Step by step
The question
Question: Juice costs $2 per litre and syrup costs $5 per litre.
A cafe mixes them to make 12 litres of drink that costs $3 per litre.
How many litres of each does it use?
1 Find what we know
What do we know?
Juice: $2 per litre.
Syrup: $5 per litre.
They make 12 litres of drink, at $3 per litre.
What do we want?
How many litres of syrup, and how many of juice.
2 Name the unknown
We don't know how many litres of syrup.
So we call it x.
The rest is juice.
12 litres in all, take away the x litres of syrup: 12 − x.
Let's put it in a table: how much, the price, and the cost.
Syrup: x litres at $5.
Juice: 12 − x litres at $2.
Cost = how much times the price.
5 times x is 5x.
2 times all of 12 − x is 2(12 − x).
We keep 12 − x together in brackets.
The mix: 12 litres at $3.
12 times 3 is $36.
3 Write the equation
In words: the syrup cost plus the juice cost makes the cost of the mix.
So the equation is 5x + 2(12 − x) = 36.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (2 + 5) ÷ 2 = $3.50 per litre, not $3.
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.
5x − 2x = 3x.
Now we have 3x + 24 = 36.
We want x by itself.
Two things are stuck to x: times 3 and + 24.
The + 24 is on the outside, so it comes off first.
Then the times 3.
6 Remove the + 24
To remove + 24, we take away 24.
We do it on both sides, so they stay equal.
+ 24 and − 24 make zero.
Cross them out.
36 − 24 = 12.
Now we have 3x = 12.
7 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.
3 divided by 3 is 1.
So just x is left.
12 divided by 3 is 4.
So x = 4!
8 Answer the question
Juice: 12 − 4 = 8.
So 4 litres of syrup and 8 litres of juice.
9 Check
Let's check in the story.
4 litres and 8 litres make 12 litres, like the story says.
The cost: $20 plus $16 is $36.
That is 12 litres at $3, like the story says.
Does it make sense?
The mix costs between the two prices, $2 and $5.
That makes sense.
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.