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Tricky average speed calculation

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utkalnayak Rising GMAT Star
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Tricky average speed calculation Post Sat Jan 24, 2015 5:15 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    Susan drove at an average speed of 30 miles per hour for the first 30 miles of a trip and then at an average speed of 60 miles per hour for the remaining 30 miles of the trip. If she made no stops during the trip, what was Susan's average speed, in miles per hour for the entire trip ?

    A. 35
    B. 40
    C. 45
    D. 50
    E. 55

    OA: B

    Is this a trick question, what is the distractor here?

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    Post Sat Jan 24, 2015 5:25 pm
    Hi utkalnayak,

    To figure out Average Speed for an entire trip, we need TOTAL Distance and TOTAL Time:

    Total Distance = (Average Speed)(Total Time)

    Here, we're given all of the numerical information we need to calculate each part of the journey:

    1st part: 30 miles/hour for 30 miles

    30 miles = (30mph)(Time)
    30/30 = T hours
    T = 1 hour

    2nd part: 60 miles/hour for 30 miles
    30 miles = (60mph)(Time)
    30/60 = T hours
    T = 1/2 hour

    Total Distance = 30 + 30 = 60 miles
    Total Time = 1 + 1/2 = 1.5 hours

    Using these values, we can plug in and find the Average Speed for the entire trip:

    60 miles = (Average Speed)(1.5 hours)
    60/1.5 = Average Speed

    Average Speed = 40 mph

    Final Answer: B

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    Post Sat Jan 24, 2015 5:27 pm
    utkalnayak wrote:
    Susan drove at an average speed of 30 miles per hour for the first 30 miles of a trip and then at an average speed of 60 miles per hour for the remaining 30 miles of the trip. If she made no stops during the trip, what was Susan's average speed, in miles per hour for the entire trip ?

    A. 35
    B. 40
    C. 45
    D. 50
    E. 55
    Average speed = (TOTAL distance)/(TOTAL time)

    TOTAL distance
    Susan traveled 30 miles at a speed of 30mph, and 30 miles at a speed of 60mph.
    So, TOTAL distance = 60 miles

    TOTAL time
    TOTAL time = (time spent driving 30 mph) + (time spent driving 60 mph)
    time = distance/speed
    - Time spent driving 30 mph = 30 miles/30mph = 1 hour
    - Time spent driving 60 mph = 30 miles/60mph = 0.5 hours
    TOTAL time = (1 hour) + (0.5 hours) = 1.5 hours
    ----------------------------------

    So, Average speed = (TOTAL distance)/(TOTAL time)
    = 60 miles/1.5 hours
    = 40 mph
    = B

    Cheers,
    Brent

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    Post Sat Jan 24, 2015 5:38 pm
    If there is a trick to the question it's to realize that you cannot just average the two speeds to get the average speed.

    Average speed is not the average over distance, it's the average over time.

    So even though she went 30 miles per hour for the first thirty miles and sixty miles per hour for another 30 miles, her average speed was not 45 miles per hour, because she spent less time going sixty miles per hour.

    That's why you need to calculate Total Distance/Total Time as discussed above.

    To keep it all straight it can help to remember that Rate x Time = Distance. So Rate = Distance/Time.

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    Post Sat Jan 24, 2015 8:10 pm
    utkalnayak wrote:
    Susan drove at an average speed of 30 miles per hour for the first 30 miles of a trip and then at an average speed of 60 miles per hour for the remaining 30 miles of the trip. If she made no stops during the trip, what was Susan's average speed, in miles per hour for the entire trip ?

    A. 35
    B. 40
    C. 45
    D. 50
    E. 55
    When the SAME DISTANCE is traveled at two different speeds, the average speed for the entire trip will be A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.
    Here, the same distance -- 30 miles -- is traveled at 30mph and 60mph.
    Since (30+60)/2 = 45, the average speed for the entire trip must be A LITTLE LESS THAN 45.

    The correct answer is B.

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    Post Sat Jan 24, 2015 10:28 pm
    For uniform speeds and linear problems,

    You can directly calculate the total average speed by the following formula:

    Total Average Speed = Total Distance/(((Distance 1)/(Avg. Speed 1))+((Distance 2)/ (Avg. Speed 2))+...)
    where Total Distance = Distance 1 + Distance 2 +...

    Here we know that,
    Distance 1 = 30miles, Distance 2 = 30miles
    Total Distance = 30miles + 30miles = 60miles
    Avg. Speed 1 = 30 miles/hr
    Avg. Speed 2 = 60 miles/hr

    Therefore by substituting,

    Total Average Speed = 60/((30/30)+(30/60))=60/(1+(1/2))=60/(3/2)=60*2/3= 40 miles/hr

    Post Sun Jan 25, 2015 1:27 pm
    Hi, utkalnayak.

    One more way to attack this problem: we can think of the calculation as a weighted average that we can evaluate on the number line. If we spend 1 hour going 30 mph, and 1/2 hour going 60mph, then we're spending twice as much time traveling at 30 mpg than at 60 mph. (Put another way, keeping distance constant, if we travel at half the speed, we'll have to travel for twice the time.) Let's call our overall average speed 'A.' Now we can evaluate like so:

    30---A------60

    Gap from 30 to A: x
    Gap from A to 60: 2x

    x + 2x = 60 - 30
    3x = 30
    x = 10
    30 + 10 = 40
    (Or 60 - 20 = 40)

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    Post Sun Jan 25, 2015 3:21 pm
    A handy formula to shelve away: if you travel half the distance at speed x and half the distance at speed y, your average speed for the whole trip is 2/(1/x + 1/y), which can also be expressed as 2xy/(x+y). This isn't the most important formula on the GMAT, but this concept comes up often enough that it's worth noting.

    If you're wondering why this is, let's do a quick demo. We'll start with the example in your problem (Susan's trip), then generalize to ANY trip.

    Susan's trip has three parts:

    first half:
    D = 30
    R = 30
    Time = 30/30 = 1

    second half
    D = 30
    R = 60
    Time = 30/60 = 1/2

    Now that we have her distance and her trip, we can apply them to the whole trip.
    D = 60
    T = 3/2
    Rate = 60 / (3/2) = 40

    As you probably guessed, this is easy to generalize to any trip. Suppose I take a trip and spend half the distance at speed x and half the distance at speed y. This gives us

    first half
    D = d
    R = x
    Time = d/x

    second half
    D = d
    R = y
    Time = d/y

    whole trip
    D = 2d
    T = d/x + d/y
    Rate = 2d / (d/x + d/y) = 2/(1/x + 1/y) = 2xy/(x + y)

    Pretty neat, and obviously useful on these sorts of questions.

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