Question

A tree grows at the rate of 1/5th of its height annually. By how much height will it grow after 2 years, if its present height is 75 cm?

• A. 108 cms
• B. 90 cms
• C. 144 cms
• D. 112 cms

Hint:

Repeated percentage growth compounds: value after $n$ years at rate $r%$ equals initial $\times\,(1+\tfrac{r}{100})^n$.

Answer 1

The Correct Option is A

Growth rate $= \dfrac{1}{5}$ of current height each year $=20%$ per annum, so the growth is compounded.
After 1 year: $75\times(1+0.20)=75\times1.20=90$ cm.
After 2 years: $90\times1.20=108$ cm.
Therefore the height after two years is 108 cm. (The net increase is $108-75=33$ cm, but the options list the final height.)

Answer 2

The question asks for a tree's height after 2 years of growth at 1/5th of its height per year, starting from 75 cm. Since the growth compounds (each year's growth is 1/5th of that year's height, not the original height), the height each year is multiplied by 1.2. We can check each option by working backwards, dividing by 1.2 twice and seeing if we land back on 75 cm.

1. 108 cms: Dividing once, \( 108 \div 1.2 = 90 \) cm, which would be the height after 1 year. Dividing again, \( 90 \div 1.2 = 75 \) cm, exactly the starting height, so this option is consistent with two years of growth.
2. 90 cms: Dividing once, \( 90 \div 1.2 = 75 \) cm, which already reaches the starting height after only one division. That means 90 cm is the height after just 1 year, not after 2 years, so this option does not fit.
3. 144 cms: Dividing once, \( 144 \div 1.2 = 120 \) cm, and dividing again, \( 120 \div 1.2 = 100 \) cm, which does not equal the starting height of 75 cm, so this option is ruled out.
4. 112 cms: Dividing once, \( 112 \div 1.2 \approx 93.3 \) cm, and dividing again, \( 93.3 \div 1.2 \approx 77.8 \) cm, which is close to but not exactly 75 cm, so this option does not satisfy the growth pattern either.

Working backwards from each option, only 108 cm returns exactly to the starting height of 75 cm after two divisions by 1.2, confirming two full years of 20% compounded growth.

So the correct answer is 108 cms.

Answer 3

Rather than multiplying by 1.2 twice, it helps to track the actual centimetres added each year one at a time, since the growth each year is 1/5th of that year's height, not of the original 75 cm.

1. 108 cms: In year one, the tree grows by 1/5th of 75 cm, which is 15 cm, taking it to 90 cm. In year two, the growth is 1/5th of the new height of 90 cm, which is 18 cm, taking the total to \( 90 + 18 = 108 \) cm, matching this option.
2. 90 cms: This value is reached after only the first year's growth of 15 cm, so it represents just one year of growth, not the full two years the question asks for.
3. 144 cms: Reaching this height would require the second year's growth to add 54 cm on top of 90 cm, far more than the 18 cm that 1/5th of 90 actually gives, so this overstates the growth.
4. 112 cms: This would need the second year's growth to be 22 cm instead of the correct 18 cm, which does not match 1/5th of the height reached after year one.

Adding the actual centimetre growth for each of the two years in turn confirms the final height reaches 108 cm.

So the correct answer is 108 cms.