Question 898143: At noon Jason left his house to walk towards Katie’s house at a speed of 3 miles per hour. 10 minutes later Katie left her house to walk towards Jason’s house at a speed of 2.5 miles per hour. If the houses are 4 miles apart, at what time do Katie and Jason meet each other?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first thing you need to do is convert everything to hours or everything to minutes.
i will convert everything to hours.
10 minutes is equal to 1/6 hours.
this is a very important step that you need to make sure of at the beginning of any of these problems because not doing so will bite you in the end if the units are not compatible.
since the rate is in miles per hour, the time has to be in hours.
you could also have converted the rate to miles per minute and left the time in minutes.
either way would have been ok as long as the units were compatible with each other.
the formula you will use will be rate * time = distance
for jason, that will become 3*t = d
for katie that will become 2.5*(t-1/6) = 4-d
katie's time is t-1/6 because katie left 10 minutes later and so will have traveled for 10 less minutes when they meet.
katie's distance is 4-d is because they will meet somewhere in the middle and their total distance traveled has to be equal to 4. the total distance they will have traveled is d + 4 - d which is equal to 4.
you want to solve these two equations simultaneously.
one way is to solve by substitution.
since you have d = 3*t, you can replace d in the second equation with 3*t and solve for t.
start with:
first equation is equal to 3*t = d and second equation is equal to 2.5*(t-1/6) = 4-d
replace d with 3*t in the second equation to get:
2.5 * (t - 1/6) = 4 - 3 * t
simplify to get:
2.5 * t - 2.5 * 1/6 = 4 - 3 * t
add 3 * t to both sides of the equation to get:
5.5 * t - 2.5 * 1/6 = 4
add 2.5 * 1/6 to both sides of the equation to get:
5.5 * t = 4 + 2.5 * 1/6
simplify to get:
5.5 * t = 26.5 / 6
solve for t to get t = 26.5 / 33 which is equivalent to 53/66.
t - 1/6 is therefore equal to 53/66 - 1/6 which is equal to 53/66 - 11/66 which is equal to 42/66.
t = 53/66
t-1 = 42/66
now you can solve for d, but you don't have to.
we'll solve for d anyway.
d = 3*t which is equal to 3 * 53/66 which is equal to 159/66.
4 - d is equal to 264/66 - 159/66 which is equal to 105/66.
3 * t must be equal to 159/66.
you get 3 * 53/66 = 159/66 which is good.
2.5 * (t-1) must be equal to 105/66.
you get 2.5 * 42/66 = 105/66 which is also good.
solution appears to be good.
now to the time.
jason left at 12:00 noon.
katie left 10 minutes later at 12:10 pm.
they met when jason had walked for 53/66 hours and katie had walked for 42/66 hours.
53/66 hours * 60 = 48.18181818 minutes.
42/66 hours * 60 = 38.18181818 minutes.
round this to the nearest minute and you get:
jason walked 48 minutes.
katie walked 38 minutes.
12:00 pm + 48 = 12:58 pm.
12:10 pm + 38 = 12:58 pm.
they met at 12:58 pm.
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