Step by step
The question
Question: Two trains leave the same station at the same time and go in opposite directions.
One train is 20 km/h faster than the other.
After 2 hours they are 440 km apart.
How fast is each train?
1 Find what we know
What do we know?
They go opposite ways.
One is 20 km/h faster than the other.
Both go for 2 hours.
Then they are 440 km apart.
What do we want?
The speed of each one.
2 Name the unknown
We know the faster speed from the slower one.
So we call the slower speed x km/h.
The other is 20 km/h faster: x + 20.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Train 1: x km/h, for 2 hours.
Train 2: x + 20 km/h, for 2 hours.
Distance = speed times time: 2x.
And for the other: 2(x + 20).
We keep x + 20 together in brackets.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is 2x + 2(x + 20) = 440.
Now we solve it.
Careful!
All of the speed x + 20 is times 2.
So it is 2(x + 20), not 2x + 20.
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 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.
2x + 2x = 4x.
Now we have 4x + 40 = 440.
We want x by itself.
Two things are stuck to x: times 4 and + 40.
The + 40 is on the outside, so it comes off first.
Then the times 4.
6 Remove the + 40
To remove + 40, we take away 40.
We do it on both sides, so they stay equal.
+ 40 and − 40 make zero.
Cross them out.
440 − 40 = 400.
Now we have 4x = 400.
7 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.
400 divided by 4 is 100.
So x = 100!
8 Answer the question
The faster one: 100 + 20 = 120.
So the speeds are 100 km/h and 120 km/h.
9 Check
Let's check in the story.
Train 1: 100 km/h for 2 hours is 200 km.
Train 2: 120 km/h for 2 hours is 240 km.
Together: 440 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.