Cyclist "A" traveld 60 km. Cyclist "B" travels 5 km/hr faster than "A" and travels the same distance in 2 hours less.
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Let A be the speed that cyclist A travels.
That means that it takes cyclist A 60/A hours to travel 60 km.
Cyclist B travels at A+5 km/hr, so does the same distance in 60/(A+5) hours.
And you know that it took 2 hours less.
Therefore:
60/A = 2 + 60/(A+5)
Multiply the 2 by (A+5)/(A+5) to make it over the same base:
60/A = 2(A+5)/(A+5) + 60/(A+5)
Add the two fractions:
60/A = [ 2(A+5) + 60 ] / (A+5)
Multiply throughout by (A+5)
60(A+5)/A = 2(A+5) + 60
Multiply throughout by A
60(A+5) = A[2(A+5) + 60]
60(A+5) = A[2A+10+60]
60A + 300 = 2A² + 70A
2A² + 10A - 300 = 0
Divide throughout by 2:
A² + 5A - 150 = 0
Because the coefficient of A² is 1, we're looking for two numbers that when ADDED make +5, but when MULTIPLIED make -150. What about 15 and -10?
(A+15)(A-10) = 0
Now we have two terms that multiply to make 0. That means that one of the terms must be zero:
A+15=0 or A-10=0.
That means A=-15 or +10
Since a negative speed doesn't make sense, cyclist A travels at 10 km/h.
Quick check: 60 km @ 10 km/hr = 60/10 = 6 hours
60 km @ (10+5)=15 km/hr = 60/15 = 4 hours, which is 2 hours less.
Check!
That means that it takes cyclist A 60/A hours to travel 60 km.
Cyclist B travels at A+5 km/hr, so does the same distance in 60/(A+5) hours.
And you know that it took 2 hours less.
Therefore:
60/A = 2 + 60/(A+5)
Multiply the 2 by (A+5)/(A+5) to make it over the same base:
60/A = 2(A+5)/(A+5) + 60/(A+5)
Add the two fractions:
60/A = [ 2(A+5) + 60 ] / (A+5)
Multiply throughout by (A+5)
60(A+5)/A = 2(A+5) + 60
Multiply throughout by A
60(A+5) = A[2(A+5) + 60]
60(A+5) = A[2A+10+60]
60A + 300 = 2A² + 70A
2A² + 10A - 300 = 0
Divide throughout by 2:
A² + 5A - 150 = 0
Because the coefficient of A² is 1, we're looking for two numbers that when ADDED make +5, but when MULTIPLIED make -150. What about 15 and -10?
(A+15)(A-10) = 0
Now we have two terms that multiply to make 0. That means that one of the terms must be zero:
A+15=0 or A-10=0.
That means A=-15 or +10
Since a negative speed doesn't make sense, cyclist A travels at 10 km/h.
Quick check: 60 km @ 10 km/hr = 60/10 = 6 hours
60 km @ (10+5)=15 km/hr = 60/15 = 4 hours, which is 2 hours less.
Check!
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Cyclist A - 60km
Cyclist B - 5km/hr faster than A
- 2 hours less than A
The easiest way I could think to go about this is guess and check. If you plugin 20km/hr for B and 15km/hr for A, you get 3 hours for B and 4 hours for A. That's not it.
After a little while, I got the right pair. Cyclist B is traveling 15km/hr and Cyclist A is traveling 10km/hr. Cyclist B would travel 60km in 4 hours and Cyclist A would travel 60 km in 6 hours. That's a 2 hour difference.
Cyclist B - 5km/hr faster than A
- 2 hours less than A
The easiest way I could think to go about this is guess and check. If you plugin 20km/hr for B and 15km/hr for A, you get 3 hours for B and 4 hours for A. That's not it.
After a little while, I got the right pair. Cyclist B is traveling 15km/hr and Cyclist A is traveling 10km/hr. Cyclist B would travel 60km in 4 hours and Cyclist A would travel 60 km in 6 hours. That's a 2 hour difference.