The strength of a school increases and decreases every alternate year by 15%. If it started with an increase in 2012, then the strength of the school in the beginning of 2015, as compared to that in 2012, had (correct to two decimal places):

The correct option is Increased by 12.41 %.

Step-by-Step Explanation:

1. Understand the timeline and changes:

Let the initial strength of the school at the beginning of 2012 be $S$.

The problem states that the strength increases and decreases every alternate year by 15%.

• Year 2012 (1st year): An increase of 15% occurs during 2012.

• Year 2013 (2nd year): A decrease of 15% occurs during 2013.

• Year 2014 (3rd year): An increase of 15% occurs during 2014.

Thus, by the beginning of 2015, three successive annual changes (2012, 2013, and 2014) have occurred in sequence: +15%, -15%, and +15%.

2. Calculate the strength at the beginning of 2015:

Let the initial strength at the beginning of 2012 be 100.

• After 2012 (Increase by 15%):

$100\times (1+\frac{15}{100})=100\times 1.15=115$

• After 2013 (Decrease by 15%):

$115\times (1-\frac{15}{100})=115\times 0.85=97.75$

• After 2014 (Increase by 15%):

$97.75\times (1+\frac{15}{100})=97.75\times 1.15=112.4125$

3. Calculate the overall percentage change:

The strength at the beginning of 2015 is 112.4125, compared to the starting strength of 100 in 2012.

Net Change = $112.4125-100=+12.4125$

Percentage Increase = $\frac{12.4125}{100}\times 100=12.4125\%$

Rounding off to two decimal places, the strength of the school at the beginning of 2015 has Increased by 12.41 %.