algebra question

Question

Jabari owns 100 shares of Stock A and 45 shares of Stock B. For the past month, his stocks have been fluctuating inversely. Stock A decreased by m cents per share and Stock B increased by n cents per share. Which equation can be used to find the total change in value of Jabari's shares per month?

Answer 1

Let's denote V_TOT as the total value of Jabari's shares at any given time, V_A as the per share value for Stock A and V_B as per share value of Stock B.

Then, at the beginning of the month, we can write V_TOT as follows -

V_TOT(0) = 100 * V_A(0) + 45 * V_B(0) (where "0" denotes the time as the starting point)

Similarly at the end of the month,

V_TOT(1) = 100 * V_A(1) + 45 * V_B(1)

Now you have a framework to calculate monthly change in value which is simply the value at the end of the month minus the value at the beginning of the month

VTOT(1) - VTOT(0)

= 100 * V_A(1) + 45 * V_B(1) - [ 100 * V_A(0) + 45 * V_B(0) ] Now, let's group all V_A and V_B values together

= 100 * [V_A(1) - V_A(0)] + 45 * [V_B(1) - V_B(0)]

We can write the change in individual share values in terms of m (for Stock A, this is declining) and n (for Stock B, this is increasing)

= 100 * (-m) + 45 * n

Note: m cents per share per month inherently implies a positive value for m. In order to represent that it is indeed declining, we have a affix the negative sign next to it.

So the change in value of Jabari's total shares can be represented by the equation

= 45n - 100m

The solution ends at this point. But let's add some intuition to the answer.

A. If n and m were equal, does Jabari's share value increase or decrease overall?

Mathematically,

Let's use m = n = 1 for simplicity, then change in value = 45 * 1 - 100 * 1 = 45 - 100 = -55 (this means the value declines).

Logically you can understand this as follows:

Since both stocks move inversely and in equal amount, and as Jabari has more quantity of shares that are declining, the total value of his shares would end up declining as well. He just doesn't have enough of the growing Stock B to offset declines in his declining Stock A

B. So if you were Jabari and didn't want your stock portfolio to lose money, how many more of Stock B do you need to add?

Mathematically,

You want to achieve the point where the total value remains unchanged. Let the number of Stock B required for that be N_B

Then you can represent the new change in overall value as N_B * n - 100 * m which you want to be = 0. We can now solve for N_B

N_B * n - 100 * m = 0

N_B * n = 100 * m

N_B = 100 * m / n

Logically, this can be explained as your stock quantities should be aligned with the rate of change of the increasing stock versus the declining stock so the overall value remains unchanged. Specifically, the ratio (m/n) is important here as it is the ratio of the decline in Stock A versus the growth in Stock B which sets the equilibrium point (point where overall stock value remains unchanged). If the growth "n" is greater than decline "m" then you will need fewer than 100 of Stock B and vice versa.

Interestingly, you can recognize the equilibrium also as the ratio of stock quantities should equal the inverse ratio of their change rates.

By rewriting N_B = 100 * m / n

As, N_B / 100 = m / n

C. A slightly more advanced question is how should Jabari re-balance his stock allocation in terms of m and n so his portfolio doesn't lose money (Hint: In this case number of Stock of A as well as number of Stock of B are unknown)

PRO TIP: Every problem needs to be looked at from multiple angles to fully grasp the mechanics of the concepts involved and how they interact. But the beauty of Math is that once you've grasped this way of thinking you can and will always apply this confidently everywhere else. Specifically these types of portfolio problems are essential for careers in Finance and Business among other professions.

Answer 2

100a-100m=100x

45b+45n=45y