Step by step
The question
Question: Two cars start 100 km apart and drive toward each other.
One drives 10 km/h faster than the other.
They meet after 40 minutes.
How fast is each one?
1 Find what we know
What do we know?
They go toward each other.
One is 10 km/h faster than the other.
Both go for 40 minutes.
At the start, they are 100 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.
Speed is in km per hour.
So the time must be in hours too.
1 hour is 60 minutes.
So 40 minutes is 40/60, which is 2/3 of an hour.
The other is 10 km/h faster: x + 10.
Careful!
The speeds are in km per hour.
So the time must be in hours too: 2/3, not 40.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Car 1: x km/h, for 2/3 of an hour.
Car 2: x + 10 km/h, for 2/3 of an hour.
Distance = speed times time: 2/3 times x.
And for the other: 2/3 times all of 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 2x/3 + 2(x + 10)/3 = 100.
Now we solve it.
4 Remove the fractions
The bottom number is 3.
Times 3 will remove it.
We multiply every part by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
3 divided by 3 is 1.
Write what is left.
Do the multiplications.
No more fractions!
5 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.
6 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 + 20 = 300.
We want x by itself.
Two things are stuck to x: times 4 and + 20.
The + 20 is on the outside, so it comes off first.
Then the times 4.
7 Remove the + 20
To remove + 20, we take away 20.
We do it on both sides, so they stay equal.
+ 20 and − 20 make zero.
Cross them out.
300 − 20 = 280.
Now we have 4x = 280.
8 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.
280 divided by 4 is 70.
So x = 70!
9 Answer the question
The faster one: 70 + 10 = 80.
So the speeds are 70 km/h and 80 km/h.
10 Check
Let's check in the story.
Car 1: 70 km/h for 2/3 of an hour is 46.67 km.
Car 2: 80 km/h for 2/3 of an hour is 53.33 km.
Together: 100 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.