Khalil drove 120 kilometers in a certain amount of time. What was his

We know that the distance is 120. Let r be his average speed. The original question: r=?

1) At the old speed, his driving time is 120/r. At the new speed, his driving time would be 120/(r+5). We can set up an equation about his new driving time, which is 1/3 hour less than his old driving time.

\(\frac{120}{r+5}=\frac{120}{r}-\frac{1}{3}\)

\(360r=360r+1800-r^2-5r\)

\(r^2+5r-1800=0\)

Since c/a is negative for the quadratic equation above, its two roots have different signs. However, only its positive root applies to the context of the problem.

Thus, we could get a unique value to answer the original question. \(\implies\) Sufficient

2) At his new speed, his driving time would be 120/(1.25r). We can set up an equation about his new driving time, which is 20% less than his old driving time.

\(\frac{120}{\frac{5}{4}r}=\frac{4}{5}\cdot \frac{120}{r}\)

The above equation is an identity. Since r can be any positive number, we can't get a unique value to answer the original question. \(\implies\) Insufficient

Answer: A

given : \(vt = 120\)

from statement (1), \((v+5)(t-\frac{1}{3}) = 120\)
\(v + 5 = 15t = 15(\frac{120}{v})\)
so \(t = 3\) and \(v = 40\) ---> sufficient

from statement (2), \((\frac{5}{4})v*(\frac{4}{5})t = 120\) ---> no use because the fractions cancel each other (insufficient)

given total distance = 120 km
actual speed =x
target total distance / total time = avg speed
#1
If Khalil had driven at an average speed that was 5 kilometers per hour faster, his driving time would have been reduced by 20 minutes.
120/x -120/x+5 = 20/60
we can solve for x ; sufficient
#2
If Khalil had driven at an average speed that was 25% faster, his driving time would have been reduced by 20%.
120/1.25x = 20/100 * time
here we have two un knowns insufficient
OPTION A sufficient

Can someone help me understand why this equation I made is wrong, for statement 2?

(2) If Khalil had driven at an average speed that was 25% faster, his driving time would have been reduced by 20%.

Average speed = Total Distance / Total Time.

1.25(avg. speed)=.8(total time) -> 1.25(120/X)=.8X

ZoltanBP the highlighted above is a really neat insight. thanks for sharing!

How does V + 5 = 15t? I can't seem to wrap my head around it.

Bunuel, kindly assist.

ZoltanBP Bunuel My question is specifically around this statement. What happens in other cases(If C/A is positive etc). I am aware of discriminant properties but not heard of this before. Can you please help. This will save a lot of time as I tried to solve the quadric because I wasn't aware of this rule. Do we have any post around this?

"Since c/a is negative for the quadratic equation above, its two roots have different signs. However, only its positive root applies to the context of the problem."

distance = rate * time
rate = distance / time
average speed = total distance / total time

120 = rt

(1)

original time = \(\frac{120}{r}\)
new time = \(\frac{120}{r+5}\)

\(\frac{120}{r+5} = \frac{120}{r} - \frac{1}{3}\)
\(360r = (r+5)(360-r)\)

We can determine an average speed. SUFFICIENT.

(2) We're told that if Khalil had driven 25% faster, his driving time would be reduced by 20%. We can't derive a value from these percentages. INSUFFICIENT.

Answer is A.
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Alternate approach:

Statement 1:
If the actual speed = 1 kph (yielding a time of 120 hours for the 120-mile distance) and the greater speed = 6 kph (yielding a time of 20 hours for the 120-mile distance), the time difference = 120-20 = 100 hours.
If the actual speed = 5 kph (yielding a time of 24 hours for the 120-mile distance) and the greater speed = 10 kph (yielding a time of 12 hours for the 120-mile distance), the time difference = 24-12 = 12 hours.
As the actual speed INCREASES, the time difference DECREASES.
Implication:
If we keep increasing the actual speed, eventually we will discover the speed required to yield a time difference of 20 minutes.
SUFFICIENT.

Statement 2:
Here -- because rate and time have a RECIPROCAL relationship -- ANY SPEED is possible.
If Khalil travels 25% faster -- in other words, if he travels at 5/4 the actual speed -- he will require 4/5 the actual time, a time decrease of 20%.
Since the information in Statement 2 holds true for any speed, INSUFFICIENT.


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