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Math Expert V
Joined: 02 Sep 2009
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How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 65% (01:12) correct 35% (01:25) wrong based on 811 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.
(2) Betty's average speed for the trip was 50 miles per hour.

Data Sufficiency
Question: 71
Category: Arithmetic Distance/ rate problems
Page: 158
Difficulty: 600

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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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SOLUTION

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

Time = Distance/Rate ?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.

(3/2*Rate)*2 = Distance;

Distance/Rate = 3. Sufficient.

(2) Betty's average speed for the trip was 50 miles per hour. Only the rate is clearly not sufficient, to get the time. Not sufficient.

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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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Answer A is sufficient.

it is the same distance - so S1*T1=S2*T2

let us assume that the original speed is x and original time t

x*t=3/2x*2
t=3x/x
t=3

the trip took 3 hours.

The answer B is not sufficient because we only know the speed but not distance.
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GMAT 3: 640 Q49 V29 Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.
(2) Betty's average speed for the trip was 50 miles per hour.

(1)
We need to know how long it took Betty to drive, so time=distance/speed.

Let's denote:
s1 is average speed
t1 is the time needed to complete the journey when travelling at speed s1
s2 is the average speed when travelling at faster speed
t2 is the time needed to complete the journey when travelling at speed s2.

Then,
t1=d/s1

But also as per (1):
s1=3/2*s2
or s2=2/3*s1 (equation #1)

And:
t2=d/s2=2hours (from statement 1)

But also as per (eq. #1):
t2= d/s2 = d/((2/3)*s1) = 2 hours --> t2 = (2/3)*(d/s1) = 2 --> (2/3)*t1 = 2 --> t1= 3 hours

Hence,
(1) is SUFFICIENT

(2) Nothing is said about distance between "her home" and Colorado, hence we have no means to determine t or d/s.
(2) IS INSUFFICIENT

Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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SOLUTION

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

Time = Distance/Rate ?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.

(3/2*Rate)*2 = Distance;

Distance/Rate = 3. Sufficient.

(2) Betty's average speed for the trip was 50 miles per hour. Only the rate is clearly not sufficient, to get the time. Not sufficient.

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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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Even though you can plug in numbers to solve this question, it is important to notice one important algebraic caveat in this type of problems.

Statement 1 is sufficient. However it provides only 1 equation with 2 variables.
R (1.5) (2) = D

At first glance, it looks like you cannot solve this using the Equation Rule of Sufficiency (the one that states that "you need n number of distinct, linear equations to solve for n variables..."). The catch is that all Distance problems are already giving us 1 equation and 3 variables, namely R * T = D.

So when you look at statement 1 you actually have 2 equations and 3 variables

(1.5) (2) = D/R
T = D/R

If you substitute, you kill one variable and thus you can solve.

(1.5) (2) = T
3 = T

Statement 2 is insufficient.
The equations you have are:
T = D/R
R = 50
3 Variables, 2 Equations --> Insufficient.
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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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1
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.
(2) Betty's average speed for the trip was 50 miles per hour.

In the original condition, from vt=d, there are 3 variables(v,t,d) and 1 equation(vt=d), which should match with the number of equations. So you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. When 1) &2), you can easily find out that C is the answer. However, in 1), you can get 2 equations(time and velocity). That is, use vt=d from (3/2)v*2=d and you can get t=3, which is sufficient. Therefore, the answer is A.

 For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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st 2: gives the speed but we are missing the distance
St 1: gives us Distance = 3/2 speed * 2
and we know that same Distance = Speed * time.
Hence time = 3 hours
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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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st 2: gives the speed but we are missing the distance
St 1: gives us Distance = 3/2 speed * 2
and we know that same Distance = Speed * time.
Hence time = 3 hours

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Re: How long did it take Betty to drive nonstop on a trip from h  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

How long did it take Betty to drive nonstop on a trip from her home to Denver, Colorado?

(1) If Betty's average speed for the trip had been 3/2 times as fast, the trip would have taken 2 hours.
(2) Betty's average speed for the trip was 50 miles per hour.

Target question: What was Betty's travel time?

This is a great candidate for rephrasing the target question

Let D = The distance driven
Let R = Betty's average speed

Since Time = Distance/Rate, we can rephrase the target question. . .

REPHRASED target question: What is the value of D/R?

Statement 1: If Betty's avg speed for the trip had been 1.5 times as fast, the trip would have taken 2 hours.
The distance is still the same (D), but the rate is 1.5R
So, statement 1 tells us that D/1.5R = 2
Is this enough information to find the value of D/R?
Sure, just take the equation D/1.5R = 2 and multiply both sides by 1.5 to get D/R = 3
Since we're able to answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: Betty's avg speed for the trip was 50 miles per hour.
This tells us that R=50
However, we cannot find the value of D/R, since we don't know the value of D (distance)
So, statement 2 is NOT SUFFICIENT

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_________________ Re: How long did it take Betty to drive nonstop on a trip from h   [#permalink] 24 Apr 2019, 17:12
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