Step by step
The question
Question: Two cars leave the same town at the same time and drive in opposite directions.
One car is 10 km/h faster than the other.
After 3 hours they are 390 km apart.
How fast is each car?
1 Find what we know
What do we know?
They go opposite ways.
One is 10 km/h faster than the other.
Both go for 3 hours.
Then they are 390 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 10 km/h faster: x + 10.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Car 1: x km/h, for 3 hours.
Car 2: x + 10 km/h, for 3 hours.
Distance = speed times time: 3x.
And for the other: 3(x + 10).
We keep x + 10 together in brackets.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is 3x + 3(x + 10) = 390.
Now we solve it.
Careful!
All of the speed x + 10 is times 3.
So it is 3(x + 10), not 3x + 10.
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 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.
3x + 3x = 6x.
Now we have 6x + 30 = 390.
We want x by itself.
Two things are stuck to x: times 6 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 6.
6 Remove the + 30
To remove + 30, we take away 30.
We do it on both sides, so they stay equal.
+ 30 and − 30 make zero.
Cross them out.
390 − 30 = 360.
Now we have 6x = 360.
7 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just x is left.
360 divided by 6 is 60.
So x = 60!
8 Answer the question
The faster one: 60 + 10 = 70.
So the speeds are 60 km/h and 70 km/h.
9 Check
Let's check in the story.
Car 1: 60 km/h for 3 hours is 180 km.
Car 2: 70 km/h for 3 hours is 210 km.
Together: 390 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.