Step by step
The question
Question: Rice costs €2 per kg and quinoa costs €8 per kg.
A shop mixes them to make 18 kg of grain mix that costs €4 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Rice: €2 per kg.
Quinoa: €8 per kg.
They make 18 kg of grain mix, at €4 per kg.
What do we want?
How many kg of quinoa, and how many of rice.
2 Name the unknown
We don't know how many kg of quinoa.
So we call it x.
The rest is rice.
18 kg in all, take away the x kg of quinoa: 18 − x.
Let's put it in a table: how much, the price, and the cost.
Quinoa: x kg at €8.
Rice: 18 − x kg at €2.
Cost = how much times the price.
8 times x is 8x.
2 times all of 18 − x is 2(18 − x).
We keep 18 − x together in brackets.
The mix: 18 kg at €4.
18 times 4 is €72.
3 Write the equation
In words: the quinoa cost plus the rice cost makes the cost of the mix.
So the equation is 8x + 2(18 − x) = 72.
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 + 36 = 72.
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.
72 − 36 = 36.
Now we have 6x = 36.
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.
36 divided by 6 is 6.
So x = 6!
8 Answer the question
Rice: 18 − 6 = 12.
So 6 kg of quinoa and 12 kg of rice.
9 Check
Let's check in the story.
6 kg and 12 kg make 18 kg, like the story says.
The cost: €48 plus €24 is €72.
That is 18 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.