Step by step
The question
Question: Peanuts cost $4 per kg and cashews cost $10 per kg.
A shop mixes them to make 12 kg of nut mix that costs $6 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Peanuts: $4 per kg.
Cashews: $10 per kg.
They make 12 kg of nut mix, at $6 per kg.
What do we want?
How many kg of cashews, and how many of peanuts.
2 Name the unknown
We don't know how many kg of cashews.
So we call it x.
The rest is peanuts.
12 kg in all, take away the x kg of cashews: 12 − x.
Let's put it in a table: how much, the price, and the cost.
Cashews: x kg at $10.
Peanuts: 12 − x kg at $4.
Cost = how much times the price.
10 times x is 10x.
4 times all of 12 − x is 4(12 − x).
We keep 12 − x together in brackets.
The mix: 12 kg at $6.
12 times 6 is $72.
3 Write the equation
In words: the cashews cost plus the peanuts cost makes the cost of the mix.
So the equation is 10x + 4(12 − x) = 72.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (4 + 10) ÷ 2 = $7 per kg, not $6.
4 Open the brackets
The 4 in front of the bracket multiplies every part inside.
So we write 4 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.
10x − 4x = 6x.
Now we have 6x + 48 = 72.
We want x by itself.
Two things are stuck to x: times 6 and + 48.
The + 48 is on the outside, so it comes off first.
Then the times 6.
6 Remove the + 48
To remove + 48, we take away 48.
We do it on both sides, so they stay equal.
+ 48 and − 48 make zero.
Cross them out.
72 − 48 = 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
Peanuts: 12 − 4 = 8.
So 4 kg of cashews and 8 kg of peanuts.
9 Check
Let's check in the story.
4 kg and 8 kg make 12 kg, like the story says.
The cost: $40 plus $32 is $72.
That is 12 kg at $6, like the story says.
Does it make sense?
The mix costs between the two prices, $4 and $10.
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.