Triathlon and Fitness
Problem
Triathlon and Fitness printable worksheet
Adam, Ben and Charles all want to qualify for the Olympic triathlon.
Here are their results from a trial race:

Swimming 1500m 
Cycling 40km 
Running 10km 
TOTAL 
Adam 
19:12 
1:04:33 
32:13 
1:55:58 
Ben 
21:19 
1:05:28 
31:54 
1:58:41 
Charles 
22:31 
1:03:22 
30:12 
1:56:05 
Which athlete do you think burned the most calories?
Adam, Ben and Charles weigh approximately 70kg each
It is estimated that athletes of this weight burn energy at the following rates:
 Swimming at 4.5km/h burns 600 kcal per hour
 Cycling at 30km/h burns 900 kcal per hour
 Running at 15km/h burns 1100 kcal per hour
Assuming that the rate at which calories are burned is directly proportional to the athletes' speed, estimate how many calories each of the three athletes burned during his race.
Are you surprised?
How can you explain your results?
Do you think the modelling assumptions are valid?
Though several versions of the triathlon exist, in the Olympic Games, the athletes swim for 1500 meters, then cycle for 40 km and finally run for a further 10 km. As you can imagine, the athletes who compete for the triathlon must have high endurance in order to be able to complete all three races.
Getting Started
What do you need to know to work out each athlete's speed?
How can you work out the speed in kilometres per hour?
If his swimming speed is slower than 4.5km/h, would you expect him to burn more or less than 600 kcal per hour?
Once you know the rate at which he burns calories, how will you work out how many calories he burns in 19 minutes and 12 seconds?
Student Solutions
Thanks for all your solutions! This was a tough one to get right...
Preeti thought that Charles would burn the most calories, because:
He took the most time, and the longer you run the more calories you burn.
Niamh, from Baston, thought that Ben would burn the most calories, and offered this reason:
Ben took the most time overall and the most time in the cycling, which burned the most calories per hour.
Patrick, from Otterbourne, said the following:
I think they will all burn the same number of calories, as an athlete who completes a stage faster burns more calories per hour, but for proportionately less time.
I think that the model is inaccurate because somebody completing the same distance, but in more time, would have to work harder. Consequently, they would burn more calories. I think that the calorie burn rate squared should be directly proportionate to the speed.
Olie, from BPS, said this:
I thought that the faster you go the more calories you burn, but I worked out that it was actually the other way round!
Whereas Karen, from Pent Valley, said the opposite:
Charles burnt the most calories  it is easier to think that the slower the time, the more calories are burnt but this is not true! The quicker your speed, the more calories you are burning.
Who is making the correct assumptions?
Amber, from Bealings, gave her thoughts on who had burned the most calories:
Adam, because he started off quickly and then he slowed down, and he was faster in the first race. In the other two races he slowed down and came last. So if Adam had kept his stamina up he would not have burned the most calories. But he was cycling and running for the longest period of time, which means Adam burned off more calories  then it was Ben, then Charles.
Margo, from ACS Egham, made these comments:
I'm not surprised that running burns most calories because it challenges your legs AND arms. I think the modelling assumptions are valid, because my Uncle cycles every day, my Dad runs every day and my great Auntie swims every day, and my dad burns the most calories.
Interesting thoughts! Do you think the numbers will agree?
Chloe from Baston Primary School said that:
Adam burned 2134 calories.
Ben burned 2127 calories.
Charles burned 2131 calories.
How did Chloe arrive at these numbers?
Jasmyn, from Baston, used percentages to claim that:
Charles wins whilst burning 3412.26.
Lucas, also from Baston, calculated the number of calories burnt per unit time:
Calories burnt  Swimming  Cycling  Running 
Per minute  10  15  18.33 
Per second  0.166  0.25  0.31 
Using these, even though Adam won the race, Ben burnt the most calories.
Is this the correct conclusion?
Preeti, from Twyford C of E, offered some advice:
Firstly, you have to make sure you do all your calculations in the correct units. I converted the 1500m into 1.5km at the start, and converted the times from hours:minutes:seconds into hours.
I then worked out their speeds in km/h.
So, for example, Adam swam at 4.6875km/h; this allowed me to deduce that he burned (4.6875/4.5)*600 kcal per hour. Then I multiplied by the number of hours he swam for.
Is this the right strategy?
Emily, from St Helen's, carried out the following calculations providing convincing evidence for her surprising conclusion:
Adam's average swimming speed was 4.6875 km/h, burning 625 calories per hour. So in 19 minutes and 12 seconds he burnt 200 calories.
His average cycling speed was 37.180481 km/h, burning 1115.4144 calories per hour. So in 1 hour 4 minutes and 33 seconds, he burnt 1200 calories.
His average running speed was 18.623901 km/h, burning 1365.7527 calories per hour. So in 32 minutes and 13 seconds he burnt 733.3 calories.
So, in total Adam burnt 2133.3 calories throughout his entire triathlon race.
If you repeat the same calculations for Ben and Charles, it turns out that they all burnt the same number of calories on each of the three sections, so they all burnt the same number in total.
Bet you weren't expecting that!
Teachers' Resources
Why do this problem?
This problem provides a reallife context for working with proportionality, speed, rates, and units of measurement. The final answer may be surprising and leads to interesting questions about the validity of the model suggested.
Possible approach
Split the class into groups of two or three, and give each group a copy of this worksheet.
"Your task is to work out which athlete burned the most calories in a triathlon. Before you do any calculations, discuss which athlete you think it might be, together with your reasons why, and make a note of it."
As they are working, circulate and listen to the conversations that students are having, to identify anyone with particular insights that it would be useful to share.
Here are some key questions that could be used to prompt groups who get stuck:
What do you need to know to work out each athlete's speed?
How can you work out the speed in kilometres per hour?
If his (swimming) speed is slower than (4.5)km/h, would you expect him to burn more or less than (600) kcal per hour?
Once you know the rate at which he burns calories, how will you work out how many calories he burns in (19 minutes and 12 seconds)?
Towards the end of the lesson, bring the class together and invite different groups to explain how they worked out the number of calories burned for each stage of the triathlon. Allow some time for discussion of the perhaps surprising answer to the question "Who burned the most calories?".
Possible support
Mixing Lemonade offers an introduction to ratio and proportion in a simple context. It might be useful preparation for working on this problem.
Possible extension
Ratios and Dilutions provides students with an opportunity to explore ratio and proportion in a different context.