# Bus Traveling At Different Rates

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#### mathdad

##### Full Member
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:

I need to use D = rt.
I will divide the situation into two cases.

Case 1

rate = 50
time = 6
Distance = 50(6)

Case 2

rate = 45
time = t
Distance = 45t

Distance for Case 1 = Distance for Case 2.

50(6) = 45t

300 = 45t

Solve for t.

300/45 = t

6.6666666667 = t

I am stuck here.

Note: HOW MANY MORE MINUTES, tells me that traveling at 45 k/hr would have taken 6 hours plus additional minutes. How do I find the additional minutes?

#### Romsek

##### Full Member
$$\displaystyle d=50km/hr \cdot 6hr= 300km$$

$$\displaystyle t=\dfrac{300km}{45km/hr} = \dfrac{20}{3}hr$$

$$\displaystyle \dfrac{20}{3}hr - 6hr = \dfrac 2 3 hr = 40min$$

#### HallsofIvy

##### Elite Member
There are 60 minutes in an hour! 6.6666666... is 6 and 2/3 hours. 2/3 of an hour is 40 minutes.

Thank you.

#### MarkFL

##### Super Moderator
Staff member
I would likely approach this by observing we have:

$$\displaystyle t=\frac{d}{v}$$

Hence:

$$\displaystyle \Delta t=d\left(\frac{1}{v_2}-\frac{1}{v_1}\right)=\frac{v_1t_1(v_1-v_2)}{v_1v_2}=\frac{t_1(v_1-v_2)}{v_2}$$

Notice we now have the time difference in terms of given data only.

Now, plug in the given data:

$$\displaystyle \Delta t=\frac{(6\text{ hr})(50-45)\frac{\text{km}}{\text{hr}}}{45\frac{\text{km}}{\text{hr}}}=\frac{2}{3}\text{ hr}\cdot\frac{60\text{ min}}{1\text{ hr}}=40\text{ min}$$

#### mathdad

##### Full Member
I would likely approach this by observing we have:

$$\displaystyle t=\frac{d}{v}$$

Hence:

$$\displaystyle \Delta t=d\left(\frac{1}{v_2}-\frac{1}{v_1}\right)=\frac{v_1t_1(v_1-v_2)}{v_1v_2}=\frac{t_1(v_1-v_2)}{v_2}$$

Notice we now have the time difference in terms of given data only.

Now, plug in the given data:

$$\displaystyle \Delta t=\frac{(6\text{ hr})(50-45)\frac{\text{km}}{\text{hr}}}{45\frac{\text{km}}{\text{hr}}}=\frac{2}{3}\text{ hr}\cdot\frac{60\text{ min}}{1\text{ hr}}=40\text{ min}$$
Very few people like Mark. He does not argue, criticize or misread my replies. Mark goes straight to the question.

#### Jomo

##### Elite Member
Very few people like Mark. He does not argue, criticize or misread my replies. Mark goes straight to the question.
Very few people like Mark. He does not argue, criticize or misread my replies. Mark goes straight to the question.
So why does not many people like Mark? I like Mark and I even think that Denis likes him.

#### Romsek

##### Full Member
So why does not many people like Mark? I like Mark and I even think that Denis likes him.
I'm going to hazard a guess that he means "There are very few people similar to Mark"

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