years} \]

### Final Answer:
Somi's age three years ago was **17 years**.

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