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# Did You Find Out? Get the Answer to the Riddle of E-News Issue No. 2, March 2023

## Outpaced

How long do the men need for one lap?

Tom and Mike are hobby runners, and like to run their laps on the track. Tom started running 30 seconds before Mike, and Mike is the better of the two runners. After running for exactly 10 minutes, Mike outruns Tom for the first time.

How many seconds do both runners need to complete 1 lap, 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, for example, he 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 outruns Tom, Mike is 30 seconds ahead (570 seconds), and runs one more lap than the slower Tom.

x = Number of laps
t = Time for one of Tom's lap

Step 1: Equation for Tom

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

Step 2: Equation for Mike

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

Step 3: Insert equation 1 into equation 2

(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

Step 4: Find the solution with the help of the quadratic 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, or 51.15 seconds. By the time he outruns Tom, Mike has run (570/51,15) 11.1 laps and Tom has run (600/59,15) 10.1 laps.

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