DRT Problem - Verify that I did it correctly?

[markl77]

The question is :
Competing in an endurance race, Mary cycled for 120 km, then swam for 12 km. Her average cycling speed was eight times faster than her average swimming speed. The race took 9 hours. What was her average swimming speed?

cycling
D : 120 km
R : 8x
T : 9

swimming
D : 12 km
R : x
T : 9

I assumed that since we already have all the values for each of the sections in the DRT graph then I can just solve for x using only the "swimming" section.

(12/9)=x
x=1.67 km/h

Am I supposed to do that or make (120/9)+(12/9)=x ?

 

[mmm4444bot]

Competing in an endurance race, Mary cycled for 120 km, then swam for 12 km. Her average cycling speed was eight times faster than her average swimming speed. The race took 9 hours. What was her average swimming speed?

cycling
D : 120 km
R : 8x
T : 9

swimming
D : 12 km
R : x
T : 9

Mary did not cycle nine hours, and she did not swim nine hours. It's the entire endurance race that took nine hours.

We don't know how much of the nine hours she spent cycling or swimming, so we need another variable.

Let t = swimming time

Then what expression represents cycling time?

Now write the D=RT equation for each part of the race.

What can you do with those two equations? :cool:

 

[markl77]

Mary did not cycle nine hours, and she did not swim nine hours. It's the entire endurance race that took nine hours.

We don't know how much of the nine hours she spent cycling or swimming, so we need another variable.

Let t = swimming time

Then what expression represents cycling time?

Now write the D=RT equation for each part of the race.

What can you do with those two equations? :cool:

120= 8x(9)
x=1.666666667

12=9x
x=0.75

(0.75)+(1.666666667) = 2.416666666667 km/h

Is this correct then?

 

[ksdhart2]

120= 8x(9)
x=1.666666667

12=9x
x=0.75

(0.75)+(1.666666667) = 2.416666666667 km/h

Is this correct then?

Well, in the future if you're ever uncertain as to whether an answer is correct or not, you can always check it yourself by plugging it back in and seeing if the original constraints are satisfied. Let's start with your original answer of 12/9, which you've (wrongly decided) is equal to 5/3 (~1.67).

x represents Mary's average swimming speed, so 8x represents her average cycling speed. She swam for 12 km at 12/9 = 4/3 km/h, so she spent 9 hours swimming. But, hold on... you're told that the entire race took 9 hours, so that only leaves her just 0 hours to complete the cycling portion. Therefore, this answer cannot be correct.

Now let's try your new proposed solution of x = 2.416666666667 = 29/12 km/h. At that pace, Mary would complete the swimming portion in ~4.96 hours, leaving her ~4.94 hours for the cycling portion. Her cycling speed is 8x = 58/3 km/h. At that pace, she'd need ~6.21 hours to complete it. But 4.96 + 6.21 > 9. Oops! That solution is not correct either.

Instead, let's now actually use the hint provided to you by mmm4444bot. Using the variable t to stand for the time Mary spent on the swimming portion of the race and revisiting your original table:

cycling
D : 120 km
R : 8x
T : 9 - t

swimming
D : 12 km
R : x
T : t

You know that Distance = Rate * Time, so let's make two equations out of this information. 120 = 8x(9 - t) and 12 = xt. Now you try finishing up from here.

 

[mmm4444bot]

120= 8x(9)

12=9x

Is this correct then?

No. You're still using 9 hours for the time spent cycling, and you're still using 9 hours for the time spent cycling.

You do not know how many hours Mary cycled, so you cannot state that she spent the entire 9 hours of the race cycling.

You do not know how many hours Mary swam, so you cannot declare that she spent the entire 9 hours of the race swimming, either.

ksdhart showed you how to represent these unknown times symbolically.

Time spent swimming = t

Time spent cycling = 9 - t

Does this make sense? The expression 9 - t gives the difference between 9 hours and the hours spent swimming. In other words, if you subtract the amount of swimming time from the total time, what's left over must be the cycling time.

ksdhart also gave you the two D=RT equations, using x and t:

120 = (8x)(9 - t)

12 = (x)(t)

We call this a "System of Two Linear Equations in Two Variables". I'm not sure what your class has covered, so far, but this system can be solved using "The Substitution Method". This method involves solving the second equation for t, and then substituting the resulting expression for t in the first equation. You will then have an equation containing only the variable x, and you can solve for that.

If you're stuck trying to solve this system, please show us what you tried. If none of this looks familiar, then tell us what your class has been talking about, lately.

Cheers :cool:

 

[markl77]

No. You're still using 9 hours for the time spent cycling, and you're still using 9 hours for the time spent cycling.

You do not know how many hours Mary cycled, so you cannot state that she spent the entire 9 hours of the race cycling.

You do not know how many hours Mary swam, so you cannot declare that she spent the entire 9 hours of the race swimming, either.

ksdhart showed you how to represent these unknown times symbolically.

Time spent swimming = t

Time spent cycling = 9 - t

Does this make sense? The expression 9 - t gives the difference between 9 hours and the hours spent swimming. In other words, if you subtract the amount of swimming time from the total time, what's left over must be the cycling time.

ksdhart also gave you the two D=RT equations, using x and t:

120 = (8x)(9 - t)

12 = (x)(t)

We call this a "System of Two Linear Equations in Two Variables". I'm not sure what your class has covered, so far, but this system can be solved using "The Substitution Method". This method involves solving the second equation for t, and then substituting the resulting expression for t in the first equation. You will then have an equation containing only the variable x, and you can solve for that.

If you're stuck trying to solve this system, please show us what you tried. If none of this looks familiar, then tell us what your class has been talking about, lately.

Cheers :cool:

Okay, thankyou!

My teacher has taught us to never use more than one variable for these questions so I never used one, but I can see that it makes more sense if you do use more than one. The thing is, she wants us to incorporate rational expressions/equations in the word problems so we are supposed to set them up as fractions, which I kind of get but not really.
We learned substitution/elimination early on in the year so I forgot about it, but I remember how to do it still I'm pretty sure.

Here is my work from using the 2 variables and the equations for the DRT.

120 = 8x(9-t)
120= 72x-96
x=3km/h
12/3 = 4 hours

120/24
=5 hours

5+4 = 9 hours, Mary spent 4 hours swimming and 5 hours cycling. The rate at which she swam at was 3 km/h and the rate at which she cycled at was 24 km/h

I really hope that's right ...

 

[stapel]

Here is my work from using the 2 variables and the equations for the DRT.

120 = 8x(9-t)
120= 72x-96
x=3km/h
12/3 = 4 hours

120/24
=5 hours

5+4 = 9 hours, Mary spent 4 hours swimming and 5 hours cycling. The rate at which she swam at was 3 km/h and the rate at which she cycled at was 24 km/h

I really hope that's right ...

Please try following the instructions provided earlier, regarding how to check your answers.

 

[mmm4444bot]

My teacher has taught us to never use more than one variable for these questions … she wants us to incorporate rational expressions/equations in the word problems so we are supposed to set them up as fractions … We learned substitution/elimination early on in the year …

It seems a bit odd to first teach methods related to systems of two equations in two variables but then instruct students to never use two variables.

Perhaps, your instructor expects you to memorize and use the following three equations (each of which relates distance traveled, rate of travel, and traveling time).

D = RT

R = D/T

T = D/R

In this case, yes, you could mentally jump to the third equation, and then say:

Time spent swimming = 12/x

Therefore:

120 = (8x)(9 - 12/x)

Doing this mental work avoids having to create an extra variable and write a system of two equations, but the end result is the same as what we've showed you.

My viewpoint is that students who correctly reason their way through an exercise and do good work ought to be congratulated first and shown other methods second. It's true: many word problems that can be solved by setting up a system of two equations can also be solved by using shortcuts involving only one variable. For me, when I did not realize or understand shortcuts, I worked problems in a way that already made sense. I see no issue with using a system of two equations, to reason through your exercise and create an equation in x to solve. :cool:

Here is my work from using the 2 variables and the equations for the DRT.

120 = 8x(9-t)
120= 72x-96
x=3km/h
12/3 = 4 hours

120/24
=5 hours

5+4 = 9 hours, Mary spent 4 hours swimming and 5 hours cycling. The rate at which she swam at was 3 km/h and the rate at which she cycled at was 24 km/h

I really hope that's right

Let's check it!

D: 120
R: 24
T: 5

120 = (24)(5)
120 = 120
Check!

D: 12
R: 3
T: 4

12 = (3)(4)
12 = 12
Check!

Mary cycles 8 times faster than swimming
(8)(3) = 24
24 = 24
Check!

The entire race took 9 hours
4 hr + 5 hr = 9 hr
Check!

Good work!

Oops. I did not see stapel's last post. (I did the check for you.)

 

[Steven G]

My viewpoint is that students who correctly reason their way through an exercise and do good work ought to be congratulated first and shown other methods second.

I agree with you 100%. Students need to learn how to think mathematically on their own. Teacher should never tell students how they must do a problem. They should allow students to think for themselves. My students sadly tell me that they were never asked to think in their math classes before coming to my class. I tell my students NEVER to believe what any of their teachers told them (including me) but rather go home and convince themselves that what they were told is correct. The main reason that students do poorly in math is that they never understand the basics.

My daughter is in 6th grade and she just learned how to compute the are of a rectangle. Given a 3in by 2in rectangle she would say that the area is 6sq inches or sometimes 6 inches. The reason is that her incompetent teacher never showed the class a 2by 3 rectangle with the 6 square inches in them. NEVER showed this! That is amazing.