Rebecca runs at a constant rate on the treadmill and it takes her 2 ho

3/T+3/(T-2)=5/4


Why the speed is not included in the formula ?

Hi Bunuel, would you please release an official solution for this one? Tried to understand other's explanation but still very lost.. thanks!

IMHO E
Idk how much correct I was when arriving at the solution, still lost a little.

Let the time taken for Jeff = t
Let the time taken for Rebecca = t+2
Distance till first milestone be = 100km

Distance ___________100km___________ | __25km__ | __25km__ | __25km__ |

Jeff ---------------t hrs-------------------|---t+3 hrs--|---t+6 hrs--|--t+9 hrs----|
Rebecca ---------------t+2 hrs----------------|---t+5 hrs--|---t+8 hrs--|--t+11 hrs----|

Eliminate a,b,c,d for t for rebecca > 11

The first method would be to make use of choices..

The choices give us time of R to cover 2 milestone, so half of each choice will give us time taken by R to cover 1 milestone, and THAT half -2 will give us time taken by J..

(A) 4
So R takes 4/2 or 2 hrs and J takes 2-2 =0...NOT possible

(B) 6
So R takes 6/2 or 3 hrs and J takes 3-2 =1, so J alone will cover 3/3 or 3 milestones in 3 hrs, but we are looking a combined distance of 1.25 milestone in 3 hrs...NOT possible

(C) 8
So R takes 8/2 or 4 hrs and J takes 4-2 =2, so J alone will cover 3/2 or 1.5 milestones in 3 hrs, but we are looking a combined distance of 1.25 milestone in 3 hrs...NOT possible

(D) 10
So R takes 10/2 or 5 hrs and J takes 5-2 =3, so J alone will cover 1 milestones in 3 hrs and R will cover 3/5 or 0.6 milestones. Combined they cover 1.6, but we are looking a combined distance of 1.25 milestone in 3 hrs...NOT possible

(E) 12
Only choice left, so our answer, but let us check
So R takes 12/2 or 6 hrs and J takes 6-2 =4, so J alone will cover 3/4 or 0.75 milestones in 3 hrs and R will cover 3/6 or 0.5 milestones in 3 hrs. Combined they cover 0.75+0.5=1.25 milestones..... Our answer

E

Second method could be..

R takes R hrs for 1 milestone, so in 3 hours, distance covered will be 3/R milestone
Similarly J will take R-2 hrs for 1 milestone, so in 3 hours, distance covered will be 3/(R-2) milestone
These 2 should equal 1.25 milestone.

So, \(\frac{3}{R}+\frac{3}{R-2}=1.25=\frac{5}{4}..............................\frac{3R-6+3R}{R(R-2)}=\frac{5}{4}\)
=> \(4*6(r-1)=5*R(R-2)\)
Either use choices to see.....E gives R as 12/2=6.....\(4*6*(6-1)=5*6*(6-2).........4*6*5=5*6*4\)..YES
OR solve further
\(24R-24=5R^2-10R...........5R^2-34R+24=0.....5R^2-30R-4R+24=0.....(R-6)(5R-4)=0\)
So, R can be 6 or 4/5, but R cannot be 4/5 as J will become negative then..
Thus R=6, and twice of R or 12 hrs is required to cover 2 milestones..

E

Let t be the time taken by Jeff to complete first milestone. J be jeff's speed and R be Rebeca's speed

jt = R(t+2) = d = firs ilestone distance

in 3 hrs,

3j + 3R = 1.25(R[t+2])

on solving t = 4 hrs
therefore, rebecca's time to complete one milestone is 6 hrs. Hence, to complete twice the milestone time taken by her would be
6*2 hrs = 12 ---> E

Start with what you have so that need for variables is limited.

Together they reach 25% beyond milestone in 3 hrs.
Ratio of 25% beyond milestone distance : milestone distance = 5 : 4
Ratio of time taken to reach 25% beyond milestone : time taken to reach milestone = 5 : 4 = 3 : ??
Time taken to reach the milestone together = 4*3/5 = 12/5 hrs

Let t be the time taken by Jeff alone to reach the milestone. Their rates are additive.
1/t + 1/(t+2) = 5/12
t = 4 (No calculations needed. To get LCM 12, the denominators would be 3 and 4 or 4 and 6 or 2 and 12 ... look for the pair that has a difference of 2)

Rebecca takes 6 hrs for one milestone so she takes 12 hrs for 2 milestones

Answer (E)

Let the milestone distance be D
Let Rebecca's speed be R
Let Jeff's speed be J
Let the time taken by Jeff be T
Now D = R(T+2) = J(T)
Similarly as per the other statement both combine to make 1.25 D in 3 hours
=> 3R + 3J = 1.25 D
=> 3R + 3D/T = 1.25D
=> 3R = 1.25D - 3D/T
=> 3R = D(1.25T - 3)/T
=> 3RT = R(T+2)(1.25T - 3)
=> 3T = (T+2)(1.25T - 3)
=> 3T = 1.25T^2 - 3T + 2.5T - 6
=> 1.25T^2 - 3.5T - 6 = 0
=> 5T^2 - 14T - 24 = 0
=> (5T + 6)(T - 4) = 0

Now T cannot be negative hence T = 4
Now Rebecca takes T+2 = 6 hours 1 milestone
For 2 milestones Rebecca would take 6 * 2 = 12 hours

Option E