Stair climb
Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?
Age
11 to 14
Challenge level
Problem
Boris, Spike and Percival are going to race up the 99 steps that lead from the beach to the car park at the top of the cliff.
Boris can run up five steps in the same time as Spike can run up four steps, which is the same time as Percival can run up three steps.
It is agreed that Boris starts from the bottom, Spike starts 21 steps up and Percival 38 steps up.
If they all start at the same time, in what order will they reach the top?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: Spike, Boris, Perceval
Scaling the number of stairs run
Boris: 5 steps 100 - finished
Spike: 4 steps $\times$20 80 - finished (21 + 80 $\gt$ 99)
Percival: 3 steps 60 - not finished (38 + 60 $\lt$ 99) Last place
$\times$19 95 (4 steps to go)
76 (21 + 76 = 97 $\Rightarrow$ 2 steps to go) winner
Working out how long each person takes
Boris: $99$ steps
Spike: $99 - 21 = 78$ steps
Percival: $99 - 38 =61$ steps
In each time unit, Boris runs $5$ steps, Spike runs $4$ steps, Percival runs $3$ steps.
Boris: $99$ steps, $99 \div 5 = 19\frac{4}{5}$
Spike: $78$ steps, $78 \div 4 = 19\frac{1}{2}$
Percival: $61$ steps, $61 \div 3 = 20\frac{1}{3}$
Therefore they finish in the order Spike, Boris, Percival.