it takes Mars 1.88 years to travel around the sun.It takes Mercury 0.24 years
to orbit the sun. If the planets are in a straight line with the sun now, how
long will it be before this is the case again? The answer must be accurate to
the nearest hundredth of a year.

We must find the smallest positive integer N 
such that when we multiply both 1.88 and .24 by N 
we get a positive integer, that is

1.88N = P , whener P is a positive integer
 and

0.24N = Q , whener Q is also a positive integer

Multiply both equations by 100 to remove decimals:

188N = 100P

24N = 100Q

Solve each for N

N = 

N = 

N = 

N = 

So we have

 = 

Divide both sides by 25

 = 

Cross multiply:

 = 

Divide both sides by 6Q

 =  

 = 

The smallest positive integer values of P and Q
which make that equation true are P = 47 and
Q = 6.

Substitute 47 for P in

1.88N = P

1.88N = 47

N = 

N = 25

As a check:

Substitute 6 for Q in

0.24N = Q

0.24N = 6

N = 

N = 25

So the answer is 25 years. That's the next time when 
both planets will have each completed a whole number 
of revolutions.

Edwin