## Introduction

The situation at hand is one where has two alternative offers for purchasing school supplies for a classroom. One alternative is a company that is offering a discount for every dollar amount spent on the purchase, while the other alternative is a company that is offering a discount for every dollar amount spent above $20.

The teacher needs to compare the two offers mathematically to determine which one of them would result in the greatest economic advantage.

This paper seeks to identify the company that will offer a better price, given the dollar amount that will be spent on the supplies, and therefore offers the greatest economic advantage.

## Evaluation of each option

### Option 1

- Let X be the dollar amount spent on the supplies
- C be the percentage of discount offered
- The amount of discount will be C * X, because discount is offered on every dollar spent (Lang, S. 2002).

Therefore the net cost that would be spent on the supplies = X – (C * X) = X – CX, that is, the amount that the teacher would have spent, less the discount received.

### Option 2

- Let X be the dollar amount spent on the supplies
- D be the percentage of discount offered

The amount of discount will be d * (x – 20), because discount is offered on every dollar spent after 20 dollars. For example if $ 100 is spent, the discount would only be calculated on $80.

Therefore the net cost that would be spent on the supplies = X – = X – DX + 20 D, that is, the amount that the teacher would have spent, less the discount received.

Assuming that $ 100 is to be spent on the supplies, and the percentage of discount offered by company A and company B are 10 % and 15 % respectively,

Then the net cost that would be spent on the supplies under option 1 would be as follows:

- Net cost = X – (10 % * X) = 100 – (0.1 * 100) = 90

And the net cost that would be spent on the supplies under option 2 would be as follows:

- Net cost = 100 – = 100 – 0.15 * 80 = 100 – 12 = 88

At the point where the two costs are equivalent, the two equations would be equal. Therefore at that point, X – CX = X – DX + 20 D

- If C = 0.1 and D = 0.15, at this point, X – 0.1 X = X – 0.15 X + (20 * 0.15)
- 0.9 X = 0.85 X + 3
- 0.9 X – 0.85 X = 3
- 0.05 X = 3
- X = 3 / 0.05 = 60

Therefore the two costs would be equivalent at the point where the teacher would have spent $ 60.

If the amount to be spent is $ 10, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 10 = 9

Since Company B’s discount is only applicable to amounts higher than $ 20, Net Cost would be 10 – 0 = $ 10.

If the amount to be spent is $ 20, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 20 = 18

Since Company B’s discount is only applicable to amounts higher than $ 20, Net Cost would be 20 – 0 = $ 20.

If the amount to be spent is $ 30, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 30 = 27
- Net Cost = 0.85 X + 3 = (0.85 * 30) + 3 = 25.5 + 3 = 28.5

If the amount to be spent is $ 50, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 50 = 45
- Net Cost = 0.85 X + 3 = (0.85 * 50) + 3 = 42.5 + 3 = 45.5

If the amount to be spent is $ 70, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 70 = 63
- Net Cost = 0.85 X + 3 = (0.85 * 70) + 3 = 59.5 + 3 = 62.5

If the amount to be spent is $ 80, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 80 = 72
- Net Cost = 0.85 X + 3 = (0.85 * 80) + 3 = 68 + 3 = 71

If the amount to be spent is $ 90, then the net cost that would be spent on the supplies under each option would be as follows:

- Net Cost = 0.9 X = 0.9 * 90 = 81
- Net Cost = 0.85 X + 3 = (0.85 * 90) + 3 = 76.5 + 3 = 79.5

From the calculations above, we can see that if the amount to be spent on the supplies is less than $ 60, Company A would offer a higher discount than Company B. if the amount to be spent on the supplies is equal to $ 60, both companies would offer the same amount of discount. However if the amount to be spent on the supplies is more than $ 60, Company B would offer a higher discount than Company A.

## Graphical representation of the situation

The following table shows the net costs under each alternative, given the dollar amount that would have been spent, X.

The information on the table can be presented graphically as follows:

## Recommendation

From the graph and calculations above, it is clear that Company B offers a better deal if the amount to be spent on the supplies is more than $ 60, but Company A offers a better deal if the amount to be spent on the supplies is less than $ 60. Therefore I would advise the teacher to go for option 2, if the amount to be spent is less than $ 60, the teacher should go for Company A. Otherwise, the teacher should go for Company B.

## Reference

Lang, S. (2002). *Algebra*. Berlin: Springer.