SOLUTION: Antonio can go around a track in 40 seconds and his little brother Guillermo in 50 seconds. If Antonio gives Guillermo 3 seconds of advantage in the start, find out how long it wil

Question 1161824: Antonio can go around a track in 40 seconds and his little brother Guillermo in 50 seconds. If Antonio gives Guillermo 3 seconds of advantage in the start, find out how long it will take him to reach it.
Found 4 solutions by ikleyn, Theo, greenestamps, MathTherapy:
Answer by ikleyn(52879)   (Show Source): You can put this solution on YOUR website!
.

Let d be the length of the track, in meters. Then Antonio' speed is m/s. The Guilermo' speed is m/s. The difference of the speeds is m/s = = = m/s. (the catching speed) The time to catch is : = = = 3*4 = 12 seconds. ANSWER

Solved.

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Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
antonio takes 40 seconds to go around the track.
guillermo takes 50 seconds to go around the same track.
rate * time = distance.
let x = the rate of antonio.
let y = the rate of guillermo.
antonio equation is x * 40 = d
guillermo equation is y * 50 = d
since they are both equal to d, you get:
x * 40 = y * 50
solve for x to get:
x = 1.25 * y
this says that antonio is traveling 1.25 times as fast as guillermo.

i think you want to know how long it will take antonio to catch up to guillermo if he gives guillermo a 3 second head start.
rate * time = distance.
the equation for antonio becomes:
x * (T - 3) = d
the equation for guillermo becomes:
y * T = d
antonio has caught up to guillermo when they have both traveled the same distance, which i call d.
this d is not the same d we started with.
the first d was the distance around the track.
this d is the distance traveled when antonio has caught up to guillermo.
since they both equal to d, the you get:
x * (T - 3) = y * T
since x = 1.25 * y, you get:
1.25 * y * (T - 3) = y * T
simplify to get:
1.25 * y * T - 1.25 * y * 3 = y * T
simplify further to tget:
1.25 * y * T - 3.75 * y = y * T
subtract y * T from both sides of the equation and add 3.75 * y to both sides of the equation and combine like terms to get:
.25 * y * T = 3.75 * y
divide both sides by y to get:
.25 * T = 3.75
solve for T to get:
T = 3.75 / .25 = 15
this says that they will both have traveled the same distance in 15 seconds.
guillermo has traveled for 15 seconds.
antonio has traveled for 12 seconds.
your solution is that it took 12 seconds for antonio to catch up to guillermo when he gave guillermo a 3 second head start.

you didn't need to know what d was originally to solve this problem, but it would have helped to understand what was going on.
for example:
if d = 200 feet (the track is 200 feet around), then:
40 * x = 200 for antonio
50 * y = 200 for guillermo.
solve for x to get x = 5 feet per second for antonio
solve for y to get y = 4 feet per second for guillermo
now you want to know how long for antonio to catch up to guillermo if he gives guillermo a 3 second head start.
the formula for antonio becomes 5 * (T - 3) = d
the formula for guillermo becomes 4 * T = d
d represents the distance both will need to travel for antonio to catch up to guillermo.
since they goth = d, then you get:
5 * (T - 3) = 4 * T
simplify to get 5T - 15 = 4T
solve for T to get T = 15.
antonio will have caught up to guillermo after 15 seconds have elapsed.
guillermo will have traveled for 15 seconds.
antonio will have traveled for 12 seconds.
your solution is that it took antonio 12 seconds to catch up to guillermo.

since we assumed the distance around the track was 200 feet, then the distance required for antonio to catch up to guillermo can now be calculated.
rate * time = distance.
x * 12 = d
x = 5 feet per second, so d = 60 feet.
y * 15 = d
y = 4 feet per second, so d = 4 * 15 = 60 feet.
antonio has caught up to guillermo in 12 seconds after traveling for a distance of 60 feet.

if the original d is assumed to be 200 feet around the track, then the d required for antonio to catch up to guillermo was 60 feet.

without the assumption of what the distance around the track was, you couldn't solve for the distance required for antonio to catch up to guillermo, but you were still able to solve for the time required.

Answer by greenestamps(13209)   (Show Source): You can put this solution on YOUR website!

Let t be the number of seconds Antonio runs to catch up to Guillermo.
Then t+3 is the number of seconds Guillermo runs before Antonio catches up to him.

The ratio of the times it takes the two of them to run around the track is 4:5, so the ratio of the times it takes them to run any equal distances is 4:5.

Antonio catches up to Guillermo when the times they have run are in the ratio 4:5.

ANSWER: Antonio catches up to Guillermo when he has run t=12 seconds and Guillermo has run t+3=15 seconds.

Answer by MathTherapy(10556)   (Show Source): You can put this solution on YOUR website!
Antonio can go around a track in 40 seconds and his little brother Guillermo in 50 seconds. If Antonio gives Guillermo 3 seconds of advantage in the start, find out how long it will take him to reach it.

Let time Antonio takes to catch-up to Guillermo, NOT “reach it,” be T, and distance, D
We then get Antonio’s speed as , and Guillermo’s as:
We also get Antonio’s and Guillermo’s times as T and T + 3
We then get the following DISTANCE equation:
------ Factoring out GCF, D, in the numerator, and 10 in the denominator
5T = 4T + 12 ------- Multiplying by LCD, 20
5T - 4T = 12
T, or time Antonio takes to catch-up to Guillermo is:
If I were you, I'd never even LOOK at @Theo's "NOVEL." This problem is NOT that complex.
From the start, it'd make almost all readers' brains hurt!
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