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3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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Updated on: 21 Jan 2014, 04:22
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3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy? A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles
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Originally posted by guerrero25 on 20 Jan 2014, 10:31.
Last edited by guerrero25 on 21 Jan 2014, 04:22, edited 1 time in total.




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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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20 Jan 2014, 11:14
guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles
I do not have the OA right now . I will post the answer this evening . Beth gives Amy a head start of 2 hours, thus Beth gives Amy a head start of 2*3 = 6 miles. To overtake Amy, Beth will need (time) = (distance)/(relative rate) = 6/(43) = 6 hours. From above we have that Amy runs for 2 + 6 = 8 hours, thus runs total of 8*3 = 24 miles. To run 24 miles Cassie needs 24/6 = 4 hours. Therefore Cassie gave Amy a head start of (time of Amy)  (time of Cassie) = 8  4 = 4 hours. Answer: B.
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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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21 Jan 2014, 03:23
head start Beth gave to Amy = 2 hours => 3mph *2 =>6 mile Head start Cassie gave to Amy = x mile
in time t distance travelled Cassie x+6 miles = 3t [ relative speed of Cassie w.r.t to Amy * t] Beth  6 miles = 1t [ relative speed of Cassie w.r.t to Amy * t] equations: x+6=3t and 6=1t x = 12 miles Head start = x/ speed of Amy => 12/3 = 4 hours Answer B



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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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21 Jan 2014, 22:23
guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles The most important clue here is this: If both Beth and Cassie overtake Amy at the same time This means that all A, B and C meet together i.e. at one instant, they all had covered the same distance. For easy calculations, let's assume that that distance was 12 miles. A would have been running for 4 hrs, B would have been running for 3 hrs and C would have been running for 2 hrs. But actually, B started 2 hrs after A so the distance must have been 24 miles. A would have been running for 8 hrs, B would have been running for 6 hrs and C would have been running for 4 hrs. So C gave A a head start of 4 hrs. Answer (B)
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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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07 Feb 2014, 20:17
guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles I used following approach: ( A,B and C denote Amy,Beth and Cassie respectively) Let the total distance be x then time of A= x/3 , B= X/4 Given that, x/3  x/4 = 2 So, x = 24 Now time taken by A = 8 h B= 6h C= 4h Thus, C gave head start to A for 84= 4h



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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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18 Aug 2014, 09:02
VeritasPrepKarishma wrote: guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles The most important clue here is this: If both Beth and Cassie overtake Amy at the same time This means that all A, B and C meet together i.e. at one instant, they all had covered the same distance. For easy calculations, let's assume that that distance was 12 miles. A would have been running for 4 hrs, B would have been running for 3 hrs and C would have been running for 2 hrs. But actually, B started 2 hrs after A so the distance must have been 24 miles. A would have been running for 8 hrs, B would have been running for 6 hrs and C would have been running for 4 hrs. So C gave A a head start of 4 hrs. Answer (B) Hi Karishma, Do you have any blog on this concept of race and circular tracks? I have trouble understanding them.



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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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18 Aug 2014, 20:42
maggie27 wrote: VeritasPrepKarishma wrote: guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles The most important clue here is this: If both Beth and Cassie overtake Amy at the same time This means that all A, B and C meet together i.e. at one instant, they all had covered the same distance. For easy calculations, let's assume that that distance was 12 miles. A would have been running for 4 hrs, B would have been running for 3 hrs and C would have been running for 2 hrs. But actually, B started 2 hrs after A so the distance must have been 24 miles. A would have been running for 8 hrs, B would have been running for 6 hrs and C would have been running for 4 hrs. So C gave A a head start of 4 hrs. Answer (B) Hi Karishma, Do you have any blog on this concept of race and circular tracks? I have trouble understanding them. I have a post on circular motion: http://www.veritasprep.com/blog/2012/08 ... ncircles/But not one on races. I will put something up in the next few weeks.
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3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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14 Sep 2015, 15:46
Let A=Amy's time to meeting Let A2=Beth's time to meeting Let AC=Cassie's time to meeting 3A=4(A2) A=8 8(3)= 4(6)=24 miles to meeting 6(8C)=24 miles C=4 hours head start given by Cassie to Amy



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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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06 Aug 2016, 11:33
I'm unable to understand this question. Are they three running together or what? This is a really confusing question.



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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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06 Aug 2016, 12:05
sudhirgupta93 wrote: I'm unable to understand this question. Are they three running together or what? This is a really confusing question. What is the confusion? Question clearly states that A started first. B started after 2 hours and we need to determine When did C start. As per the question, we know that All three meet at the same point. So, we can determine when did A and B meet. The point they will meet would be the same at which C will be present at that time. So, A and B will meet after t = 6/(43) = 6 hours. In these 6 hours, B would have covered 4*6=24 miles. => C would also have covered 24 miles and that would be in 24/6=4 hours. Thus, C started 2 hours after B or 4 hours after A. hence, answer is B.
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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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06 Aug 2016, 12:12
abhimahna wrote: sudhirgupta93 wrote: I'm unable to understand this question. Are they three running together or what? This is a really confusing question. What is the confusion? Question clearly states that A started first. B started after 2 hours and we need to determine When did C start. As per the question, we know that All three meet at the same point. So, we can determine when did A and B meet. The point they will meet would be the same at which C will be present at that time. So, A and B will meet after t = 6/(43) = 6 hours. In these 6 hours, B would have covered 4*6=24 miles. => C would also have covered 24 miles and that would be in 24/6=4 hours. Thus, C started 2 hours after B or 4 hours after A. hence, answer is B. I've been doing distancerate questions since morning. I guess my glass is full. Got it from your explanation. Thanks..



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3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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10 Aug 2018, 00:45
My Approach:
x 1h 2h 3h 4h 5h 6h 7h 8h A 3 6 9 12 15 18 21 24 B 0 0 4 8 12 16 20 24 C 0 0 0 0 6 12 18 24
we start filling out the distance travelled per hour by A, Since B gives A, 2 hours headstart, we have B starting 3 hours at the end of 8 hour A and B catch up. In order that C also catches up at the same time we start filling at the last column and start reducing at 6 per hour(speed of C) Doing so C shoud start at 5th hour. Hence C must give 4 hours head start.
This is alternative approach. Might work for some.



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3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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22 Jan 2019, 07:04
guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles
Excellent opportunity for RELATIVE VELOCITY (speed) and UNITS CONTROL , two powerful tools covered in our course! \(\left( * \right)\,\,\,2{\rm{h}}\,\, \cdot \,\,{{3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,6\,\,{\rm{miles}}\,\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{distance}}\,\,{\rm{A}}\,\,{\rm{starts}}\,\,{\rm{ahead}}\,\,{\rm{of}}\,\,{\rm{B}}\,} \right]\) \({{\rm{V}}_{{\rm{B}} \to {\rm{A}}}} = {{4  3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,{{6\,\,{\rm{miles}}} \over {{T_B}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_B} = 6\,{\rm{h}}\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{B}}\,\,{\rm{to}}\,\,{\rm{overtake}}\,\,A\,} \right]\) \(? = x\,\,{\rm{h}}\) \(\left( {**} \right)\,\,\,x\,\,{\rm{h}}\,\, \cdot \,\,{{3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,3x\,\,{\rm{miles}}\,\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{distance}}\,\,{\rm{A}}\,\,{\rm{starts}}\,\,{\rm{ahead}}\,\,{\rm{of}}\,\,{\rm{C}}\,} \right]\) \({{\rm{V}}_{{\rm{C}} \to {\rm{A}}}} = {{6  3\,\,{\rm{miles}}} \over {1\,\,{\rm{h}}}}\,\,\, = \,\,\,{{3x\,\,{\rm{miles}}} \over {{T_C}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{T_C} = x\,{\rm{h}}\,\,\,\,\,\,\,\,\,\left[ {\,{\rm{C}}\,\,{\rm{to}}\,\,{\rm{overtake}}\,\,A\,} \right]\) \({\rm{Stem}}\,\,\,\, \Rightarrow \,\,\,\,\, {T_C} + x = {T_B} + 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = x = 4\) We follow the notations and rationale taught in the GMATH method. Regards, Fabio.
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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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27 Jan 2019, 19:29
guerrero25 wrote: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph, 4 mph and 6 mph respectively. They run a race in which Beth gives Amy a head start of 2 hrs. If both Beth and Cassie overtake Amy at the same time, what head start did Cassie give Amy?
A. 3 hours B. 4 hours C. 5 hours D. 9 miles E. 10 miles We can let t = the time Amy has run before Beth and Cassie overtake her. Thus, we can create the following equation: 4t = 3(2 + t) 4t = 6 + 3t t = 6 Thus, Amy has run 3(2 + 6) = 24 miles in 8 hours before Beth and Cassie overtake her. Since Cassie only needs 4 hours to run 24 miles then she has to give Amy 4 hours for the head start. Answer: B
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Re: 3 people, Amy, Beth, and Cassie, have speeds of 3 mph
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