A polyhedron in the time: New Year's resolutions ... in algebraic language!

A few days after the start of the New Year, many people have in mind the New Year's resolutions. The number of people who intend to change their life is unknown and so is the number of purposes they want to fulfill. However, in these cases, the “constant” known is the type of purposes that most people want to follow.

The most common resolutions are the following as well as how to fulfill them with the help of algebraic language:

Save money, pay debts

For fulfill this purpose you should know, first of all:

C = how much money do you have?

D = how much money do you owe?

X = how much money do you want to have for the end of the year?

I = I_c + I_v, how much money do you have as a constant (Ic) and variable (Iv) income amount?

G = Gc + Gv = how much money do you have as a constant (Gc) and variable (Gv) expenses amount?

The savings equation is therefore:

X = C – D + (I_c + I_v) – (G_c + G_v)

It means, the savings is equal to the initial capital C minus your debt D plus the incomes less the expenses.

But not all are unknown in this equation and so it will be much easier for you to reach your goal. Let us suppose that your initia capital is $ 20, your debt is 50 and what you expecto to save at the end of the year is $ 2000. Well, everyone would want to save a different amount and start from different conditions, right? Now, if your constant monthly expenses are $ 500 and your income is $ 700, the equation for saving is the balance between variable expenses and variable income:

5000 = 20 - 50 + (700x12 + I_v) - (500x12 + G_v)

5000 = 20 - 50 + 700x12 + I_v - 500x12 - G_v

5000 = 2370 + I_v - G_v

It says that the balance between income and variable expenses in the year must be $ 2630:

I_v - G_v = 2630

That is, the monthly savings must be over $219.

Of course, there is a way to optimize savings, perhaps with investments that generate interest. Of course, it has more algebraic language and more equations with the use of exponentials. Indeed, if you want to fulfill this purpose you must perform accounts.

Lose weight

One of the most common purposes of the beginning of the year is to lose weight. Many people (again the number is an unknown) choose diets while other people (the number is an unknown as well) translate this purpose into exercising.

To fulfill this purpose, the mathematical balance is very clear. The food C you eat has calories and the activity A you do requires calories. To live, an average sedentary adult requires no more than 1800 calories (kilocalories, actually, but the word calories is more used for kcal). Staying at your weight can be expressed, algebraically as:

Stay at your weight = C - A

Stay at your weight = 1800 - 1800 = 0

It means that you are allowed to eat food with C = 1800 calories because while you are living, sleeping and moving a little bit, your body consumes that energy without accumulating it.

How is it possible to lose weight then? Well, the equation should be something like this:

Lose weight = (C – D) – (A + E)

You can reduce your calories consumption with a diet D. Let us say that you choose to ingest C – D = 1500 calories, that is, you reduce your intake by 300 calories. On the other hand, to lose weight, you also may decide to add some exercise E. Many people (an unknown for us) start jogging for 30 minutes so as you decide. Such exercise consumes about 100 calories so the weight loss equation is:

Lose weight = 1500 – (1800 + 100)

Lose weight = – 400 cal

Therefore, the monthly number of reduced calories will be about 400x30 = 12,000. Now the crucial question is how many pounds of weight will be lost with this reduction? Well, to lose a pound of weight, you need about 15,840 calories, so with this plan, you can lose almost a pond of body mass per month.

If the routine only considers a reduction of the consumption (a diet), mass is lost but your body is not shaped. If you exercise, depending of the exercise, some parts of your body are shaped. Jogging or running stylizes the body, weightlifting shapes your body and other exercises such as rowing, skiing or mountaineering have a strong impact on the daily calories burned.

To accomplish this purpose, the important thing is constancy without waiting for a miracle to replace the mathematics of caloric balance, for sure!

Travel

One of the most common wishes for most people (is it another unknown?) is to have the opportunity to travel more. In order to travel there are two fundamental factors: having money and time.

Sometimes you do not have money but you have time and sometimes there is money but you do not have any time. Of course, there is always the worst case in which you do not have money or time or the ideal situation in which the stars aligned so that you have both, money and time.

For mortal and common beings, the problem is always a matter of money or time. Having money is equal to saving, the first purpose of the list, already discussed. While having time could be solved in a more or less similar way.

What do you do 24 hours a day, 7 days a week? Perhaps your most important set of activities can be summarized as:

t₁ = time to sleep or rest

t₂ = time spent working or studying

t₃ = time to perform obligations (cleaning, order, purchases, etc.)

t₄ = time for your family

t₅ = time for some physical activity

t₆ = time to socialize (go out with your friends, parties)

t₇ = time for you (hobbies)

t₈ = spiritual time (meditation)

t₉ = time for other activities

By day, week and month, you can see how much time you spend on each activity. Let us suppose that you sleep t₁ =28 hours a week, you work or study t₂ = 40 hours, you dedicate about t₃ = 7 hours to your obligations, t₄ = 7 hours to your family (which coincides with the meal time t₉, maybe, but this year you want to spend more time, so you will increase about 3 hours on Saturdays and another 3 on Sundays that give a total of t₄ = 13 hours). You do not exercise (t₅ = 0) although your purpose this year is to run a daily time (correcting, t₅ = 7 hours). For the time to socialize the things are clear: about 2 hours a day in weekdays with your study or work partners, plus about 5 hours on Fridays and another 5 hours on Saturdays that give a total of t₆ = 20 hours. You have practically no hobbies, but your purpose this year is to take a something course (an unknown) to which you may devote t₇ = 4 hours per week. You do not usually meditate although your purpose is to spend at least 10 minutes a day (t₈ = 1.1 hours per week). A week has 168 hours and with these assumptions, your time follows this equation:

168 = t₁ + t₂ + t₃ + t₄ + t₅ + t₆ + t₇ + t₈ + t₉

168 = 28 + 40 + 13 + 7 + 20 + 4 + 1.1 + t₉

168 = 113.1 + t₉

So t₉ = 54.9 hours is dedicated to other activities. If you want to go out a couple of days to travel and readjust your schedules, you would still have 4.9 hours left for your other activities.

This is an example, of course, but for many people (the unknown) simply the accounts do not work out. Time is not enough! The problem here may be the interpretation of time for other activities. To travel, as well as to save money is a matter to review in an analytic way a balance. If you spend a lot of time at work, you will not be able to spend time in other activities or if you spend many hours at parties and outings, then other activities, including travel, will be affected.

Travel time should not necessarily be weekly, it can be planned along the year. We must consider when and the cost, both of them, a matter of accounts and of course could be solved with the use of algebraic language, whether or not we are aware of it.

Other common resolutions at the beginning of the year are: spending more time with the family, stope smoking (or some other activity that you do not like), finding a partner and learning something new.

For some of them, it is clear that you must have an idea of what you have and how you want to reach your goal. Spending more time with the family is a time balance equation as the case of traveling. Stop smoking is similar to exercise or diet; it is a matter of determination and perseverance. But it is also a matter of accounts that tell you how much you will save on buying cigarettes and the years added to your life for a healthy life. Finding a partner is not a matematical issue, of course, but dedicating time to activities that can help you to find it requires similar analyzes for previous purposes.