Number of Minutes

[harpazo]

A bus traveling at an average rate of 50 kilometers per hour made the trip to town in 6 hours. If it had traveled at 45 kilometers per hour, how many more minutes would it have taken to make the trip?

Solution

Let 6 hours = 360 minutes.
Let m = number of minutes

I think a proportion can be used to solve for m.

50/45 = 360/m

A. Is this the correct proportion set up?
B. How can I use the formula D = rt to find the number of minutes?

Note: D = distance, r = rate, t = time (in minutes)

 

[Dr.Peterson]

Your proportion claims that speeds are directly proportional to times. Does that sound right? How do you know it isn't inversely proportional?

I'd probably use d = rt to make sure I got things right. That would require either converting units to match (i.e. if you use km/h, you need to keep time in hours and convert at the end), or just using the formula to remind yourself of the appropriate proportion.

 

[Otis]

… Let m = number of minutes …

That definition is not clear, harpazo. We need to assign the specific time interval that we're asked to find.

Let m = the number of additional minutes to make the trip at 45kph

50/45 = 360/m

… Is this the correct proportion set up? …

No.

50/45 = (360 + m)/360

Recall that proportion A/B = C/D represents A·D = B·C

… How can I use the formula D = rt to find the number of minutes?

D is the same for both trips, so set the product r·t from the 50kph trip equal to the product r·t from the 45kph trip, and then solve that equation for m. (That equation will match the form A·D=B·C, from the proportion above.)

?

 

[harpazo]

That definition is not clear, harpazo. We need to assign the specific time interval that we're asked to find.

Let m = the number of additional minutes to make the trip at 45kph


No.

50/45 = (360 + m)/360

Recall that proportion A/B = C/D represents A·D = B·C


D is the same for both trips, so set the product r·t from the 50kph trip equal to the product r·t from the 45kph trip, and then solve that equation for m. (That equation will match the form A·D=B·C, from the proportion above.)

?

Are you saying to set it up like this?

50(6) = 45t, where t in time in hours

 

[Steven G]

Please do not write let this = that if it is an identity. That is please don't write let 6hrs = 360 minutes, instead write note that 6 hrs = 360 minutes.

r1 * t1 = d1 and r2 * t2 = d2.

 

[harpazo]

Please do not write let this = that if it is an identity. That is please don't write let 6hrs = 360 minutes, instead write note that 6 hrs = 360 minutes.

r1 * t1 = d1 and r2 * t2 = d2.

50(6) = 45t, where t in time in hours.

Yes?

 

[Dr.Peterson]

50(6) = 45t, where t in time in hours.

Yes?

Yes, though you may need to be more careful in defining what you mean by t. (It isn't the answer to the problem.)

Jomo's comment, whose meaning you may have missed, is that when you say, "Let 6 hours = 360 minutes", you are saying that you are choosing to define this as so. (That's what "let" means, though we seldom explain it.) You are merely saying that this is a fact, so you should omit the word "let". Just say, perhaps, "Note that 6 hours = 360 minutes".

 

[Otis]

Are you saying to set it up like this?

50(6) = 45t, where t in time in hours

Almost, harpazo. When I said 'solve for m", it was a hint to express the longer time in terms of m (not t).

You'd already picked a symbol to represent the answer. You let m = the number of additional minutes. So, we need to form an equation containing symbol m.

This is the usual process, for setting up and solving basic word problems: First assign a symbol to represent the requested number, then use it to obtain an equation and to solve for it.

Here's another hint: Using your definition for symbol m, we can easily express the longer time as m minutes more than the slower time.

?

 

[harpazo]

Yes, though you may need to be more careful in defining what you mean by t. (It isn't the answer to the problem.)

Jomo's comment, whose meaning you may have missed, is that when you say, "Let 6 hours = 360 minutes", you are saying that you are choosing to define this as so. (That's what "let" means, though we seldom explain it.) You are merely saying that this is a fact, so you should omit the word "let". Just say, perhaps, "Note that 6 hours = 360 minutes".

Here is my work for m.

50(6) = 45m

300 = 45m

300/45 = m

6.7 = m

Note: I rounded to the nearest tenths.

The number of additional minutes to make the trip at 45kph is 6.7. Can we have minutes as decimals or fractions?

 

[Dr.Peterson]

Here is my work for m.

50(6) = 45m

300 = 45m

300/45 = m

6.7 = m

Note: I rounded to the nearest tenths.

The number of additional minutes to make the trip at 45kph is 6.7. Can we have minutes as decimals or fractions?

But the LHS is the length of the normal trip in kilometers, and the RHS is a speed in km/hour times a number of minutes, which is not the distance in kilometers!

And even if you went back to the correct equation you had before, with t in hours, that would not be the number of additional hours, but the total time in hours. You have to subtract the normal time from that to get the additional time (and convert to minutes to answer the question).

This is why we emphasize defining variables fully:

m = additional time in minutes​

vs

t = total time in hours​

You claim m is the former, but the equation is appropriate for the latter.

 

[harpazo]

But the LHS is the length of the normal trip in kilometers, and the RHS is a speed in km/hour times a number of minutes, which is not the distance in kilometers!

And even if you went back to the correct equation you had before, with t in hours, that would not be the number of additional hours, but the total time in hours. You have to subtract the normal time from that to get the additional time (and convert to minutes to answer the question).



This is why we emphasize defining variables fully:

m = additional time in minutes​

vs

t = total time in hours​

You claim m is the former, but the equation is appropriate for the latter.

Are you saying I must subtract 6.7 minutes from 6 hours?

 

[Dr.Peterson]

No. It isn't 6.7 minutes in the first place! And if you subtracted the new total time from the original, that would be how many fewer minutes, not how many more!

Don't try to guess what I said; go back to the problem and compare what you did to what you had to do.

 

[Otis]

50(6) = 45m

Ah, I didn't notice that you'd switched from 360 minutes to 6 hours. (Sorry about that.)

50(360) = 45(???)

The trip time is 360 minutes, at 50kph.

At 45kph, the trip time is m minutes longer.

Here's another way to get that equation. Start with the corrected proportion from earlier (shown below) and then write A·D=B·C.

50/45 = (360 + m)/360

Recall that proportion A/B = C/D represents A·D = B·C

?

 

[harpazo]

No. It isn't 6.7 minutes in the first place! And if you subtracted the new total time from the original, that would be how many fewer minutes, not how many more!

Don't try to guess what I said; go back to the problem and compare what you did to what you had to do.

We add 6.7 minutes to 360 minutes.

 

[harpazo]

Ah, I didn't notice that you'd switched from 360 minutes to 6 hours. (Sorry about that.)

50(360) = 45(???)

The trip time is 360 minutes, at 50kph.

At 45kph, the trip time is m minutes longer.

Here's another way to get that equation. Start with the corrected proportion from earlier (shown below) and then write A·D=B·C.


?

Is my answer correct? Must I now add 6.7 to 360 minutes?

 

[MarkFL]

A bus traveling at an average rate of 50 kilometers per hour made the trip to town in 6 hours. If it had traveled at 45 kilometers per hour, how many more minutes would it have taken to make the trip?

I would begin by writing:

[MATH]d=vt\implies t=\frac{d}{v}[/MATH]
And so:

[MATH]\Delta t=\frac{d}{45\dfrac{\text{km}}{\text{hr}}}-\frac{d}{50\dfrac{\text{km}}{\text{hr}}}=\frac{d}{450}\,\frac{\text{hr}}{\text{km}}[/MATH]
We know:

[MATH]d=\left(50\frac{\text{km}}{\text{hr}}\right)\left(6\text{ hr}\right)=300\text{ km}[/MATH]
Hence:

[MATH]\Delta t=\frac{300}{450}\,\text{hr}=\frac{2}{3}\,\text{hr}\cdot\frac{60\text{ min}}{1\text{ hr}}=40\text{ min}[/MATH]

 

[Dr.Peterson]

No.

In this new equation, 50(360) = 45m, m is the total time in minutes at 45 km/h. The solution is not 6.7, but 50(360)/45 = 400 minutes.

That is the total time. How much more is that than 360? 400 - 360 = 40 minutes. That is the answer.

Using your old equation, the 6.7 (actually 6 2/3) was the total time in hours. Subtracting the original 6 hours, this is 2/3 of an hour more. That converts to 40 minutes.

And MarkFL has just given you a third way to get the same result. All of these are valid. But in all of them you have to pay close attention to the definition of the variable.

 

[harpazo]

No.

In this new equation, 50(360) = 45m, m is the total time in minutes at 45 km/h. The solution is not 6.7, but 50(360)/45 = 400 minutes.

That is the total time. How much more is that than 360? 400 - 360 = 40 minutes. That is the answer.

Using your old equation, the 6.7 (actually 6 2/3) was the total time in hours. Subtracting the original 6 hours, this is 2/3 of an hour more. That converts to 40 minutes.

And MarkFL has just given you a third way to get the same result. All of these are valid. But in all of them you have to pay close attention to the definition of the variable.

I will pat attention to the definition of variables.

 

[Jeyanthi R]

Distance = speed * time
=50*6=300 km
...
Time = distance / speed
Time = 300 /45
time = 6.66 hours
..
It takes 0.66 hours more
= 0.66 *60
= 39.6 minutes
= 40 minutes

 

[harpazo]

Distance = speed * time
=50*6=300 km
...
Time = distance / speed
Time = 300 /45
time = 6.66 hours
..
It takes 0.66 hours more
= 0.66 *60
= 39.6 minutes
= 40 minutes

Your reply is much easier to grasp.