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How much time did it take a certain car to travel 400

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How much time did it take a certain car to travel 400  [#permalink]

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New post 12 Oct 2018, 14:30
DSGB wrote:
How much time did it take a certain car to travel 400 kilometers?

(1) The car traveled the first 200 kilometers in 2.5 hours.
(2) If the car's average speed had been 20 kilometers per hour greater than it was, it would have traveled the 400 kilometers in 1 hour less time than it did.

\(400\,{\rm{km}}\,\,\,\,\,\left\{ \matrix{
\,\,\left( {{\rm{real}}\,\,{\rm{speed}}\,,\,\,{\rm{real}}\,\,{\rm{time}}} \right)\,\,\,{\rm{ = }}\,\,\,\left( {{V_R}\,\,,\,\,{T_R}} \right) \hfill \cr
\,\,\left( {{\rm{hypothetical}}\,\,{\rm{speed}}\,,\,\,{\rm{hypothetical}}\,\,{\rm{time}}} \right)\,\,\,{\rm{ = }}\,\,\,\left( {{V_H}\,\,,\,\,{T_H}} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\,\,\,\left[ {\,{{{\rm{km}}} \over {\rm{h}}}\,} \right]\,\,\,,\,\,\,\,\left[ {\,{\rm{h}}\,} \right]\,\,\,} \right)\)

\(? = {T_R}\,\,\,\left[ {\rm{h}} \right]\)


\(\left( 1 \right)\,\,\left\{ \matrix{
\,{\rm{If}}\,\,{\rm{it}}\,\,{\rm{took}}\,\,0.5\,{\rm{h}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{last}}\,\,200\,{\rm{km}}\,\,\,\,\, \Rightarrow \,\,\,\,?\,\, = \,\,3\, \hfill \cr
\,{\rm{If}}\,\,{\rm{it}}\,\,{\rm{took}}\,\,1\,{\rm{h}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{last}}\,\,200\,{\rm{km}}\,\,\,\,\, \Rightarrow \,\,\,\,?\,\, = \,\,3.5\,\, \hfill \cr} \right.\)


\(\left( 2 \right)\,\,{V_H} = {V_R} + 20\,\,\,\left( * \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{T_R} - {T_H} = 1\,\,\,\,\,\,\left[ {\rm{h}} \right]\)

Now it´s time for UNITS CONTROL, one of the most powerful tools of our method:

\({{{\rm{km}}} \over {\,\,\,{{{\rm{km}}} \over {\rm{h}}}\,\,}} = {\rm{h}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\{ \matrix{
{T_R} = {{400} \over {{V_R}}} \hfill \cr
{T_H}\,\mathop = \limits^{\left( * \right)} \,\,{{400} \over {{V_R} + 20}} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{{400} \over {{V_R}}} - {{400} \over {{V_R} + 20}} = 1\)

\({{400\left( {{V_R} + 20} \right)} \over {{V_R}\,\,\left( {{V_R} + 20} \right)}} - {{400\,\,{V_R}} \over {\left( {{V_R} + 20} \right)\,\,{V_R}}} = 1\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,400 \cdot 20 = {V_R}\,\,\left( {{V_R} + 20} \right)\)

\({V_R}^2 + 20{V_R} - 400 \cdot 20 = 0\,\,\,\,\,\,\mathop \Rightarrow \limits_{{\rm{of}}\,\,{\rm{roots}}}^{{\rm{product}}} \,\,\,\,\,\,\,\left( {{c \over a} = } \right)\,\,\,{{ - 400 \cdot 20} \over 1} < 0\,\,\,\, \Rightarrow \,\,\,\,\,{V_R}\, > 0\,\,\,\,{\rm{unique}}\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\)

\(\left( {\,\,\,\left. \matrix{
{V_R}\,\,{\rm{unique}}\,\, \hfill \cr
{\rm{400}}\,{\rm{km}} \hfill \cr} \right\}\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,{T_R}\,\,\,{\rm{unique}}\,\,\,} \right)\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 13 Oct 2018, 07:58
Top Contributor
DSGB wrote:
How much time did it take a certain car to travel 400 kilometers?

(1) The car traveled the first 200 kilometers in 2.5 hours.
(2) If the car's average speed had been 20 kilometers per hour greater than it was, it would have traveled the 400 kilometers in 1 hour less time than it did.


Target question: How long did it take to travel 400km
To find the travel time, we need to know the average speed traveled.
Let x = the average speed traveled.

REPHRASED target question: What is the value of x?

Statement 1: The car traveled the first 200 km in 2.5 hrs
No info about the 2nd half of the trip, so we can't determine the overall average speed (aka x).
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: If the car’s average speed had been 20 km/h faster, it would have traveled the 400 km in 1 hour less time.
Let's start with a word equation:
(travel time at x km/h) - 1 = (travel time at x+20 km/h)
Since time = distance/speed, we can now write:
(400/x) - 1 = 400/(x+20)

IMPORTANT: At this point, we need only determine whether this equation will yield 1 or 2 valid values of x. If it yields only 1 valid value, then statement 2 is sufficient. If it yields 2 valid values, then statement 2 is not sufficient.

Rewrite as: (400-x)/x = 400/(x+20)
Cross multiply: (400)(x) = (400-x)(x+20)
Simplify: 400x = 8000 + 380x - x²
Rewrite: x² + 20x - 8000 = 0
Factor: (x + 100)(x - 80) = 0
So, x = -100 or 80
Since x cannot be negative, it must be the case that x = 80
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 13 Oct 2018, 09:48
VeritasKarishma wrote:
Yellow22 wrote:
How much time did it take a certain car to travel 400 km?

1) The car travelled the first 200 KM in 2.5 hrs?
2) If the Car's average speed was 20 KM per hour greater than it was, it would have travelled the 400 KM in 1 hour less time than it did.


I guess the solution is already clear to you.

Just for intellectual purposes, look at an alternative method:

If the car actually took t hours to cover 400 km, in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km
This must be the distance it covered in each hour since we are considering average speed.
400 = 20*(t - 1)*t
t(t - 1) = 20
t = 5 hrs




Hi VeritasKarishma
The solution appears to be way shorter, but I don't understand a certain aspect.

"in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km"
THis is the part I am confused about. The speed has been increased by 20. SO the time taken has been reduced by 1 hour. So the distance that is covered from this increase in speed is:20(t-1).
This is the total extra distance. so then what do you mean when you say:
1." IN the last 1 hour"
2.400 = 20*(t - 1)*t (why are we then multiplying by t again)

Regards
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 15 Oct 2018, 04:56
nitesh50 wrote:
VeritasKarishma wrote:
Yellow22 wrote:
How much time did it take a certain car to travel 400 km?

1) The car travelled the first 200 KM in 2.5 hrs?
2) If the Car's average speed was 20 KM per hour greater than it was, it would have travelled the 400 KM in 1 hour less time than it did.


I guess the solution is already clear to you.

Just for intellectual purposes, look at an alternative method:

If the car actually took t hours to cover 400 km, in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km
This must be the distance it covered in each hour since we are considering average speed.
400 = 20*(t - 1)*t
t(t - 1) = 20
t = 5 hrs




Hi VeritasKarishma
The solution appears to be way shorter, but I don't understand a certain aspect.

"in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km"
THis is the part I am confused about. The speed has been increased by 20. SO the time taken has been reduced by 1 hour. So the distance that is covered from this increase in speed is:20(t-1).
This is the total extra distance. so then what do you mean when you say:
1." IN the last 1 hour"
2.400 = 20*(t - 1)*t (why are we then multiplying by t again)

Regards


Current case: Car travels 400 km in t hrs

Hypothetical case: Car travels 20 km extra for every (t - 1) hrs and covers the distance in (t - 1) hrs. Why did the car take 1 hr less here? Because it travelled some extra distance for (t - 1) hrs. What was that extra distance? 20*(t - 1) km. This is the distance it travels in its current case in the last hour (because the current case speed is lower).

If it travels 20*(t - 1) km in its last hour in our current case, it travels the same 20*(t - 1) in each previous hour too because average speed is considered.

So speed of the car = 20*(t - 1) km/hr (in current case)

20*(t - 1)*t = 400 (speed * time = distance in current case)
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 15 Oct 2018, 05:30
1
fskilnik wrote:
VeritasKarishma wrote:
Just for intellectual purposes, look at an alternative method:

If the car actually took t hours to cover 400 km, in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km
This must be the distance it covered in each hour since we are considering average speed.
400 = 20*(t - 1)*t
t(t - 1) = 20
t = 5 hrs

This is simply beautiful VeritasKarishma. You got my kudos for this reasoning. If it were possible, I would have given you two.

Kind Regards,
Fabio.


Thanks Fabio! Your feedback is much appreciated!
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 16 Oct 2018, 01:02
nitesh50 wrote:
VeritasKarishma wrote:
Yellow22 wrote:
How much time did it take a certain car to travel 400 km?

1) The car travelled the first 200 KM in 2.5 hrs?
2) If the Car's average speed was 20 KM per hour greater than it was, it would have travelled the 400 KM in 1 hour less time than it did.


I guess the solution is already clear to you.

Just for intellectual purposes, look at an alternative method:

If the car actually took t hours to cover 400 km, in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km
This must be the distance it covered in each hour since we are considering average speed.
400 = 20*(t - 1)*t
t(t - 1) = 20
t = 5 hrs




Hi VeritasKarishma
The solution appears to be way shorter, but I don't understand a certain aspect.

"in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km"
THis is the part I am confused about. The speed has been increased by 20. SO the time taken has been reduced by 1 hour. So the distance that is covered from this increase in speed is:20(t-1).
This is the total extra distance. so then what do you mean when you say:
1." IN the last 1 hour"
2.400 = 20*(t - 1)*t (why are we then multiplying by t again)

Regards


Try this:

Attachment:
Ques1.jpeg
Ques1.jpeg [ 47.89 KiB | Viewed 53 times ]

Focus on the (t - 1)th hr. In the hypothetical case, the total distance covered is 20*(t - 1) extra. This is the distance that in current case would have been covered in the last hour. That is why we need 1 hr extra in current case. Since speed S is same every hour, in the last hour too, distance covered in current case would be S, same as distance covered in every hour.
So S = 20*(t - 1) = Distance covered every hour in our current case.

Since we take t hrs to cover the entire distance in current case, 400 = 20*(t - 1)*t
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Re: How much time did it take a certain car to travel 400  [#permalink]

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New post 16 Oct 2018, 07:52
nitesh50 wrote:
VeritasKarishma wrote:
Yellow22 wrote:
How much time did it take a certain car to travel 400 km?

1) The car travelled the first 200 KM in 2.5 hrs?
2) If the Car's average speed was 20 KM per hour greater than it was, it would have travelled the 400 KM in 1 hour less time than it did.


I guess the solution is already clear to you.

Just for intellectual purposes, look at an alternative method:

If the car actually took t hours to cover 400 km, in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km
This must be the distance it covered in each hour since we are considering average speed.
400 = 20*(t - 1)*t
t(t - 1) = 20
t = 5 hrs




Hi VeritasKarishma
The solution appears to be way shorter, but I don't understand a certain aspect.

"in the last 1 hour, the car travels a distance which is equal to 20*(t - 1) km"
THis is the part I am confused about. The speed has been increased by 20. SO the time taken has been reduced by 1 hour. So the distance that is covered from this increase in speed is:20(t-1).
This is the total extra distance. so then what do you mean when you say:
1." IN the last 1 hour"
2.400 = 20*(t - 1)*t (why are we then multiplying by t again)

Regards


Try this:

Attachment:
Ques1.jpeg

Focus on the (t - 1)th hr. In the hypothetical case, the total distance covered is 20*(t - 1) extra. This is the distance that in current case would have been covered in the last hour. That is why we need 1 hr extra in current case. Since speed S is same every hour, in the last hour too, distance covered in current case would be S, same as distance covered in every hour.
So S = 20*(t - 1) = Distance covered every hour in our current case.

Since we take t hrs to cover the entire distance in current case, 400 = 20*(t - 1)*t[/quote]




I cannot begin to explain how this picture has made this question so much easier. Kudos to you!!!
I hope that I can atleast recognise these patterns whenever I see one.
IN any case, this explaination is just too good.!
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Re: How much time did it take a certain car to travel 400 &nbs [#permalink] 16 Oct 2018, 07:52

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