Some times ago the height of my son 90 cm at the end of 1996. When I have observed that his height is increasing equally every year. After seven years in the last month of 2003. I have found that his height was `1/9 th` more than that at the end of 2002. Can you find the height of my son at the end of 2008 ?

Options

A. 360 cm
B. 450 cm
C. 250 cm
D. 270 cm

Answer

To find the height of the son at the end of 2008, we will follow these steps:

### Step 1: Determine the height increase per year
We know that the son's height was 90 cm at the end of 1996. We will assume that his height increases by a constant amount \( x \) cm each year.

### Step 2: Calculate the height at the end of 2002
From 1996 to 2002, there are 6 years. Therefore, the height at the end of 2002 can be expressed as:
\[ \text{Height in 2002} = 90 + 6x \]

### Step 3: Calculate the height at the end of 2003
According to the problem, the height at the end of 2003 is \( \frac{1}{9} \) more than the height at the end of 2002. Thus, we can express the height at the end of 2003 as:
\[ \text{Height in 2003} = 90 + 7x \]
This is equal to:
\[ \text{Height in 2003} = \text{Height in 2002} + \frac{1}{9} \times \text{Height in 2002} \]
Substituting the height in 2002:
\[ 90 + 7x = (90 + 6x) + \frac{1}{9}(90 + 6x) \]

### Step 4: Simplify the equation
Now we will simplify the equation:
\[ 90 + 7x = 90 + 6x + \frac{1}{9}(90 + 6x) \]
To eliminate the fraction, multiply through by 9:
\[ 9(90 + 7x) = 9(90 + 6x) + (90 + 6x) \]
This simplifies to:
\[ 810 + 63x = 810 + 54x + 90 + 6x \]
Combining like terms:
\[ 810 + 63x = 900 + 60x \]

### Step 5: Solve for \( x \)
Now, we will isolate \( x \):
\[ 63x - 60x = 900 - 810 \]
\[ 3x = 90 \]
\[ x = 30 \text{ cm} \]

### Step 6: Calculate the height at the end of 2008
From 1996 to 2008, there are 12 years. Therefore, the height at the end of 2008 can be expressed as:
\[ \text{Height in 2008} = 90 + 12x \]
Substituting \( x = 30 \):
\[ \text{Height in 2008} = 90 + 12 \times 30 \]
\[ \text{Height in 2008} = 90 + 360 = 450 \text{ cm} \]

### Final Answer
The height of the son at the end of 2008 is **450 cm**.

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