Worked examples
The question
Question: A car drives 150 km in 3 hours.
What is its speed in km/h?
1 Find what we know
What do we know?
The car goes 150 km in 3 hours.
What do we want?
The speed, in km per hour.
2 Name the unknown
We don't know the speed.
So we call it x km/h.
x = speed (km/h)
Careful!
Speed in km/h means how many km in 1 hour.
Careful: km/h = km in 1 hour
How far we go: speed times time.
distance = speed × time
3 Write the equation
In words: speed times time makes the distance.
3x = 150
So the equation is 3x = 150.
Now we solve it.
3x = 150
4 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3x3 = 1503
3 divided by 3 is 1.
So just x is left.
150 divided by 3 is 50.
1503 = 50
So x = 50!
x = 50
5 Answer the question
So the speed is 50 km/h.
6 Check
Let's check in the story.
50 km/h for 3 hours: 50 times 3 is 150 km, like the story says.
So the answer is right!
50 × 3 = 150 km
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.
▶ Watch it solved
The question
Question: A bus takes 4 hours for a trip.
A car takes 3 hours for the same trip.
The car is 20 km/h faster than the bus.
How fast is each one?
1 Find what we know
What do we know?
The bus takes 4 hours.
The car takes 3 hours for the same trip.
The car is 20 km/h faster.
What do we want?
The speed of each one.
2 Name the unknown
We know the car's speed from the bus's speed.
So we call the bus's speed x.
x = bus's speed
The car is 20 km/h faster: x + 20.
x + 20 = car's speed
How far each one goes: speed times time.
distance = speed × time
Let's put it in a table: speed, time, distance.
The bus: x km/h, for 4 hours.
The car: x + 20 km/h, for 3 hours.
Distance = speed times time: 4x.
And 3 times all of x + 20: 3(x + 20).
We keep x + 20 together in brackets.
3 Write the equation
In words: the bus's distance is the same as the car's distance.
4x = 3(x + 20)
So the equation is 4x = 3(x + 20).
Now we solve it.
4x = 3(x + 20)
Careful!
The faster car takes less time: 3 hours.
So 3 goes with x + 20.
Each speed with its own time.
Careful: not 4(x + 20) ✗
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
4x = 3 × x + 3 × 20
Work out each multiplication.
4x = 3x + 60
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 3x from the right side.
To remove 3x, we take away 3x.
We do it on both sides, so they stay equal.
4x − 3x = 3x + 60 − 3x
3x and − 3x make zero.
Cross them out.
4x − 3x = x.
4x − 3x = x
So x = 60!
x = 60
6 Answer the question
The car: 60 + 20 = 80.
x + 20 = 60 + 20 = 80
So the bus goes 60 km/h and the car goes 80 km/h.
7 Check
Let's check in the story.
The bus: 4 hours at 60 km/h is 240 km.
bus: 4 × 60 = 240 km
The car: 3 hours at 80 km/h is 240 km.
The same trip, like the story says.
car: 3 × 80 = 240 km
Does it make sense?
The car is faster, and it takes less time.
That makes sense.
So the answer is right!
makes sense: 80 km/h > 60 km/h
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.
▶ Watch it solved
The question
Question: Two walkers start 18 km apart and walk toward each other.
One walks 2 km/h faster than the other.
They meet after 2 hours.
How fast is each one?
1 Find what we know
What do we know?
They go toward each other.
One is 2 km/h faster than the other.
Both go for 2 hours.
At the start, they are 18 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.
x = walker 1's speed
The other is 2 km/h faster: x + 2.
x + 2 = walker 2's speed
How far each one goes: speed times time.
distance = speed × time
Let's put it in a table: speed, time, distance.
Walker 1: x km/h, for 2 hours.
Walker 2: x + 2 km/h, for 2 hours.
Distance = speed times time: 2x.
And for the other: 2(x + 2).
We keep x + 2 together in brackets.
3 Write the equation
In words: the two distances add up to the whole distance.
2x + 2(x + 2) = 18
So the equation is 2x + 2(x + 2) = 18.
Now we solve it.
2x + 2(x + 2) = 18
Careful!
All of the speed x + 2 is times 2.
So it is 2(x + 2), not 2x + 2.
Careful: not 2x + 2 ✗
4 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
2x + 2 × x + 2 × 2 = 18
Work out each multiplication.
2x + 2x + 4 = 18
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.
2x + 2x = 4x
Now we have 4x + 4 = 18.
4x + 4 = 18
We want x by itself.
Two things are stuck to x: times 4 and + 4.
The + 4 is on the outside, so it comes off first.
Then the times 4.
6 Remove the + 4
To remove + 4, we take away 4.
We do it on both sides, so they stay equal.
4x + 4 − 4 = 18 − 4
+ 4 and − 4 make zero.
Cross them out.
18 − 4 = 14.
18 − 4 = 14
Now we have 4x = 14.
4x = 14
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.
4x4 = 144
4 divided by 4 is 1.
So just x is left.
We can make 14/4 simpler.
It is the same as 7/2.
144 = 72
So x = 7/2!
x = 72
8 Answer the question
7/2 is 3.5.
72 = 3.5
The faster one: 3.5 + 2 = 5.5.
x + 2 = 3.5 + 2 = 5.5
So the speeds are 3.5 km/h and 5.5 km/h.
9 Check
Let's check in the story.
Walker 1: 3.5 km/h for 2 hours is 7 km.
walker 1: 3.5 × 2 = 7 km
Walker 2: 5.5 km/h for 2 hours is 11 km.
walker 2: 5.5 × 2 = 11 km
Together: 18 km, like the story says.
So the answer is right!
7 + 11 = 18 km
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.
▶ Watch it solved
