Step by step
The question
Question: A boat takes 6 hours for a trip.
A ship takes 4 hours for the same trip.
The ship is 8 km/h faster than the boat.
How fast is each one?
1 Find what we know
What do we know?
The boat takes 6 hours.
The ship takes 4 hours for the same trip.
The ship is 8 km/h faster.
What do we want?
The speed of each one.
2 Name the unknown
We know the ship's speed from the boat's speed.
So we call the boat's speed x.
The ship is 8 km/h faster: x + 8.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
The boat: x km/h, for 6 hours.
The ship: x + 8 km/h, for 4 hours.
Distance = speed times time: 6x.
And 4 times all of x + 8: 4(x + 8).
We keep x + 8 together in brackets.
3 Write the equation
In words: the boat's distance is the same as the ship's distance.
So the equation is 6x = 4(x + 8).
Now we solve it.
Careful!
The faster ship takes less time: 4 hours.
So 4 goes with x + 8.
Each speed with its own time.
4 Open the brackets
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
Work out each multiplication.
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 4x from the right side.
To remove 4x, we take away 4x.
We do it on both sides, so they stay equal.
4x and − 4x make zero.
Cross them out.
6x − 4x = 2x.
Now we have 2x = 32.
6 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
32 divided by 2 is 16.
So x = 16!
7 Answer the question
The ship: 16 + 8 = 24.
So the boat goes 16 km/h and the ship goes 24 km/h.
8 Check
Let's check in the story.
The boat: 6 hours at 16 km/h is 96 km.
The ship: 4 hours at 24 km/h is 96 km.
The same trip, like the story says.
Does it make sense?
The ship is faster, and it takes less time.
That makes sense.
So the answer is right!
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Then: answer the question.
Last: check.