tough DS Question(Speed Question)

[aj_gmatwizard]

Rachel drove the 120 miles from A to B at a constant speed. What was this speed?

Statement #1: If she had driven 50% faster, her new time would have been 2/3 of her original time.

Statement #2: If she had driven 20 mph faster, she would have arrived an hour sooner.

Answer:
Rachel drove the 120 miles from A to B at a constant speed. What was this speed?

Let's say the original variables are D = 120, R, and T, and the original case is 120 = RT.

Statement #1: If she had driven 50% faster, her new time would have by 2/3 of her original time.

To increase by 50%, we will multiply by the multiplier 1.5. This means that the new speed, R2, is R2 = 1.50*R = (3/2)*R. The new time is T2 = (2/3)*T. Well, we know, that 120 = (R2)*(T2) --- the new speed and time must have the small product as the original speed and time. 120 = (R2)*(T2) = [(3/2)*R]*[(2/3)*T] = (3/2)*(2/3)*RT = RT. This information leads in a big logical circle right back to the original equation 120 = RT. It gives us no new information at all. Therefore, this statement, by itself, does not provide any insight into the answer to the prompt question. This statement is insufficient.

Statement #2: If she drove 20 mph faster, she would have arrived an hour sooner.

Now, we know R2 = R + 20 and T2 = T - 1. Again, we know that 120 = (R2)*(T2), so

120 = (R + 20)*(T - 1) = RT - R + 20T - 20

120 - RT = 0 = 20T - R - 20

That is one equation with two unknowns, and 120 = RT is a second equation with two unknowns. We have two clearly different equations for the two unknowns, so even without solving, this is sufficient to determine a unique value for the variables R & T.

Answer = (B)

I don't understand the second part. we cant solve the equations 120=RT and 120-RT=0=20T-R-20

20T-R=20 this just 1 equation and two unknowns[/b]

[GMATGuruNY]

Rachel drove the 120 miles from A to B at a constant speed. What was this speed?

Statement #1: If she had driven 50% faster, her new time would have been 2/3 of her original time.

Statement #2: If she had driven 20 mph faster, she would have arrived an hour sooner.

Statement 1:
RATE and TIME are RECIPROCALS.
If Rachel were to travel at 3/2 her actual rate, the trip would take 2/3 her actual time.
This relationship will hold true for ANY RATE.
Implication:
The rate can be any value.
INSUFFICIENT.

Statement 2:
Test rates that are 20mph apart and that divide cleanly into d=120.
Look for a time difference of 1 hour.
10mph and 30mph --> 120/10 = 12 hours, 120/30 = 4 hours.
20mph and 40mph --> 120/20 = 6 hours, 120/40 = 3 hours.
40mph and 60mph --> 120/40 = 3 hours, 120/60 = 2 hours.
The option in red works.
Thus, the actual rate = 40mph.
SUFFICIENT.

The correct answer is B.

[[email protected]]

Hi aj_gmatwizard,

Since a couple of explanations have already been presented, I won't rehash any of that work here. Instead, I'm going to offer a bit more explanation as to why Fact 1 is insufficient. It's actually based on a "math truism", which in real simple terms is a "math fact that most people don't realize."

The Distance Formula is...

Distance = (Rate)(Time)

....and it's a formula that you'll likely use a couple of times on the Official GMAT (so it's worth knowing).

Fact 1 tells us that increasing the rate by 50% would lead to a time that is 2/3 of the original. Using the Distance Formula, we can prove that this occurs for ANY speed...

D = (R)(T)

Raising the Rate by 50% and decreasing the Time to 2/3 of the original time gives us...

(3/2)(R)(2/3)(T).

Since we're multiplying 'elements', we can put them in any order:

(3/2)(2/3)(R)(T)

(3/2)(2/3) = 6/6 = 1

So we have (1)(R)(T)

D = (R)(T)

In the end, the information in Fact 1 is NOT new information - it's a math fact when dealing with ANY speed and nothing more. Fact 1 is insufficient.

GMAT assassins aren't born, they're made,
Rich

[Brent@GMATPrepNow]

Rachel drove the 120 miles from A to B at a constant speed. What was this speed?

Statement #1: If she had driven 50% faster, her new time would have been 2/3 of her original time.
Statement #2: If she had driven 20 mph faster, she would have arrived an hour sooner.

Here's an algebraic solution for statement 2 that uses only 1 variable.

Target question: What was Rachel's speed?

Statement 1: If she had driven 50% faster, her new time would have been 2/3 of her original time.
Mitch has already done a great job (as usual) explaining why statement 1 is NOT SUFFICIENT, so I'll defer you to his solutions.

Statement 2: If she had driven 20 mph faster, she would have arrived an hour sooner.
Let's start with a "word equation"
When driving her regular speed, Rachel's trip time is 1 HOUR LONGER than her trip time when driving 20 mph faster.
So, let's write the following:
(Rachel's trip time at REGULAR speed) - (Rachel's trip time at FASTER speed) = 1 hour

From here, let's let R = her REGULAR speed
This means R + 20 = her FASTER speed

Since trip time = distance/speed, we can now write the following:
120/R - 120/(R+20) = 1 hour
Now solve the equation for R

Multiply both sides by (R)(R+20) to get: 120(R+20) - 120R = (R)(R+20)
Expand to get: 120R + 2400 - 120R = R² + 20R
Rearrange and simplify to get: R² + 20R - 2400 = 0
Factor: (R + 60)(R + 40) = 0
So, R = -60 or R = 40
Since Rachel's speed cannot be negative, it MUST be the case that R = 40
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

-------------------------------------------
Word problems are tricky for A LOT of students. One of the hardest steps is creating the corresponding equation(s). In many cases, I like to create a "WORD EQUATION" as an intermediary step.

Here's an article I wrote about this strategy: https://www.beatthegmat.com/mba/2014/09/ ... -equations

Here are a few questions where I've used a word equation:
- https://www.beatthegmat.com/insurance-t278612.html
- https://www.beatthegmat.com/equation-t107935.html
- https://www.beatthegmat.com/speed-distan ... 80473.html
- https://www.beatthegmat.com/stuck-with-m ... 63906.html
- https://www.beatthegmat.com/rate-distanc ... 21963.html
- https://www.beatthegmat.com/distance-wor ... 81078.html

Cheers,
Brent