MATH BASICS : Fitness and Mathematics

Last time, I wrote a post about mathematics and money which started with some intermediate stuff like geometric series and eventually gave some hint at probability and numberical methods. And, I also wrote a post about mathematics and a girl's tears explaining the basics of differential equations, which are considered as "advanced" by few people. This time, I'll be dealing with neither intermediate stuff, nor advanced stuff. I'll be dealing with something very basic and trivial. Hope you have fun reading this post.

When I see symbols like + - x / %, I see a lot of words. And some people see only those symbols. Through out this post, a statement is framed and represented twice...

• One time in plain English language.
• The other time in Mathematics.

Generally, Mathematical form takes less space. But, when some extra explaination is needed, I've added some extra lines. Hope you change the way you see the mathematical symbols after reading this post

ADDITION

A person walked for 5 kilometers in the morning, and 5 kilometers in the evening. Total how many kilometers were walked by that person?

5km + 5km = 10km

SUBSTRACTION

The same person weighs 125 kilograms. And that person's target is to reach 75 kilograms. How many kilograms should be lost?

125kg - 75kg = 50kg

COMPARISION OPERATORS: (greater than, less than)

A person walks for 120 minutes a day and lost 6 kilograms in 30 days. In the next 30 days, the person did lunges for 30 minutes a day, and lost 9 kilograms. In these two exercises, which one needed more time to finish? Which one gave faster results?

Since 120 > 30, walking requires more time.
Since 6 < 9, lunges give quicker results.

MULTIPLICATION

The same person walked for 10 kilometers a day for a period of 30 days. How many kilometers were walked in these 30 days?

30 x 10km = 300km

DIVISION

In these 30 days, the person has lost 6 kilograms. How much weight can be lost by walking a kilometer?

6kg / 300km = 0.02 kg per km = 20 grams per kilometer

PERCENTAGE

It takes 15 minutes for a person to travel to the gym. That person spends 30 minutes in the gym to do lunges, and the return journey from gym is also 15 minutes. What is the total time spent by that person? And how much percentage of the time is efficiently spent in doing the exercises?

The percentage is ...
30/(15 + 30 + 15)
= 30 / 60
= 0.5
= 0.5 x 100%
= 50% efficiency

HEALTH AND MATH

Two people are of equal height and weight and body fat. One of them tried to lose weight through exercises. The other person tried to lose fat through dieting. Both of them started when they weighed 100 kgs, with 40% bone weight, 30% muscle weight, 30% fat weight.

After 6 months, 1st person lost 20 kg fat and gained 5 kg muscle due to exercises, and the 2nd person lost 10 kg fat and 10 kg muscle due to fasting.

Who is more successful in weight loss?
Who is more successful in fat loss (both in terms of percentage)?
What is the new body fat % for both of them?

total weight lost by the 1st person = 20 - 5 = 15 kg
total weight lost by the 2nd person = 10 + 10 = 20 kg

weight-loss percentage of the 1st person = 15 / 100 = 15%
weight-loss percentage of the 2nd person = 20 / 100 = 20%

MEANING: When it comes to weight loss, the 2nd person has lost more weight due to dieting.

Now, fat loss..

original fat weight of both the people = 30% of 100kg = 30 kg

New fat weight of the 1st person = 30 - 20 = 10 kg
New fat weight of the 2nd person = 30 - 10 = 20 kg

changed weight of 1st person = 100 - 15 = 85 kg
changed weight of 2nd person = 100 - 20 = 80 kg

bodyfat percentage of the 1st person = 10 / 85 = 11.7% (approximately)
bodyfat percentage of the 2nd person = 20 / 80 = 25.0%

MEANING: Even though dieting resulted in more weight loss, dieting resulted in more than two times the fat!

PHYSICS FORMULAE: (E = mgh)

A Joule is the unit of energy. The energy required to lift an object of mass m kg, to a height of h meters, against the gravity is ... E = mgh.When a person weighing 100 kg is 10 meters above the ground, against the gravity of 10m/sec², what is the work done in lifting him?

E = mgh = 100 x 10 x 10 = 10,000 = 10 kilo Joules

It is estimated that when a 100 kg person climbs up a stair of 15 cm height, approximately 0.25 kilo calories of energy is burnt by the muscles. What is the efficiency of the muscles? (Approximate 4.2 Joules = 1 calorie)

Energy required to lift = mgh = 100 x 10 x 0.15 = 150 Joules = 0.15 kilo Joules

Energy spent by the muscles = 0.25 kilo Calories = 0.25 x (4.2 kilo Joules) = 1.05 kilo Joules

Efficiency of the muscles = work_done / energy_spent = 0.15 / 1.05 = 0.1428... = 15% (approximately)

A kilogram of fat can fuel around 7700 kilo Calories of energy. Assuming a 100 kg person's muscles are working at 10% efficiency while climbing a stair of 15 cm, and a floor can have 20 stairs, calculate how many floors the person should climb a day to burn 2 kilograms of fat in a month.

Target calories = 2 kg of fat
= 2 x 7700 kilo calories
= 15,400 kilo calories in 30 days
= 513 kilo calories per day (approximately)

efficiency = work_done / energy_spent
=> 0.1 = mgh / energy_spent
=> energy_spent = mgh x 4
=> energy_spent = 100 x 10 x 0.15 x 10
= 1500 Joules
= 350 calories
= 0.35 kilo calories (approximately) per each stair.

=> energy spent in climbing up a floor of 20 stairs
= 20 x energy_spent in climbing up a stair
= 20 x 0.35
= 7 kilo calories

No. of floors = target_energy / energy_per_floor
= 513 / 7
= 73 floors (approximately)

MEANING: If that person climbs 73 floors a day, 2 kilograms of fat can be lost in a month.

SIMULTANEOUS LINEAR EQUATIONS:
(a₁x + b₁y + c₁ = a₂x + b₂y + c₂ = 0)

An upcoming actor got a really great opportunity to act with two superstars in an action movie. He was informed that the shooting would be for a week, and will start as soon as both those superstars are of equal weight, and the upcoming actor is supposed to weigh exactly the same weight as those superstars.

If he fails to weigh the same as those stars, he would be relaced with some other upcoming actor. Because of their other movie commitments, both these stars are working on their body transformation. One star weighs 100 kg now and is reducing the weight at a pace of 5 kg per month. Another star weighs 70 kg, and is increasing his weight at a pace of 1kg per month.

Find out in how many months the shooting will start. Also, if the upcoming actor weighs 85 kg, calculate the target weight he should aim at - to get ready for the shoot.

Let the shooting starts in "t" no. of months from now. (t for time).

The weight equation of the 1st star would be .... weight = 100kg - 5kg x t months

The weight equation of the 2nd star would be .... weight = 70kg + 1kg x t months

These two are in the simultaneous linear equations form.

y = 100 - 5x
y = 70 + x

Solving these simultaneous linear equations, we get....

x = 5
y = 75

MEANING: In 5 more months from now, both those stars would weigh 75 kg.

FINALLY: The upcoming actor should transform his body from 85 kg to 75 kg. He has 5 months time to lose those extra 10 kg.

CONCLUSION:

So... my dear fellow math enthusiasts, next time, when you see any student who says... "My interest is not in mathematics, but in acting/sports/something-else. I don't want to study mathematics"... Hope you would find posts like these useful in explaining why mathematics is important, irrespective of the career they are interested in.

I hope that such students would understand that there is a lot more hidden behind the boring/scary mathematical symbols they see! And try to understand how these mathematical symbols and concepts can be related to the field they want to pursue.