let walking rate be x mph
let biking rate be x+10
time for walking = 12/x
time biking = 8/(x+10)
isn't 12/x + 8(x+10) = 5 ?
multiply each term by x(x+10)
you will get quadratic, solve for x
let biking rate be x+10
time for walking = 12/x
time biking = 8/(x+10)
isn't 12/x + 8(x+10) = 5 ?
multiply each term by x(x+10)
you will get quadratic, solve for x
12/(v+10) + 8/v = 5 is the sum of the times spent bidking and walking.
Solve that equation for v. You will have to start with a common denominator.
[12v + 8(v+10)]/[v(v+10)] = 5
5 v^2 + 50 v = 20 v + 80
v^2 + 6v - 16 = 0
(v + 8)(v -2) = 0
Choose the positive root, v = 2 miles/hr
The bike speed was then v + 10 = 12 mph.
Since we know that his rate when riding a bicycle was 10 miles per hour faster than his rate walking, his rate when riding a bicycle would be "x+10" miles per hour.
The time it takes for the man to ride a bicycle for 12 miles can be calculated using the formula: time = distance / rate.
So, the time taken for the bicycle ride is 12 / (x+10).
Similarly, the time taken for the hike of 8 miles can be calculated as 8 / x.
We are given that the total time for the trip was 5 hours. So, we can set up the equation:
12 / (x+10) + 8 / x = 5
To solve this equation, we can start by clearing the fractions by multiplying throughout by the common denominator, which is x(x+10):
12x + 8(x+10) = 5x(x+10)
Now, simplify the equation:
12x + 8x + 80 = 5x^2 + 50x
Combine like terms:
20x + 80 = 5x^2 + 50x
Rearrange the equation to set it equal to zero:
5x^2 + 50x - 20x - 80 = 0
Combine like terms:
5x^2 + 30x - 80 = 0
This is now a quadratic equation in the form ax^2 + bx + c = 0, where a = 5, b = 30, and c = -80.
To solve the quadratic equation, we can either factor or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-30 ± √(30^2 - 4 * 5 * -80)) / (2 * 5)
Calculating the discriminant:
x = (-30 ± √(900 + 1600)) / 10
x = (-30 ± √2500) / 10
Now, let's simplify further:
x = (-30 ± 50) / 10
We have two solutions:
x1 = (-30 + 50) / 10 = 20 / 10 = 2
x2 = (-30 - 50) / 10 = -80 / 10 = -8
Since rates cannot be negative, the solution x2 = -8 can be disregarded.
Therefore, the man's rate walking (x) is 2 miles per hour, and his rate riding a bicycle (x + 10) is 2 + 10 = 12 miles per hour.