This is a straightforward Question Answer type discussion focused on one problem related to fitness planning and some variations to it that takes deep into the concept of average and exposes some of the errors people make while using it.
Question: Rohan is on a fitness drive. He has planned to burn 300 kcal in one exercise session is Gym. He has been suggested to do warmup, cardio, strength training and cooling. He divided his target kcal into 4 equal parts of 75 kcal each. His fitness tracker displays few data including calorie burnt.
Warmup burns 4 calories per minute, cardio exercise burns 8 kcal per minute, strength exercises 10 kcal per minute and cooling 2 kcal per minute.
How much time he would take in the gym to complete his exercise.
Answer: Since Rohan wants to burn 75 kcal in each of the four parts, calculate the time for each activity separately.
- Warm-up: Burns 4 kcal/min
- Time = 75 ÷ 4 = 18.75 minutes (18 min 45 sec)
- Cardio: Burns 8 kcal/min
- Time = 75 ÷ 8 = 9.375 minutes (9 min 22.5 sec)
- Strength training: Burns 10 kcal/min
- Time = 75 ÷ 10 = 7.5 minutes (7 min 30 sec)
- Cooling: Burns 2 kcal/min
- Time = 75 ÷ 2 = 37.5 minutes (37 min 30 sec)
Total time:
18.75 + 9.375 + 7.5 + 37.5 = 73.125 minutes
So, Rohan would take 73.125 minutes, i.e. 73 minutes 7.5 seconds (approximately 1 hour 13 minutes) to complete his entire gym session.
Question: As he was planning to burn equal calories in each part, he thought that taking average of 4 exercise data would make the computation simple. He got average of 4, 8, 10 and 2 as 24/4=6 calories per minute. He expected to finish in 300/6=50 minutes. How many calories he had burnt by the end of 50 minutes.
Answer: After 50 minutes, Rohan would have completed:
- Warm-up: 18.75 minutes → 75 kcal burned.
- Cardio: 9.375 minutes → 75 kcal burned.
- Strength training: 7.5 minutes → 75 kcal burned.
Time used so far:
18.75 + 9.375 + 7.5 = 35.625 minutes
Remaining time out of 50 minutes:
50 − 35.625 = 14.375 minutes
He spends this remaining time in the cooling phase, which burns 2 kcal/min.
Calories burned during cooling:
14.375 × 2 = 28.75 kcal
Therefore, total calories burned by the end of 50 minutes:
75 + 75 + 75 + 28.75 = 253.75 kcal
Answer: 253.75 kcal.
This also shows why his estimate of 50 minutes was incorrect. Although the arithmetic mean of the burn rates is 6 kcal/min, the correct overall average burn rate is:
300/73.125 = approx 4.10 kcal/min,
which is much lower because more time is spent in the slower activities, especially the cooling phase.
This is a nice illustration of why averaging rates directly can give the wrong answer when the work (calories) is divided equally rather than the time.
Question: What Rohan should do if he wants to achieve 300 kcal in 50 minutes. Will the average be 6 kcal/min in that case? How much calorie he will burn in different exercises then? Are there many options?
Answer: Now 50 minutes is fixed. Rohan is free to distribute his time among the four activities.
Suppose he spends
- 10 min warming up (4 kcal/min)
- 15 min doing cardio (8 kcal/min)
- 20 min strength training (10 kcal/min)
- 5 min cooling (2 kcal/min)
The total time is
10 + 15 + 20 + 5 = 50 minutes.
Calories burned are
- Warm-up: 40 kcal
- Cardio: 120 kcal
- Strength: 200 kcal
- Cooling: 10 kcal
Total = 370 kcal.
Too much.
So, we adjust.
But do we need to do trials like this or there is some better way?
Here, the time to be spend in four activities are variable. Let them be denoted by t1, t2, t3 and t4 for warmup, cardio, strength and cooling activities.
4*t1 + 8*t2 + 10*t3 + 2*t4 = 300
with
t1 + t2 + t3 + t4 = 50.
Since there are only two equations and variables are four, there are infinite possible values satisfying both the equations. Assume any values for any two variables and solve for remaining two. Only care needs to be taken that these values are positive.
So, if Rohan takes 10 minutes for warm-up and 10 minutes for cardio, he burns 10*4 = 40 kcal on warm-up and 10*8 = 80 kcal on cardio. Remaining 180 kcal he has to burn in Strength and Cooling. He has already spent 20 minutes so 30 minutes are left.
We have 4*10 + 8*10 + 10*t3 + 2*t4 = 300
and 10 + 10 + t3 + t4 = 50
So, 10*t3 + 2*t4 = 180 and t3 + t4 = 30
Solving these two equations gives t3 = 15 minutes and
t4 = 15 minutes
One can find several solutions this way.
Question: But the average kcal per minute is 300/50 = 6. The average of calorie rate for four exercises 4, 8, 10 and 2 is also 6. These averages seem simple. So, is there some elegantly simple solution.
Answer: The Arithmetic Mean commonly known as average of 4, 8, 10 and 2 is 6. This comes from (4+8+10+2)/4. Equal weight is given to the four numbers representing calorie burning rates. Since the target is 300 kcal with average 6 kcal/minute, time to be given is 300/6 = 50 minutes. Equal weight given to four exercises will give the desired result. So, 50/4 = 12.5 minutes each to the four exercises.
That gives, 12.5*4 + 12.5*8 + 12.5*10 + 12.5*2 = 300 calories. He burns 50 kcal in warm-up, 100 kcal in cardio, 125 kcal in strength training and 25 kcal in cooling.
Question: But Rohan plan has equal weight to the four exercises. He planned to spend equal calories in each of the four exercises. That's why he took the average. Why is this wrong?
Answer: It's important to understand here that equal weight means equal time in this situation. Rohan has done nothing wrong except interpreting conceptually wrong meaning of equal weight. He thought that assigning equal calories 75 to each of the exercises is giving equal weight.
Question: Why time, why not kcal? Is there any situation in which equal calories can mean as equal weight?
Answer: The calorie burning rates are given as kcal per unit time. The result depends on the time spent in different rates. So, weight should be based on time.
If the rate is given the other way like time per calorie and it is planned to burn equal calories in each of the exercises, then it is equal weight to the rates given.
Take 4 kcal/minute. It means 1/4 = 0.25 minute/kcal.
Similarly, 8 kcal/minute = 1/8 = 0.125 minute/kcal,
10 kcal/minute = 1/10 = 0.1 minute/kcal,
and 2 kcal/minute = 1/2 = 0.5 minute/kcal.
In this case, it's correct to take the average of these four rates given in minutes/kcal if equal calories are to be burnt on these four rates. The average is (0.25+0.125+0.1+0.5)/4 = 0.975/4 minutes/kcal.
To find time required to burn 300 calories if equal calories are burnt in each of the exercises, multiply this rate by 300.
This gives (0.975/4) * 300 = 0.975*75 = 73.125 minutes. This is same as the time required calculated earlier.
Take another situation. How fast you walk can be expressed as km/hour or as minutes/km. Like 6 km/hour or 10 minutes/km. Whether you plan your walk based on time at different speeds or distance covered at different speeds will decide what way the average has to be found. Just to relate, one is arithmetic mean and other is harmonic mean.
Question: Suppose Rohan has to burn 360 kcal using same four exercises. How can he make a quick plan?
Answer: Since rates of calorie burnt are given in kcal/minute, spending equal time in these four exercises will keep his calorie burning rate same as the average 6 kcal/minute. For 360 kcal he needs to exercise for 360/6 = 60 minutes. He needs to plan 60/4 = 15 minutes in each exercise. That way he burns 15*4 = 60 kcal in warm-up, 15*8 = 120 kcal in cardio, 15*10 = 150 kcal in strength training and 15*2 = 30 kcal in cooling. Total is 360 kcal.
This discussion shows that finding an average is not just about applying a formula. It is equally important to understand what is being averaged and what is being given equal importance. In this example, expressing the rates in different units led to two different but correct ways of averaging. Looking at the problem from different perspectives also revealed that there can be many valid workout plans instead of just one. Such questions encourage us to think beyond calculations and pay attention to the ideas behind the mathematics.
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