Worked Examples: Business Applications of Matrices
Example 1
The governing board for a Fortune 500 company is expected to collect votes of managers within the company on a potential new policy change in the way it manages clients. It needs at least 250 votes to pass the policy. From past experience, 30% of technical managers and 60% of administrative managers voted in favor of a similar policy. In total, there are 715 managers across the company. What is the least number of votes from each group necessary to pass the policy?Solution
We want to know if the combination of technical and administrative managerial votes will favor the new policy. How many of each will there be? Well, there will be x technical votes and y administrative votes made. So, let x = total # of technical votes; y = total # of administrative votes We know or assume:- 715 votes will be made
- 30% of the technical managers along with 60% of the administrative managers must total up to 250.
y = 715 – x
.30x + .60(715 – x) = 250
–.3x + 429 = 250
x ≈ 596.7
x ≈ 597
597 +y = 715
y = 118
Thus, we expect that there are 597 technical managers and 118 administrative managers, based on our assumptions. This is not important, however. We are primarily concerned with how many from each of these two groups will vote. That is, 30% of 597 means about 179 technical managers' votes are needed and 60% of 118 means that about 71 administrative managers' votes are needed.Example 2
Referencing the example above, describe how else it would be possible to achieve the necessary votes for the policy change if in fact there was not a 597 to 118 ratio of technical to administrative managers.Solution
To restate, we would like to know if there were a different split, would there still be sufficient votes. We examine what happens when the 715 managers are split up differently.Tech | Admin | Votes in Favor |
---|---|---|
605 | 110 | .30(605) + .60(110) = 247.5 |
604 | 111 | .30(604) + .60(111) = 247.8 |
603 | 112 | .30(603) + .60(112) = 248.1 |
602 | 113 | .30(602) + .60(113) = 248.4 |
601 | 114 | .30(601) + .60(114) = 248.8 |
600 | 115 | .30(600) + .60(115) = 249 |
599 | 116 | .30(599) + .60(116) = 249.3 |
598 | 117 | .30(598) + .60(117) = 249.6 |
597 | 118 | .30(597) + .60(118) = 249.9 |
596 | 119 | .30(596) + .60(119) = 250.2 |
595 | 120 | .30(595) + .60(120) = 250.5 |
594 | 121 | .30(594) + .60(121) = 250.8 |
593 | 122 | .30(593) + .60(122) = 251.1 |
= 179.1 + .30n + 70.9 – .60n
= 250 – .30n
This tells us that the number of favorable votes decreases by .30 for every additional technical manager. When n is negative, we reduce the number of technical managers, and add .30 for each decrease in one technical manager.Example 3
When moving, it can often be a questionable matter to decide among moving truck companies. U-Haul currently charges $19.95 per day for a 10' moving truck and $0.59 per mile (SOURCE: www.uhaul.com). Imagine that another company, You-Haul, moves into the market and offers the same 10' truck for $70 per day. Assume a one-day rental takes place.- Using systems of linear equations, under what mileage conditions would it make it make sense to choose one company over the other?
- Without offering a flat-rate fee, how could U-Haul create a more competitive market, from a mathematical perspective? Give a specific explanation and note that answers may vary.
Solution
To restate, we would like to know if there were a different split, would there still be sufficient votes. We examine what happens when the 715 managers are split up differentlyTech | Admin | Votes in Favor |
---|---|---|
605 | 110 | .30(605) + .60(110) = 247.5 |
604 | 111 | .30(604) + .60(111) = 247.8 |
603 | 112 | .30(603) + .60(112) = 248.1 |
602 | 113 | .30(602) + .60(113) = 248.4 |
601 | 114 | .30(601) + .60(114) = 248.8 |
600 | 115 | .30(600) + .60(115) = 249 |
599 | 116 | .30(599) + .60(116) = 249.3 |
598 | 117 | .30(598) + .60(117) = 249.6 |
597 | 118 | .30(597) + .60(118) = 249.9 |
596 | 119 | .30(596) + .60(119) = 250.2 |
595 | 120 | .30(595) + .60(120) = 250.5 |
594 | 121 | .30(594) + .60(121) = 250.8 |
593 | 122 | .30(593) + .60(122) = 251.1 |
= 179.1 + .30n + 70.9 – .60n
= 250 – .30n
This tells us that the number of favorable votes decreases by .30 for every additional technical manager. When is negative, we reduce the number of technical managers, and add .30 for each decrease in one technical manager. https://youtu.be/GNJzWSk0MDEMilos Podmanik, By the Numbers, "Systems of Linear Equations and Solutions Using Matrices," licensed under a CC BY-NC-SA 3.0 license. MathIsGreatFun, "1_1 P4 MAT217.mp4," licensed under a Standard YouTube license.