Certainly! Let's walk through the problem step-by-step.
1. Initial Value:
Assume the initial value of the shares at the end of December 2010 is [tex]\( V_0 = 100 \)[/tex]. This simplification helps make percentage calculations straightforward.
2. Value after 2011 Decrease:
In 2011, the value of the shares decreased by 20%. To find the new value, we calculate:
[tex]\[
V_1 = V_0 - (0.20 \cdot V_0) = V_0 \cdot (1 - 0.20) = 100 \cdot 0.80 = 80
\][/tex]
So, at the end of 2011, the value of the shares is [tex]\( V_1 = 80 \)[/tex].
3. Value after 2012 Increase:
In 2012, the value of the shares increased by 30%. To find this new value, we calculate:
[tex]\[
V_2 = V_1 + (0.30 \cdot V_1) = V_1 \cdot (1 + 0.30) = 80 \cdot 1.30 = 104
\][/tex]
So, at the end of 2012, the value of the shares is [tex]\( V_2 = 104 \)[/tex].
4. Overall Percentage Increase from 2010 to 2012:
To calculate the overall percentage increase from the end of 2010 to the end of 2012, we compare the final value [tex]\( V_2 \)[/tex] to the initial value [tex]\( V_0 \)[/tex]. The formula for percentage increase is:
[tex]\[
\text{Percentage Increase} = \left( \frac{V_2 - V_0}{V_0} \right) \times 100
\][/tex]
Plugging in the values:
[tex]\[
\text{Percentage Increase} = \left( \frac{104 - 100}{100} \right) \times 100 = \left( \frac{4}{100} \right) \times 100 = 4\%
\][/tex]
Therefore, over the two years from the end of 2010 to the end of 2012, the value of the shares increased by 4%.
