1. A moving van leaves a house travelling at an average rate of 35 mi/hr. The family leaves the house 3/4 hour later following the same route in a car. They travel at an average rate of 50 mi/hr. How long it will take the car to catch up with the moving van?
2. Juan drives to work. Because of traffic conditions, he averages 22 miles per hour. He returns home averaging 32 miles per hour. The total travel time is 2 1/4 hours. Find the time Juan spends driving to work.
in 3/4 hour, the van moves 26.25 mi
Since the family is going 15 mi/hr faster, it will take 26.25/15 = 1.75 hours to catch up
let the time taken by the family be t hours
then the time taken by the van = t+1/2 hours
for those times:
distance covered by family = 50t miles
distance covered by van = 35(t + 1/2) =35t + 25 miles
but at catch-up they must have gone the same distance, so
50t = 35t + 25
15t = 25
t = 25/15 hours = 5/3 hours or 1 hour and 40 minutes.
For the second, you have to base your equation on the fact that
the distance each way is the same
let that distance be d
time to go to work = distance/rate = d/22
time to return home = d/32
we are told that the total time is 2 1/4 or 9/4 hours
so ......
d/22 + d/32 = 9/4
hint: multiply each term by 352 , the LCD
and take over from there
1. To find the time it will take for the car to catch up with the moving van, we need to determine the distance that the van has already traveled by the time the car starts.
Let's start by finding the distance the van has traveled in the time it took for the car to start moving. The van has an average rate of 35 mi/hr and it starts 3/4 hour ahead of the car. So, the distance traveled by the van in that time is given by:
Distance = Rate * Time
Distance = 35 mi/hr * 3/4 hr
Distance = (35 * 3) / 4
Distance = 26.25 miles
Now, we need to find the time it will take for the car to catch up with the van. Since both vehicles are traveling on the same route, the time it will take for the car to catch up will be the same as the time it takes for the car to cover the distance traveled by the van.
Let's calculate the time it will take for the car to cover the distance traveled by the van. The car has an average rate of 50 mi/hr. So, the time it will take for the car to catch up is given by:
Time = Distance / Rate
Time = 26.25 miles / 50 mi/hr
Time = 0.525 hour (rounded to three decimal places)
Therefore, it will take the car approximately 0.525 hour, or around 31.5 minutes, to catch up with the moving van.
2. To find the time Juan spends driving to work, we need to set up an equation using the given information.
Let's assume that the time Juan spends driving to work is x hours.
We are given that Juan averages 22 miles per hour while driving to work, so the distance traveled is 22x miles.
We are also given that Juan returns home averaging 32 miles per hour. Since the total travel time is 2 1/4 hours, and Juan spends x hours driving to work, he spends (2 1/4 - x) hours driving back home. The distance traveled on the return trip is 32(2 1/4 - x) miles.
The total distance traveled is the sum of the distances for the two legs of the trip:
22x + 32(2 1/4 - x)
Since the total travel time is 2 1/4 hours, we can write:
x + (2 1/4 - x) = 2 1/4
Simplifying the equation:
2 1/4 - x + x = 2 1/4
2 1/4 = 2 1/4
The equation is true. This means that any value of x that satisfies the equation will be the time Juan spends driving to work.
Therefore, the time Juan spends driving to work is 2 1/4 hours.