Step by step
The question
Question: Oats cost $2 per kg and seeds cost $8 per kg.
A shop mixes them to make 9 kg of muesli that costs $4 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Oats: $2 per kg.
Seeds: $8 per kg.
They make 9 kg of muesli, at $4 per kg.
What do we want?
How many kg of seeds, and how many of oats.
2 Name the unknown
We don't know how many kg of seeds.
So we call it x.
The rest is oats.
9 kg in all, take away the x kg of seeds: 9 − x.
Let's put it in a table: how much, the price, and the cost.
Seeds: x kg at $8.
Oats: 9 − x kg at $2.
Cost = how much times the price.
8 times x is 8x.
2 times all of 9 − x is 2(9 − x).
We keep 9 − x together in brackets.
The mix: 9 kg at $4.
9 times 4 is $36.
3 Write the equation
In words: the seeds cost plus the oats cost makes the cost of the mix.
So the equation is 8x + 2(9 − x) = 36.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (2 + 8) ÷ 2 = $5 per kg, not $4.
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.
8x − 2x = 6x.
Now we have 6x + 18 = 36.
We want x by itself.
Two things are stuck to x: times 6 and + 18.
The + 18 is on the outside, so it comes off first.
Then the times 6.
6 Remove the + 18
To remove + 18, we take away 18.
We do it on both sides, so they stay equal.
+ 18 and − 18 make zero.
Cross them out.
36 − 18 = 18.
Now we have 6x = 18.
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.
18 divided by 6 is 3.
So x = 3!
8 Answer the question
Oats: 9 − 3 = 6.
So 3 kg of seeds and 6 kg of oats.
9 Check
Let's check in the story.
3 kg and 6 kg make 9 kg, like the story says.
The cost: $24 plus $12 is $36.
That is 9 kg at $4, like the story says.
Does it make sense?
The mix costs between the two prices, $2 and $8.
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.