A bus trip of 90 miles would have taken an hour less if the usual average speed had been increased by 3 miles per hour. Find the usual average speed of the bus.
If normally the speed is x, then
90/x = 1 + 90/(x+3)
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Let's assume the usual average speed of the bus is "x" miles per hour.
According to the given information, if the usual average speed is increased by 3 miles per hour, the new average speed becomes "x + 3" miles per hour.
The time taken to complete the trip can be calculated using the formula:
Time = Distance / Speed
The time taken with the usual average speed is:
Time with usual speed = 90 / x
The time taken with the increased speed is:
Time with increased speed = 90 / (x + 3)
It is given that the trip would have taken 1 hour less if the speed was increased. Therefore, we can set up the following equation:
Time with usual speed - Time with increased speed = 1
90 / x - 90 / (x + 3) = 1
Multiplying both sides by x(x + 3) to eliminate the fractions, we get:
90(x + 3) - 90x = x(x + 3)
Simplifying the equation:
90x + 270 - 90x = x^2 + 3x
Combining like terms:
270 = x^2 + 3x
Rearranging the equation in standard quadratic form:
x^2 + 3x - 270 = 0
We can solve this equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(x - 15)(x + 18) = 0
Setting each factor to zero:
x - 15 = 0 or x + 18 = 0
Solving for x:
x = 15 or x = -18
Since the speed of the bus cannot be negative, the usual average speed of the bus is 15 miles per hour.
To find the usual average speed of the bus, we can set up an equation based on the given information.
Let's assume the usual average speed of the bus is represented by "x" miles per hour.
According to the question, if the usual average speed had been increased by 3 miles per hour, the bus would have taken an hour less to complete the trip.
So, the time taken to travel 90 miles at the usual average speed (x mph) is 90/x hours.
The time taken to travel 90 miles at an increased speed (x + 3 mph) is 90/(x + 3) hours.
We can set up the equation:
90/x - 90/(x + 3) = 1
To solve this equation, we can multiply the entire equation by x(x + 3) to eliminate the denominators:
90(x + 3) - 90x = x(x + 3)
Now, we can simplify and solve for x:
90x + 270 - 90x = x^2 + 3x
270 = x^2 + 3x
Rearranging the equation:
x^2 + 3x - 270 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula.
Factoring:
(x - 15)(x + 18) = 0
Setting each factor equal to zero:
x - 15 = 0 or x + 18 = 0
x = 15 or x = -18
Since the average speed of a bus cannot be negative, we can discard the solution x = -18.
Therefore, the usual average speed of the bus is 15 miles per hour.