A tree increases annually by `(1)/(8)` th of its height . By how much will it increase after 2 years if stands today 64 cm high?

Answer

To solve the problem step by step, we will calculate the height of the tree after 2 years, given that it increases annually by \( \frac{1}{8} \) of its height. The current height of the tree is 64 cm.

### Step 1: Understand the annual increase
The tree increases by \( \frac{1}{8} \) of its height each year. Therefore, if the current height is \( H \), the increase in height after one year will be:
\[ \text{Increase after 1 year} = \frac{1}{8} \times H \]

### Step 2: Calculate the height after 1 year
The height after 1 year can be calculated as:
\[ \text{Height after 1 year} = H + \text{Increase after 1 year} = H + \frac{1}{8} \times H = H \left(1 + \frac{1}{8}\right) = H \left(\frac{8}{8} + \frac{1}{8}\right) = H \left(\frac{9}{8}\right) \]
Substituting \( H = 64 \) cm:
\[ \text{Height after 1 year} = 64 \times \frac{9}{8} = 72 \text{ cm} \]

### Step 3: Calculate the height after 2 years
Now, we will calculate the height after the second year using the new height:
\[ \text{Height after 2 years} = \text{Height after 1 year} + \text{Increase after 1 year} \]
Using the height after 1 year:
\[ \text{Height after 2 years} = 72 + \frac{1}{8} \times 72 = 72 \left(1 + \frac{1}{8}\right) = 72 \left(\frac{9}{8}\right) \]
Calculating this:
\[ \text{Height after 2 years} = 72 \times \frac{9}{8} = 81 \text{ cm} \]

### Step 4: Calculate the total increase in height after 2 years
To find the total increase in height after 2 years, we subtract the original height from the height after 2 years:
\[ \text{Total increase} = \text{Height after 2 years} - \text{Original height} = 81 - 64 = 17 \text{ cm} \]

### Final Answer
The tree will increase by **17 cm** after 2 years.

---