I need an equation on how they came up with the answer of 5.6 or 5 min 36 secs?

Question

This is a math problem. Marcus and Dan decide to ride their bicycles around a circular track. At the start/finish line, both riders begin riding at the same time. If Marcus completes a lap every 42 seconds, and Dan completes a lap every 48 seconds, in how many minutes will Marcus and Dan cross the start/finish line at the same time?

Answer 1

Method 1: The biggest number that divides into both 42 and 48 is 6.

The answer is the product of the two numbers divided by the greatest common factor (42 * 48 / 6) = 336

Method 2: The prime factors of 42 are 2*3*7, and the prime factors of 48 are 2*2*2*2*3

The answer is the product of the two numbers, with common factors canceled out of one of them (2*3*7 * 2*2*2 [2*3 eliminated]) = 42*8 = 336

Answer 2

Assume Marcus does 'm' laps, while Dan does 'd' laps.

Since they cross the line at the same time then ...

42m = 48d

which simplifies to

7m = 8d

this has an integer solution when m = 0 and d = 0 i.e. when they start.

The next time this happens is when m = 8, d = 7 since using substitution above

7 x 8 = 8 x 7

there is no solution lower than this because 7 and 8 have no matching factors other than 1.

So if m = 8 then then time would be

T = 42m = 42 x 8 = 336 seconds

so T = 5 minutes 36 seconds

Answer 3

It's not an equation you need; it's a Least Common Multiple
(LCM).

The answer has to be a multiple of 48 and 42, the SMALLEST MULTIPLE available.

To find the LCM methodically you factor each number into prime factors:

48 = 2^4 x 3
42 = 2x3x7

Then take each prime the MOST number of times it shows in any one number:

2^4x3x7 (= 336 seconds, which converted to minutes by dividing by 60, gives 5.6 minutes.)
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An alternative approach here would be to start multiplying the larger number by 1, then 2, 3, 5, 7, 11, 13, etc... checking after each multiplication for divisibility by 42. This works, but is more trial-and-error and somewhat less analytic, IMO.
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For information purposes we can make an ordered quad-let:

(laps-runby-Marcus-the-faster, laps-run-by-Dan-the-slower, seconds-elapsed-till-they-meet-at-start/finish, minutes-elapsed-till-they-meet-at-start/finish) =

(8, 7, 336, 5.60) and any whole number multiple of the ordered quad-let elements will yield an answer when the 2 are at the start/finish line/

Answer 4

LCM of 42s and 48s =336 s.=5min and 36s.=5.6 minutes
during this time Marcus will make 8laps and Dan will complete exactly 7 laps and so pass the start/ finish line at the same time.