Step by step
The question
Question: Two cars start 420 km apart and drive toward each other.
One goes 65 km/h and the other goes 75 km/h.
After how many hours do they meet?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find what we know
What do we know?
At the start, they are 420 km apart.
They go toward each other.
One goes 65 km in each hour.
The other goes 75 km in each hour.
What do we want?
When they meet.
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.
Car 1: 65 km/h, for x hours.
Car 2: 75 km/h, for x hours.
Distance = speed times time: 65 times x is 65x.
And 75 times x is 75x.
Let's draw it.
They start at the two ends.
Car 1 goes 65x km.
Car 2 goes 75x km.
When they meet, together they have gone the whole 420 km.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is 65x + 75x = 420.
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.
65x + 75x = 140x.
Now we have 140x = 420.
Is x alone now?
Not yet: 140x means 140 times x.
How do we remove it?
5 Remove the 140
To remove times 140, we divide by 140.
We do it on both sides, so they stay equal.
140 divided by 140 is 1.
They cancel each other.
So just x is left.
420 divided by 140 is 3.
So x = 3.
6 Answer the question
So they meet after 3 hours.
7 Check
Let's check in the story.
Car 1: 3 hours at 65 km/h is 195 km.
Car 2: 3 hours at 75 km/h is 225 km.
Together: 420 km, like the story says.
Check it with the picture.
Car 1's arrow is 195 km, car 2's is 225 km.
Together 420 km.
So the answer is right!
8 Pick the choice
Our answer is 3 hours.
That is choice A.
9 Why not the others
B is 6.
Wrong: 420 ÷ 70: used one speed in between; both cars close the gap.
C is 6.5.
Wrong: 420 ÷ 65: one car alone.
D is 42.
Wrong: 420 ÷ 10: used the difference of the speeds; they drive toward each other.
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.