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# Did You Find Out? Get the Answer to the Riddle of E-News Issue No. 6, November 2014

## Outpaced

How long do the men need for one lap?

Tom and Mike are hobby joggers, and like to run their laps in a stadium. Tom started running 30 seconds before Mike, and Mike is the better of the two runners. After running for exactly 10 minutes, Tom is outpaced by Mike for the first time.

How many seconds do both runners need to complete 1 lap in the Stadium, when Mike runs every lap 8 seconds faster than his friend Tom?

## Solution

In 10 minutes (= 600 seconds), Tom runs 600/x laps and needs t seconds to complete 1 lap. If he for example needed exactly 50 seconds for 1 lap, he would do 12 laps within 10 minutes (600/50 = 12). Mike is faster and needs 8 seconds less for 1 lap, meaning (t-8) seconds. By the time he outpaces Tom, Mike is 30 seconds less (570 seconds) on his way, and also runs one more lap compared to the slower Tom.

x = Number of run rounds
t = Time for one round of Tom

1. Step: Equation for Tom

x * t = 10 minutes (600 seconds)
x = 600 seconds/ t

2. Step: Equation for Mike

(x + 1) * (t – 8 seconds) = 10 minutes – 30 seconds = 570 seconds

3. Step: Insert 1. equation into 2. equation

(600 /t +1) * (t – 8) = 570
(600 * t – 4800)/ t + t – 8 = 570
600 – 4800/ t +t = 578
22t – 4800 + t2 = 0
t2 + 22t – 4800 = 0

4. Step: Get the solution with the help of the p,q formula

x1 = - 22/2 +  √(484/4 + 4800) = 59.15
x2 = - 22/2 - √(484/4 + 4800) = - 81.15

X2 is useless, because seconds can only be positive. Tom needed 59.15 seconds per lap, while Mike needed 8 seconds less time with 51.15 seconds in total. By the time he outpaces Tom, Mike ran (570/51,15) 11.1 laps and Tom ran (600/59,15) 10.1 laps.