Step by step
The question
Question: Cherries cost $9 per kg and plums cost $3 per kg.
A shop mixes them to make 12 kg of fruit mix that costs $5 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Cherries: $9 per kg.
Plums: $3 per kg.
They make 12 kg of fruit mix, at $5 per kg.
What do we want?
How many kg of cherries, and how many of plums.
2 Name the unknown
We don't know how many kg of cherries.
So we call it x.
The rest is plums.
12 kg in all, take away the x kg of cherries: 12 − x.
Let's put it in a table: how much, the price, and the cost.
Cherries: x kg at $9.
Plums: 12 − x kg at $3.
Cost = how much times the price.
9 times x is 9x.
3 times all of 12 − x is 3(12 − x).
We keep 12 − x together in brackets.
The mix: 12 kg at $5.
12 times 5 is $60.
3 Write the equation
In words: the cherries cost plus the plums cost makes the cost of the mix.
So the equation is 9x + 3(12 − x) = 60.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (9 + 3) ÷ 2 = $6 per kg, not $5.
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 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.
9x − 3x = 6x.
Now we have 6x + 36 = 60.
We want x by itself.
Two things are stuck to x: times 6 and + 36.
The + 36 is on the outside, so it comes off first.
Then the times 6.
6 Remove the + 36
To remove + 36, we take away 36.
We do it on both sides, so they stay equal.
+ 36 and − 36 make zero.
Cross them out.
60 − 36 = 24.
Now we have 6x = 24.
7 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just x is left.
24 divided by 6 is 4.
So x = 4!
8 Answer the question
Plums: 12 − 4 = 8.
So 4 kg of cherries and 8 kg of plums.
9 Check
Let's check in the story.
4 kg and 8 kg make 12 kg, like the story says.
The cost: $36 plus $24 is $60.
That is 12 kg at $5, like the story says.
Does it make sense?
The mix costs between the two prices, $3 and $9.
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.