### Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

### Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won \$1 200. What were the assets of the players at the beginning of the evening?

### Oh for the Mathematics of Yesteryear

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45 days against the siege of the enemy?

# Triathlon and Fitness

##### Stage: 3 Challenge Level:

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.

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:

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!