Step by step
The question
Question: Two cyclists leave the same place and ride in opposite directions.
One rides at 12 km/h and the other at 18 km/h.
After how many hours are they 120 km apart?
1 Find what we know
What do we know?
They start at the same place.
They go opposite ways.
One goes 12 km in each hour.
The other goes 18 km in each hour.
We want them 120 km apart.
What do we want?
When they are 120 km apart.
2 Name the unknown
We don't know how many hours.
So we call it x.
Both go for the same x hours.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Cyclist 1: 12 km/h, for x hours.
Cyclist 2: 18 km/h, for x hours.
Distance = speed times time: 12 times x is 12x.
And 18 times x is 18x.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is 12x + 18x = 120.
Now we solve it.
4 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
12x + 18x = 30x.
Now we have 30x = 120.
5 Remove the 30
30x means 30 times x.
We want just x.
To remove times 30, we divide by 30.
We do it on both sides, so they stay equal.
30 divided by 30 is 1.
So just x is left.
120 divided by 30 is 4.
So x = 4!
6 Answer the question
So after 4 hours they are 120 km apart.
7 Check
Let's check in the story.
Cyclist 1: 4 hours at 12 km/h is 48 km.
Cyclist 2: 4 hours at 18 km/h is 72 km.
Together: 120 km, like the story says.
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.