SOLUTION: The distance between two towns consist of two stretches of road whose length are consecutive integers. A man drives on the shorter stretch at 60 mile per hour and on the longer str

Question 936066: The distance between two towns consist of two stretches of road whose length are consecutive integers. A man drives on the shorter stretch at 60 mile per hour and on the longer stretch at 46 mile per hour, so that his average rate for the whole trip is 52 mile per hour. Find the total distance.
Found 3 solutions by josmiceli, TimothyLamb, MathTherapy:
Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
The the consecutive integers be
and
----------------------
The average rate is: ( total distance ) / ( total time )

and

The total distance is 131.434 mi
check:

close enough

Answer by TimothyLamb(4379)   (Show Source): You can put this solution on YOUR website!
x = length of short stretch of road
y = length of long stretch of road
z = distance between towns
a = time on short stretch of road
b = time on long stretch of road
---
z = x + y
y = x + 1
---
s = d/t
t = d/s
---
time on short stretch of road:
a = x/60
x = 60a
---
time on long stretch of road:
b = y/46
b = (x + 1)/46
x = 46b - 1
---
equate x's:
60a = 46b - 1
46b = 60a + 1
b = (60a + 1)/46
---
average speed:
52 = z/(a + b)
---
distance between towns:
z = 52(a + b)
z = 52a + 52b
z = 52a + 52(60a + 1)/46
z = 52a + (52*60/46)a + 52/46
z = (52*46/46)a + (52*60/46)a + 52/46
z = ((52*46 + 52*60)/46)a + 52/46
z = (5512/46)a + 52/46
x + y = (5512/46)x/60 + 52/46
x + x + 1 = (5512/(46*60))x + 52/46
2x = (5512/2760)x + 52/46 - 1
2x - (5512/2760)x = 52/46 - 46/46
(2 - 5512/2760)x = 6/46
x = (6/46)/(2 - 5512/2760)
x = 45
y = 46
---
answer:
z = 45 + 46
z = distance between towns = 91 miles
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Answer by MathTherapy(10557)   (Show Source): You can put this solution on YOUR website!
The distance between two towns consist of two stretches of road whose length are consecutive integers. A man drives on the shorter stretch at 60 mile per hour and on the longer stretch at 46 mile per hour, so that his average rate for the whole trip is 52 mile per hour. Find the total distance.

Let longer-stretch distance, be L
Then shorter-stretch distance is: L - 1
Distance between towns: L + L - 1, or 2L - 1
Thus, time taken on longer stretch is: , and time taken on shorter stretch is:
Time taken on both stretches or to complete entire trip:
------- Multiplying fractions by LCD, 1,380

Since total distance, divided by total time, equals average speed, then it can be said that: �
* ------- Changing DIVISION to MULTIPLICATION, and INVERTING DIVISOR on left-side

52(53L � 23) = 2,760L � 1,380 ------- Cross-multiplying
2,756L � 1,196 = 2,760L � 1,380
2,756L � 2,760L = � 1,380 + 1,196
� 4L = � 184
L, or longer-stretch distance = , or 46 miles
Distance between towns: 2(46) � 1, or 92 � 1, or miles
You can do the check!!
===================
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