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Study Guides > Finite Math

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.
Mathematically, x + y = 715 total votes will be made .30x + .60y = 250 votes in favor of policy needed Solving, we get

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
We see that if the technical manager group is larger than our assumption dictates, then the votes in favor would fall short of the required 250; However, if there are in fact fewer than 597 technical managers (meaning more in administrative managers), then the new policy should pass. Why is this? Observing the tradeoff, we notice that each administrative manager contributes .6 units to the total tally, whereas each technical manager only contributes .3 units to the total tally. In net, we see that the total vote drops by .3 units for each additional technical manager (under our assumed voting percentages). Mathematically, if we let be the number of additional technical managers beyond 597, then we have 597 + n technical managers and 118 – n administrative managers (since the total still has to add up to 715). Notice that 597 + n + 118 – n = 715. Thus, we rewrite the second equation we wrote to be the following: Number of favorable votes = .30(597 + n) + .60(118 – n)

  = 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.
  1. Using systems of linear equations, under what mileage conditions would it make it make sense to choose one company over the other?
  2. 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 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
We see that if the technical manager group is larger than our assumption dictates, then the votes in favor would fall short of the required 250; However, if there are in fact fewer than 597 technical managers (meaning more in administrative managers), then the new policy should pass.   Why is this? Observing the tradeoff, we notice that each administrative manager contributes .6 units to the total tally, whereas each technical manager only contributes .3 units to the total tally. In net, we see that the total vote drops by .3 units for each additional technical manager (under our assumed voting percentages).   Mathematically, if we let be the number of additional technical managers beyond 597, then we have (597 + n) technical managers and (118 - n)  administrative managers (since the total still has to add up to 715). Notice that .   Thus, we rewrite the second equation we wrote to be the following:   Number of favorable votes = .30(597 + n) + .60(118 – n)

  = 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/GNJzWSk0MDE  
Milos 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.

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  • By the Numbers, Systems of Linear Equations and Solutions Using Matrices. Authored by: Milos Podmanik. License: CC BY: Attribution.