Problem #1
Number problem. If one-half of one integer is subtracted from three-fifths of the next consecutive integer, the difference is 3. What are the two integers?
My answer is: the two integers are 24 & 25
Problem#2
Science and medicine. A passenger train can travel 325 mi in the same time a freight train takes to travel 200 mi. If the speed of the passenger train is 25 mi/h faster than the speed of the freight train, find the speed of each.
My answer is: The speed for the freight is 40mph. The speed for the passenger train is 65mph
correct.
To solve problem #1, we can set up a system of equations. Let's let the first consecutive integer be represented by x and the next consecutive integer be represented by x+1.
Given that one-half of one integer is subtracted from three-fifths of the next consecutive integer, we can write the equation as:
(3/5)(x+1) - (1/2)(x) = 3
To solve this equation, we can start by getting rid of the fractions by multiplying through by the common denominator of 10:
10*((3/5)(x+1) - (1/2)(x)) = 10*3
This simplifies to:
6(x+1) - 5(x) = 30
Now we can distribute and combine like terms:
6x + 6 - 5x = 30
x + 6 = 30
Next, we solve for x by subtracting 6 from both sides:
x = 30 - 6
x = 24
Now we can find the next consecutive integer by adding 1 to x:
x+1 = 25
So the two integers are 24 and 25.
For problem #2, let's let the speed of the freight train be represented by x mph. Since the passenger train is 25 mph faster, the speed of the passenger train can be represented by x+25 mph.
Given that the passenger train can travel 325 miles in the same time the freight train takes to travel 200 miles, we can set up the following equation using the formula:
time = distance / speed
325 / (x+25) = 200 / x
To solve this equation, we can cross multiply:
325x = 200(x+25)
Now we can distribute and combine like terms:
325x = 200x + 5000
Next, we subtract 200x from both sides:
325x - 200x = 5000
125x = 5000
Finally, we solve for x by dividing both sides by 125:
x = 5000 / 125
x = 40
So the speed of the freight train is 40 mph. To find the speed of the passenger train, we add 25 to x:
x+25 = 40+25 = 65 mph
Therefore, the speed of the freight train is 40 mph and the speed of the passenger train is 65 mph.