Question 157341
sara moves x feet in 8 minutes at the rate of r1.
the sidewalk moves x feet in 6 minutes at the rate of r2
if sara and the sidewalk move together, then sara moves x feet at the combined rate of (r1+r2)
distance = rate * time.
x = 8*r1
x = 6*r2
x = y*(r1+r2)
so 8*r1 = 6*r2 = y*(r1+r2)
now if x = 8*r1, then r1 = x/8.
similarly, if x = 6*r2, then r2 = x/6.
to solve the equation x = y*(r1+r2), we can substitute x/8 for r1 and x/6 for r2 since they are equivalent.  this allows us to solve for 1 unknown rather than 3.
the equation then becomes {{{x=y*((x/8)+(x/6))}}}
we multiply both sides of the equation to remove the fractions using a multiplier of 24 because both 8 and 6 divide into 24 without a remainder.
the equation then becomes {{{24*x = y*((24*(x/8))+(24*(x/6)))}}}
which then becomes {{{24*x = y*((3*x)+(4*x))}}}
which then becomes {{{24*x = y*(7*x))}}}
which then becomes {{{y=(24*x)/(7*x)}}}
which finally becomes {{{y=(24/7)}}} because the x cancels out.
this equals the time it takes to travel x feet at y feet per minute.
y = 24/7.
expressed as a decimal, then y = 3.428571429 minutes which is approximately 3 minutes and 26 seconds.
to prove the answer is correct, r1 and r2 should equal (r1+r2).
we don't know the distance, nor do we know the actual rate, but we can make an assumption.
say the distance is anything.  we can say the distance is anything because we're dealing with ratios.
let's use 1000 feet.
so r1 must be 1000 / 8 = 125 feet per minute.
likewise r2 must be 1000 / 6 = 166.6666666.... feet per minute.
if r3 = r1 + r2, then r3 must be 125 + 166.6666666... feet per minute.
that makes r3 = 291.66666..... feet per minute.
since rate * time = distance, then time = distance / rate, so ...........
1000 feet / 291.666..... feet per minute = 3.428571429 minutes to cover the distance which is exactly the number of minutes we calculated previously.