Step by step
The question
Question: Large cups cost $4 and small cups cost $2.
A cafe sold 3 times as many small cups as large cups.
It made $150.
How many of each cup did it sell?
1 Find what we know
What do we know?
Two kinds: large cups cost $4 each, small cups cost $2 each.
There are 3 times as many small cups as large cups.
All the money: $150.
What do we want?
How many large cups, and how many small cups.
2 Name the unknown
We know the small cups from the large cups.
So we start with the large cups.
We call how many there are x.
There are 3 times as many small cups as large cups.
3 times x: 3x.
Careful!
There are 3 times as many small cups as large cups.
So the small cups are 3x, not the large cups.
Let's put it in a table: how many, the price, and the money.
Large cups: x of them, $4 each.
Small cups: 3x of them, $2 each.
Money = how many times the price.
4 times x is 4x.
2 times all of 3x is 2(3x).
We keep 3x together in brackets.
The total: $150 in all.
3 Write the equation
In words: the large money plus the small money makes all the money.
So the equation is 4x + 2(3x) = 150.
Now we solve it.
4 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.
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.
4x + 6x = 10x.
Now we have 10x = 150.
6 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so they stay equal.
10 divided by 10 is 1.
So just x is left.
150 divided by 10 is 15.
So x = 15!
7 Answer the question
Small cups: 3x.
3 × 15 = 45.
So 15 large cups and 45 small cups.
8 Check
Let's check in the story.
3 times 15 is 45: 3 times as many, like the story says.
The money: $60 plus $90 makes $150, like the story says.
Does it make sense?
Both are whole numbers, not below 0.
That fits cups.
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.