Step by step
The question
Question: Almonds cost $12 per kg and raisins cost $4 per kg.
A shop mixes them to make 20 kg of trail mix that costs $7 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Almonds: $12 per kg.
Raisins: $4 per kg.
They make 20 kg of trail mix, at $7 per kg.
What do we want?
How many kg of almonds, and how many of raisins.
2 Name the unknown
We don't know how many kg of almonds.
So we call it x.
The rest is raisins.
20 kg in all, take away the x kg of almonds: 20 − x.
Let's put it in a table: how much, the price, and the cost.
Almonds: x kg at $12.
Raisins: 20 − x kg at $4.
Cost = how much times the price.
12 times x is 12x.
4 times all of 20 − x is 4(20 − x).
We keep 20 − x together in brackets.
The mix: 20 kg at $7.
20 times 7 is $140.
3 Write the equation
In words: the almonds cost plus the raisins cost makes the cost of the mix.
So the equation is 12x + 4(20 − x) = 140.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (12 + 4) ÷ 2 = $8 per kg, not $7.
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.
12x − 4x = 8x.
Now we have 8x + 80 = 140.
We want x by itself.
Two things are stuck to x: times 8 and + 80.
The + 80 is on the outside, so it comes off first.
Then the times 8.
6 Remove the + 80
To remove + 80, we take away 80.
We do it on both sides, so they stay equal.
+ 80 and − 80 make zero.
Cross them out.
140 − 80 = 60.
Now we have 8x = 60.
7 Remove the 8
8x means 8 times x.
We want just x.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8 divided by 8 is 1.
So just x is left.
We can make 60/8 simpler.
It is the same as 15/2.
So x = 15/2!
8 Answer the question
15/2 is 7.5.
Raisins: 20 − 7.5 = 12.5.
So 7.5 kg of almonds and 12.5 kg of raisins.
9 Check
Let's check in the story.
7.5 kg and 12.5 kg make 20 kg, like the story says.
The cost: $90 plus $50 is $140.
That is 20 kg at $7, like the story says.
Does it make sense?
The mix costs between the two prices, $4 and $12.
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.