Question

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?

Answer 1

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 xx (1 + 11/100) = X_0 xx 1.11`

`Y_1 = Y_0 xx (1 - 4/100) = Y_0 xx 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`

`=(X_0 xx 1.11) + (Y_0 xx 0.96)`

`=(X_0 xx 1.11) + ((35000 - X_0) xx 0.96)`

`=(X_0 xx 1.11) + (35000 xx 0.96) - (X_0 xx 0.96)`

`=(X_0 xx (1.11 - 0.96)) + 33600`

`=(X_0 xx 0.15) + 33600`

Subtract `33600` from both sides to get:

`X_0 xx 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 xx 1.11 = 21904`

`Y_1 = Y_0 xx 0.96 = 14656`

`X_1 + Y_1 = 21904 + 14656 = 36560`