4 years ago, ratio of Shivam's age to Deepak's age was 2: 3 and ratio of Shivam's age 4 years ago to Deepak's age 5 years hence is 8: 15. Find present age of Shivam.

Options

A. A)32 years
B. B)28 years
C. C)40 years
D. D)24 years

Answer

To find the present age of Shivam, we will follow these steps:

### Step 1: Define Variables
Let Shivam's age 4 years ago be \( S - 4 \) and Deepak's age 4 years ago be \( D - 4 \).

### Step 2: Set Up the First Ratio
From the problem, we know that the ratio of Shivam's age to Deepak's age 4 years ago was \( 2:3 \). This can be expressed as:
\[ \frac{S - 4}{D - 4} = \frac{2}{3} \]
Cross-multiplying gives us:
\[ 3(S - 4) = 2(D - 4) \]
Expanding this, we get:
\[ 3S - 12 = 2D - 8 \]
Rearranging gives us:
\[ 3S - 2D = 4 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Ratio
The problem also states that the ratio of Shivam's age 4 years ago to Deepak's age 5 years hence is \( 8:15 \). This can be expressed as:
\[ \frac{S - 4}{D + 5} = \frac{8}{15} \]
Cross-multiplying gives us:
\[ 15(S - 4) = 8(D + 5) \]
Expanding this, we get:
\[ 15S - 60 = 8D + 40 \]
Rearranging gives us:
\[ 15S - 8D = 100 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 3S - 2D = 4 \) (Equation 1)
2. \( 15S - 8D = 100 \) (Equation 2)

We can solve these equations simultaneously. First, let's express \( D \) from Equation 1:
\[ 2D = 3S - 4 \implies D = \frac{3S - 4}{2} \]

Now substitute \( D \) in Equation 2:
\[ 15S - 8\left(\frac{3S - 4}{2}\right) = 100 \]
Multiplying through by 2 to eliminate the fraction:
\[ 30S - 8(3S - 4) = 200 \]
Expanding gives:
\[ 30S - 24S + 32 = 200 \]
Combining like terms:
\[ 6S + 32 = 200 \]
Subtracting 32 from both sides:
\[ 6S = 168 \]
Dividing by 6:
\[ S = 28 \]

### Step 5: Find Present Age of Shivam
Since \( S \) is Shivam's age 4 years ago, his present age is:
\[ S + 4 = 28 + 4 = 32 \]

### Final Answer
The present age of Shivam is **32 years**.

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