Distance problem?

Question

Bill and Bob both drive 60 miles to work. By averaging 10 miles per hour faster than Bob, Bill gets to work 12 minutes earlier than Bob. How fast does each one drive?

Answer 1

First convert 12 minutes into 1/5 hour, so we are all in miles and hours.

Bill's time is t
Bob's time is t + 1/5

Bill's rate is r
Bob's rate is r + 10

D = 60 = rt = (r + 10)(t + 1/5)

Solve for r:
r = 60/t

So:
(60/t + 10)(t + 1/5) = 60
60 + 10t + 60/5t + 10/5 = 60

Simplifying:
10t + 12/t + 2 = 0

Multiply by t:
10t^2 + 2t + 12 = 0
5t^2 + t + 6 = 0

Factor:
(5t + 6)(t - 1) = 0

So t = 1 or -6/5. Since time is positive, t = 1 is the correct answer.

Now solve for r:
r = 60/t
r = 60/1
r = 60 mph
Bill's speed is 60 mph.

And Bob's speed is 10 less than this:
60 - 10 = 50
Bob's speed is 50 mph.

Checking:
1 * 60 = 6/5 * 50

Summarizing:
Bill takes 1 hour, Bob takes 1 1/5 hour.
Bill's speed is 60 mph, Bob's speed is 50 mph.

Answer 2

x=Bill's speed
y=Bob's speed

Building equations...

x-y=10 (Bill drives 10 mph faster than Bob)

60/x = 60/y - 0.2 (Bill's time is equal to Bob's time minus 0.2 hours - you need to convert to hours because the speed is in miles per hour)

Solve for x and y. Plug x=y+10 in the second equation and you get

60/(y+10) = 60/y-0.2

60/(y+10) = (60-0.2y)/y

60y = (y+10)(60-0.2y)

60y = 60y + 600 -0.2y^2 -2y

0 = -0.2y^2 - 2y + 600

The positive solution to this quadratic equation is y=50. Therefore Bill is going at 60 mph (60 miles in 1 hour) and Bob at 50 mph (60 miles in one hour and 12 minutes) .

Answer 3

d = r*t
let x = Bob's rate
60 = xt = (x + 10)(t - 12/60)
(x + 10)(60/x - 12/60) = 60
60 -12x/60 + 600/x -120/60 = 60
-12x^2 + 36000 - 120x = 0
x^2 + 10x - 3000 = 0
x = -5 ± sqrt(25 + 3000)
x = 50 mph
x + 10 = 60 mph

(50 + 10)(60/50 - 12/60) = 60
60(1.2-0.2) = 60

Answer 4

x=bob's speed
bills speed=x+10
bob takes 60/x hrs
bill takes 60/(x+10) hrs
bill takes 12 min=.2 hrs less so
60/x=60/(x+10)+.2 multoply by x(x+10)
60(x+10)=60x+.2x(x+10) subtract 60x from each side
600=2+.2x^2+2x rearange
.2x^2+2x-598=0
x=(-2+/-sqrt(.2^2+4*.2*598))/.4
x=-2/.4+/-sqrt(478.44)/.4
x=5+/-21.87/.4=5+/-54.68
since speed must be positive
Bob drives
x=59.68 roud off to 60 mph
Bill drives
x+10=70 mph.