David knew he made a mistake when he calculated that Gilda walks 123 miles to the station. Read through David's calculations:

Question

Gilda walks to the train station. If she walks at the rate of 3 mph, she misses her train by 7 minutes. However, if she walks at the rate of 4 mph, she reaches the station 5 minutes before the arrival of the train. Find the distance Gilda walks to the station.

 

David knew he made a mistake when he calculated that Gilda walks 123 miles to the station. Read through David's calculations:

Using d = rt, the distance is the same, but the rate and time are different.
If Gilda misses the train, it means the time t needs 7 more minutes so d = 3(t + 7).
If she gets to the station 5 minutes early means the time t can be 5 minutes less so d = 4(t - 5).
3(t + 7) = 4(t - 5)
3t + 21 = 4t - 20
t = 41
d = rt, so d = 3(41) = 123

Find David's mistake in his calculations. In two or more complete sentences, explain his mistake. Include the correct calculations and solutions in your answer.

Answer 1

One of the huge mistakes of D-I-R-T (Distance-Is-Rate-times-Time) problems is using inconsistent units of measurement – for example, David must work with both minutes and hours.

We could convert miles per hour to miles per minute or we could convert minutes to a fraction of an hour.  Or – if we wanted to seriously avoid factions (I have trouble with fractions), we could convert the rate (speed) to feet per minute:

 

3 mi/hr = (3 mi/hr) * (5280 ft/mi) * ( (1 hr)/(60 min) )  =  264 ft/min

4 mi/hr = (3 mi/hr) * (5280 ft/mi) * ( (1 hr)/(60 min) )  =  352 ft/min

 

The distance is the same.     [note: this makes D-I-R-Tproblems possible to solve]

 

Let T = arrival time of train (after Gilda starts walking)

   D = (264 ft/min) (T+7 min)    =   (352 ft/min)(T-5 min)

               [note: looks like David’s calculation, but with correct units]

           264T + 1848  =  352T – 1760

              20238 = 88T

              41 minutes = T

 

  We could use either formula to find D:

      D = 264(48) =  12672

 

Another big mistake students, teachers, and tutors make is failing to check their work.  Let’s check:

At 264 ft/min (3 mph), Gilda walks 12672 ft in (12672/264) 48 minutes.  She is 7 minutes late.

At 353 ft/min (4 mph), Gilda walks 12672 ft in (12672/352) 36 minutes.  She is 5 minutes early.

   

Answer 2

David measured speed in miles per hour, mph, yet calculated the time in minutes.