# How do you find the percentage of stock X originally invested?

## $35000 is invested in two stocks X and Y. At the end of the investment period, the value of stock X has increased by 11% and that of stock Y decreased by 4% with the total value of the combined stocks being$36560. What percentage of the original split was invested in stock X?

May 24, 2015

Let ${X}_{0}$ stand for the initial amount invested in stock X and ${Y}_{0}$ for the initial amount invested in stock Y.

Let ${X}_{1}$ stand for the value of the stock X holding at the end of period and ${Y}_{1}$ for the value of the stock Y holding at the end of period.

${X}_{0} + {Y}_{0} = 35000$

${X}_{1} = {X}_{0} \times \left(1 + \frac{11}{100}\right) = {X}_{0} \times 1.11$

${Y}_{1} = {Y}_{0} \times \left(1 - \frac{4}{100}\right) = {Y}_{0} \times 0.96$

${X}_{1} + {Y}_{1} = 35000 + 1560 = 36560$

We want to determine ${X}_{0}$.

Starting with the first equation, subtract ${X}_{0}$ from both sides to get:

${Y}_{0} = 35000 - {X}_{0}$

Then working backwards through our equations, we find...

$36560 = {X}_{1} + {Y}_{1}$

$= \left({X}_{0} \times 1.11\right) + \left({Y}_{0} \times 0.96\right)$

$= \left({X}_{0} \times 1.11\right) + \left(\left(35000 - {X}_{0}\right) \times 0.96\right)$

$= \left({X}_{0} \times 1.11\right) + \left(35000 \times 0.96\right) - \left({X}_{0} \times 0.96\right)$

$= \left({X}_{0} \times \left(1.11 - 0.96\right)\right) + 33600$

$= \left({X}_{0} \times 0.15\right) + 33600$

Subtract $33600$ from both sides to get:

${X}_{0} \times 0.15 = 36560 - 33600 = 2960$

Divide both sides by $0.15$ to get:

${X}_{0} = 19733.33$

As a percentage this is 100 xx 19733.33 / 35000 = 56.38%

Check:

${Y}_{0} = 35000 - 19733.33 = 15266.67$

${X}_{1} = {X}_{0} \times 1.11 = 21904$

${Y}_{1} = {Y}_{0} \times 0.96 = 14656$

${X}_{1} + {Y}_{1} = 21904 + 14656 = 36560$