ennifer and Taylor are running together. Taylor thinks Jennifer runs a little slow, so she gives Jennifer a 1 km advantage. This means Taylor is at the starting point and Jennifer is 1 km in front. Taylor runs 0.25 km per minute and Jennifer runs 0.15 km. How long will it take before Taylor catches up to Jennifer?

Question

naudiapie · Answer

helppp pls

mathstudent55 · Answer

What is the difference in their speeds?

naudiapie · Answer

0.10?

mathstudent55 · Answer

Correct.
The difference in speed is 0.10 km/minute.
How many minutes will it take for Taylor to run 1 km more than Jennifer being that Taylor runs 0.10 km/minute faster?

naudiapie · Answer

not sure

naudiapie · Answer

is there formula??

mathstudent55 · Answer

Time (min)        0    1        2        3        4        5       6         7        8         9       10
Taylor (km)       1    1.15   1.30   1.45   1.60   1.75  1.90    2.05   2.20    2.35  2.5
Jennifer (km)     0    0.25   0.50   0.75   1.00   1.25  1.50   1.75    2.00    2.25  2.5

mathstudent55 · Answer

We can write an equation.
Also, we can reason it out.
Taylor runs 0.10 km/minute faster than Jennifer.
In 10 minutes, Taylor will have run 1 km more than Jennifer.
Since Jennifer starts 1 km ahead, after 10 minutes, Taylor catches her.

naudiapie · Answer

so the answer is 10 min?

mathstudent55 · Answer

Yes.

mathstudent55 · Answer

I'll show it to you with an equation below.

naudiapie · Answer

did you get it by dividing 1 by 0.10

mathstudent55 · Answer

distance        time       speed
Jennifer                  x - 1            t            0.15
Taylor                       x               t            0.25

Taylor runs a total of x distance. Since Jennifer has a 1 km head start, she runs 1 km less than Taylor, so Jennifer runs x - 1 distance.
They both take t time until they are side by side.
The speeds are 0.15 for Jennifer and 0.25 for Taylor.

speed = distance/time

to solve for time, multiply both sides by time and divide both sides by speed

time = distance/speed

Now we write an equation for each person using the equation just above and the info from the table

Jennifer:    t = (x - 1)/0.15

Taylor:     t = x/0.25

Since t in both equations is equal, we can equate the equations:

(x - 1)/0.15 = x/0.25

$\dfrac{x - 1}{0.15} = \dfrac{x}{0.25} $

Cross multiply:

0.25(x - 1) = 0.15x

0.25x - 0.25 = 0.15x

0.1x - 0.25 = 0

0.1x = 0.25

x = 2.5

They meet at distance 2.5 km from the start.

Now we can find the time.

t = x/0.25

t = 2.5/0.25

t = 10

They meet at 10 minutes form the start.

mathstudent55 · Answer

You are correct.
Taylor has to make up 1 km of distance.
Since the difference in speeds is 0.10 km/min, when you divide distance by speed you get time, so 

(1 km)/(0.1 km/min) = 10 minutes

naudiapie · Answer

So you can do it with 2 equations>

mathstudent55 · Answer

We solved this problem three different ways.
We did it by seeing where they both are at the end of each minute until their distances are the same.
We also did it by reasoning and looking at the difference in speeds.
Then we did it with equations.

mathstudent55 · Answer

Yes.
Use the two equations as a system of equations.

naudiapie · Answer

i find it easier to simply divide 1 by 0.1

naudiapie · Answer

thank you for the help

mathstudent55 · Answer

You can set up two equations like we did above, or you can say: speed = distance/time

Jennifer:      0.15 = (x-1)/t
Taylor:        0.25 = x/t

Use the above equations and solve them simultaneously as a system of equations.

mathstudent55 · Answer

You're welcome.

naudiapie · Answer

Okay I see, noted