Step by step
The question
Question: Two cyclists leave the same place at the same time and ride in opposite directions.
One rides 6 km/h faster than the other.
After 30 minutes they are 15 km apart.
How fast is each one?
1 Find what we know
What do we know?
They go opposite ways.
One is 6 km/h faster than the other.
Both go for 30 minutes.
Then they are 15 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 30 minutes is 30/60, which is 1/2 of an hour.
The other is 6 km/h faster: x + 6.
Careful!
The speeds are in km per hour.
So the time must be in hours too: 1/2, not 30.
How far each one goes: speed times time.
Let's put it in a table: speed, time, distance.
Cyclist 1: x km/h, for 1/2 of an hour.
Cyclist 2: x + 6 km/h, for 1/2 of an hour.
Distance = speed times time: 1/2 times x.
And for the other: 1/2 times all of x + 6.
We keep x + 6 together in brackets.
3 Write the equation
In words: the two distances add up to the whole distance.
So the equation is x/2 + (x + 6)/2 = 15.
Now we solve it.
4 Remove the fractions
The bottom number is 2.
Times 2 will remove it.
We multiply every part by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
2 divided by 2 is 1.
Write what is left.
Do the multiplications.
No more fractions!
5 Open the brackets
A bracket with a plus or nothing in front can just be removed.
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.
x + x = 2x.
Now we have 2x + 6 = 30.
We want x by itself.
Two things are stuck to x: times 2 and + 6.
The + 6 is on the outside, so it comes off first.
Then the times 2.
7 Remove the + 6
To remove + 6, we take away 6.
We do it on both sides, so they stay equal.
+ 6 and − 6 make zero.
Cross them out.
30 − 6 = 24.
Now we have 2x = 24.
8 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
24 divided by 2 is 12.
So x = 12!
9 Answer the question
The faster one: 12 + 6 = 18.
So the speeds are 12 km/h and 18 km/h.
10 Check
Let's check in the story.
Cyclist 1: 12 km/h for 1/2 of an hour is 6 km.
Cyclist 2: 18 km/h for 1/2 of an hour is 9 km.
Together: 15 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.