Step by step
The question
Question: A walker takes 6 hours for a trip.
A cyclist takes 2 hours for the same trip.
The cyclist is 10 km/h faster than the walker.
How fast is each one?
1 Find what we know
What do we know?
The walker takes 6 hours.
The cyclist takes 2 hours for the same trip.
The cyclist is 10 km/h faster.
What do we want?
The speed of each one.
2 Name the unknown
We know the cyclist's speed from the walker's speed.
So we call the walker's speed x.
The cyclist is 10 km/h faster: x + 10.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
The walker: x km/h, for 6 hours.
The cyclist: x + 10 km/h, for 2 hours.
Distance = speed times time: 6x.
And 2 times all of x + 10: 2(x + 10).
We keep x + 10 together in brackets.
3 Write the equation
In words: the walker's distance is the same as the cyclist's distance.
So the equation is 6x = 2(x + 10).
Now we solve it.
Careful!
The faster cyclist takes less time: 2 hours.
So 2 goes with x + 10.
Each speed with its own time.
4 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 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 2x from the right side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
6x − 2x = 4x.
Now we have 4x = 20.
6 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
20 divided by 4 is 5.
So x = 5!
7 Answer the question
The cyclist: 5 + 10 = 15.
So the walker goes 5 km/h and the cyclist goes 15 km/h.
8 Check
Let's check in the story.
The walker: 6 hours at 5 km/h is 30 km.
The cyclist: 2 hours at 15 km/h is 30 km.
The same trip, like the story says.
Does it make sense?
The cyclist 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.