At the beginning of a walk juanita and roberto are 77 miles

At the beginning of a walk, juanita and roberto are 7.7 miles apart. If they leave at the same time and travel in the same direction roberto will overtake juanita in 11 hours. If they walk towards eachother they will meet in 1 hour. What are their speeds?

Solution

let R = rate that Roberto travels at.
let J = rate that Juanita travels at.
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if they walk towards each other they will meet in 1 hour.
therefore the total distance they traveled is 7.7 miles.
since Rate * Time = Distance,
Roberto traveled for 1 hours at rate R, and Juanita traveled for 1 hour at rate J.
equation for that is:
1*R + 1*J = 7.7
which is the same as:
R + J = 7.7 (first original equation) ************************************
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if they walk in the same direction, then they will meet in 11 hours.
Roberto will have walked 7.7 miles more than Juanita.
therefore
11 hours * rate of Roberto = D + 7.7
11 hours * rate of Juanita = D
equation for that is:
11*R = D + 7.7 (second original equation) *************
11*J = D (third original equation) ********************
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the total distance traveled by both would be 2*D + 7.7
so, .....
11*R + 11*J = 2D + 7.7
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we now have 2 equations that need to be solved simultaneously.
R + J = 7.7
11*R + 11*J = 2D + 7.7
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if R + J = 7.7, then 11*R + 11*J = 11 * 7.7
so, .....
11*R + 11*J = 11 * 7.7 = 84.7
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since 2D + 7.7 = 11*R + 11*J, and 84.7 = 11*R + 11*J, then 2D+7.7 = 84.7 because 2 things equal to the same thing are equal to each other.
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equation to solve is:
2D + 7.7 = 84.7
subtract 7.7 from both sides.
2D = 77
D = 38.5
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from above, we know that:
11*R = D + 7.7
11*J = D
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solvng for J, we get:
11*J = D = 38.5
divide both sides by 11.
D = 38.5/11 = 3.5 mph.
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solving for R, we get:
11*R = D + 7.7 = 38.5 + 7.7 = 46.2
divide both sides by 11.
R = 4.2 mph.
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answer appears to be:
R = 4.2 mph
J = 3.5 mph.
to prove, substitute in original equations:
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first original equation is:
R + J = 7.7
4.2 + 3.5 = 7.7
first equation looks good.
second original equation is:
11*R = D + 7.7
11*4.2 = 38.5 + 7.7 = 46.2
46.2 = 46.2
second original equation checks out ok.
third original equation is:
11*J = D
11*3.5 = 38.5
38.5 = 38.5
third original equation checks out ok.
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you could also have solved these as 2 simultaneous equations:
R + J = 7.7
11R + 11J = 2D + 7.7
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you would get the same answer, because:
multiply first equation by 11 to get:
11R + 11J = 84.7
11R + 11J = 2D + 7.7
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subtract top equation from bottom equation to get:
0 = 2D + 7.7 - 84.7
add 84.7 to both sides of equation.
84.7 = 2D + 7.7
subtract 7.7 from both sides of equation.
77 = 2D
divide both sides of equation by 2.
38.5 = D
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same answer.