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# Math Revolution and GMAT Club Contest! John drove on a highway at a

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Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 02:54
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10
00:00

Difficulty:

25% (medium)

Question Stats:

80% (02:31) correct 20% (02:40) wrong based on 258 sessions

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Math Revolution and GMAT Club Contest Starts!

QUESTION #12:

John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r

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Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum

We are happy to announce a Math Revolution and GMAT Club Contest

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To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 05:42
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John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r
Explanation:-
Suppose Tom drove for time t speed 4r/3(given) And John for time t+2 speed r ( given)
Since both distance are equal hence t.4r/3=(t+2).r-->t=6
There fore John travelled 8r distance ANS-C
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 07:34
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1
Relative speed of tom with reference to john=(4r/3)-r=r/3
The distance john would have travelled in 2 hours=2r
Time taken by tom to catch up with john=2r/(r/3)=6 hours
When tom catches up with john,
total Distance travelled by john=total distance travelled by tom
2r+6r= 6*(4r/3)
=8r
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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Updated on: 16 Dec 2015, 07:34
2
When Tom started driving, the distance the John had already traveled is: 2r miles
Each hour the distance between Tom and John shortened by (4r/3-r) = r/3 miles
Then it will take Tom $$\frac{2r}{(r/3)}=6$$ hours to catch up John, which means till Tom caught up with John, John had been driving for 8 hours => The distance John drove is 8r
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 10:23
1
rel =4r/3 - r =r/3
time they met = 2r/(r/3)=6
d=s*t= 6r+2r = 8r
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 10:45
1
Speed of John =r miles per hour and he started at 13:00
speed of Tom =4r/3 miles per hour and he started at 15:00
by the time Tom Started John covered r*2 miles
relative speed =4r/3-r=r+r/3-r=r/3
d=2r
time taken =2r/(r/3)=6 hours
total time John was on road =6+2 =8 hours
therefore John drove (8r)
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 12:51
1
QUESTION #12:

John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r

Solution:
The distance traveled by both John and Tom are same when they meet.
Distance traveled by John = Distance traveled by Tom
(Speed of John)*(2hr + time) = (Speed of Tome)*(time)
r*(t+2) = (4r/3)*t
t = 6
So, distance traveled = 8r

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 15:19
1
Firstly we notice that John and Tom will have travelled the same distance when they meet, John will just have been travelling for longer.

If we define $$x$$ as the amount of time John spent travelling until he met Tom, then the amount of time Tom spent travelling would be $$x-2$$. The distance John travelled would be $$x*r$$ and the distance Tom travelled would be $$(x-2)*\frac{4r}{3}$$.

Since we know that they travelled the same distance, we can write $$xr=(x-2)*\frac{4r}{3}$$ and solve for $$x$$ to give $$x=8$$.

If $$x=8$$ and we know that the distance travelled by John is $$x*r$$ then we know the distance is $$8r$$, the answer is C.
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 16:37
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John - r m/hr, starts at 1 PM. Two hours later, John has travelled 2r miles.
Tom - 4r/3 m/hr, starts at 3 PM.

First find what time does Tom meet John.

2r + r*x = 4r/3*x. x=6 hours in this case. So basically 6 hours after Tom starts, he catches up with John.

John drove 2r + r*6 = 8r. Answer is C
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 17:40
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At 15:00, The lead is 2r for John
This lead of 2r has to be covered at a relative speed of ((4r/3) - r) to catch up with John
This means 6 hours from 15:00, the 2 will meet
But John started at 13:00 i.e. he has driven for 8 hours @ rmph
Hence John's distance is 8r when they meet
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 21:56
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1
for 2 hrs, J travelled 2r miles.. now tom starts with 4r/3 speed.. suppose they meet after time T.. distance travelled by tom = 4rT/3

also John will travell rT miles in this time..
now.. 2r+rT = 4rT/3
2r=rT/3
T=6

Therefore John travels = 2r +6r = 8r miles
C
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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13 Dec 2015, 21:59
for 2 hrs, J travelled 2r miles.. now tom starts with 4r/3 speed.. suppose they meet after time T.. distance travelled by tom = 4rT/3

also John will travell rT miles in this time..
now.. 2r+rT = 4rT/3
2r=rT/3
T=6

Therefore John travels = 2r +6r = 8r miles
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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14 Dec 2015, 07:13
1
QUESTION #12:
John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?
A. 3r
B. 5r
C. 8r
D. 9r
E. 10r

Solution:

V=s/t
r=d/t

For Tom: 4r/3=d/t-2
or, r=3d/4(t-2)

d/t=3d/4(t-2)
or, 3t=4t-8
or,t=8

Since, r=d/t
r=d/8
or d=8r

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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14 Dec 2015, 10:40
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Speed of John = r mph
Speed of Tom = 4r/3 mph

Relative speed = (4r/3) - r = r/3 mph

Tom leaves 2 hrs after John --> Distance covered by John in 2 hrs = 2r miles

So Tom has to cover this additional distance to meet John.

Time taken by Tom to meet John = Relative Distance/Relative speed = (2r)/(r/3) = 6 hours.

Since John drives for 2 more hours than Tom, time taken by John to meet Tom = 6 + 2 = 8 hours

Distance driven by John = Speed * Time = 8 * r = 8r miles

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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14 Dec 2015, 18:19
1
QUESTION #12:

John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r

In 2 hours (from 13:00 to 15:00), the distance John traveled $$= d_r=2*r$$ (miles)
Relative speed of Tom and John $$= S_r=\frac{4r}{3}-r=\frac{r}{3}$$ (miles per hour) - meaning: in 1 hour, Tom can cover distance of r/3 in relative with distance that John traveled.
Tom caught up with John only when Tom covered the 2r distance that John traveled --> the time Tom need to cover $$2r = \frac{2r}{r/3} = 6$$(hrs)
--> In total, John traveled: $$6+2= 8$$ hrs or $$8*r=8r$$ miles

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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14 Dec 2015, 18:54
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Using the Eqn $$D = RT$$

$$D(John) = R * T$$
$$D(Tom) = 4/3 * R * (T - 2)$$ (Since Tom started 2 hrs late)

When Tom Caught up with John

$$D(John) = D(Tom) => R*T = 4/3 R (T-2) => 3T = 4 (T-2) => T = 8$$

Therefore John Drove R*8 Miles before Tom Caught up with him

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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15 Dec 2015, 00:03
1
Since they catch up after driving on the same highway, the two distances are the same:

$$Distance John = Distance Tom$$

$$rt = \frac{4r}{3}*(t-2)$$

$$3rt =4rt -8r$$

$$t= 8$$ so the distance for John is rt or 8r

Same distance for Tom $$\frac{4r}{3}*6 =8r$$
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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17 Dec 2015, 04:56
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option C.
This can be solved using the concept of relative speed.

We are given that Tom started driving 2 hours after John. So, by this time John must have covered distance = r*2 = 2r

Relative speed of Tom with respect to John = 4r/3 - r => r/3
Now to catch up with john this distance of 2r needs to be travelled by Tom relative to John.
So time required to travel this distance = 2*r/relative speed => 2r/(r/3) => 6hrs.

Thus it will take Tom 6 hours to catch up with John.

Now, by this time the distance travelled by John will be = r*6 = 6r.
Also, John had already travelled a distance of 2r before Tom started.

Thus total distance travelled by John will be 2r+6r = 8r. => option C.
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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17 Dec 2015, 20:31
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speed equlas to distance travelled time so
John => r=m1/t1----(eq1);
Tom => 4*r/3 = m2/t2---(eq2);
as John should meet tom m1 should be equal to m2
so m1=r*t1 from eq1 & m2= 4*r/3*t2 which on solving both results t1=4/3 t2;
As tom has started 2 hrs late so t2-t1=2; solving for t1 we get t1=8 and m1=8*r
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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a  [#permalink]

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18 Dec 2015, 23:21
1
QUESTION #12:

John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r

---

say time taken by John when John & Tom catch up with each other is "t" hours.
Since Tom has started 2 hours "later", Tom would have taken "t-2" hours (2 hours lesser than John) when they meet up.

Total Distance that John would've traveled in t hours : D1 = r*t
Total Distance that Tom would've traveled in (t-2) hours : D2 = (4r/3)*(t-2)

They would've both covered the same distance when they meet , so D1 = D2

r*t = (4r/3)*(t-2)
=> 3rt = 4rt -8r
=> rt = 8r
=> r(t-8) = 0
we know r>0 , otherwise both John and Tom would've been stationary , so are not moving at all and can never meet.
so t = 8;

Going by the previous equation we've setup ,
Total Distance traveled by John when they meet = r*y = 8r

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Re: Math Revolution and GMAT Club Contest! John drove on a highway at a   [#permalink] 18 Dec 2015, 23:21

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