Walk or run?
Can you work out how long it will take for John to walk to and from school?
Age
11 to 14
Challenge level
Problem
When John walks to school and then runs home, it takes him 40 minutes.
When he runs both ways, it takes him 24 minutes.
He has one fixed speed whenever he walks, and another fixed speed whenever he runs.
How long would it take him to walk both ways?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: 56 minutes
Using words
walk, run takes 40 minutes
run, run takes 24 minutes
running takes 12 minutes each way
walk, run takes 40 minutes total, 12 come from running on the way back
So the walking must take 28 minutes.
walk, walk takes 28$\times$2 = 56 minutes.
Using algebra
If it takes John $w$ minutes to walk to (or from) school and $r$ minutes to run to (or from) school, then we have $$\begin{align}r+w&=40\\2r&=24\\2w&=?\end{align}$$
Finding $r$ first
$2r=24\Rightarrow r=12.$ Substituting this into $r+w=40$ gives $12+w=40\Rightarrow w=28.$ So $2w=2\times28=56.$ So it takes $56$ minutes to walk both ways.
Doubling the first equation
Doubling both sides of $r+w=40$ gives $2r+2w=80.$ Since $2r=24,$ $24+2w=80\Rightarrow 2w=80-24=56.$ So it takes $56$ minutes to walk both ways.